R Joetzjer

D Roehrig

T Saint-Martin

R Oudar

A Stchepounoff

Voldoire

Dr Hervé Douville
email: herve.douville@meteo.fr

B Decharme

C Delire

J Colin

E Joetzjer

R Roehrig

D Saint- Martin

T Oudar

R Stchepounoff

A Voldoire

Drivers of the enhanced decline of land near-surface relative humidity to abrupt-4xCO 2 in CNRM-CM6-1

Keywords: climate change, relative humidity, land, drying, drivers

Drivers of the enhanced decline of land near-surface relative humidity to abrupt 4xCO2 in CNRM-CM6-1

Introduction

Under enhanced atmospheric CO2 concentration, relative humidity (RH) is expected to remain approximately constant on global and climatological time scales [START_REF] Held | Robust responses of the hydrologic cycle to global warming[END_REF]). Yet, global climate models show regional disparities in projected changes in near surface RH, especially over land where such changes may have policy-relevant impacts on human health (e.g., heat stress) and terrestrial ecosystems (e.g., wildfires). RH is also a key variable for understanding the projected landsea warming contrast [START_REF] Byrne | Link between land-ocean warming contrast and surface relative humidities in simulations with coupled climate models[END_REF], regional changes in surface evaporation [START_REF] Laîné | A diagnostic study of future evaporation changes projected in CMIP5 climate models[END_REF], land-ocean shifts in tropical precipitation [START_REF] Lambert | Land-ocean shifts in tropical precipitation linked to surface temperature and humidity change[END_REF], and global changes in vegetation growth [START_REF] Yuan | Increased atmospheric vapor pressure deficit reduces global vegetation growth[END_REF].

According to the fifth assessment report (AR5) of the Intergovernmental Panel on Climate Change (IPCC), there is medium confidence that "reductions in near-surface RH over many land areas are likely", mainly due to the larger warming rate projected over land than over ocean [START_REF] Collins | Long-term Climate Change: Projections, Commitments and Irreversibility Pages 1029 to 1076[END_REF]. Conversely, projected changes in RH were shown to contribute to the land amplification of global warming. A theory that assumes equal changes in equivalent potential temperature over land and ocean captures the projected land-sea warming contrast in the tropics [START_REF] Byrne | Link between land-ocean warming contrast and surface relative humidities in simulations with coupled climate models[END_REF]. According to this theory, land surface RH changes and the land-sea contrast in the control climate contribute equally to the tropical warming contrast, but the intermodel spread in the land-sea warming contrast is primarily linked to changes in land surface RH thereby emphasizing the need to better constrain them in climate projections.

Conceptual models, focusing on atmospheric moisture transport between the land and ocean, have been proposed to investigate the decrease in land surface RH as the climate warms [START_REF] Byrne | Understanding decreases in land relative humidity with global warming: conceptual model and {G}{C}{M} simulations[END_REF][START_REF] Chadwick | A simple moisture advection model of specific humidity change over land in response to {S}{S}{T} warming[END_REF]. The simplest model assumes equal fractional increases in specific humidity over land and ocean and captures the main features of the projected changes in RH over land given the limited increase of RH over the oceans and the land-sea contrast in global warming. Yet, evapotranspiration was also found to contribute to the simulated decrease in RH over land, possibly due to the stomatal closure effect of increased atmospheric CO2 and thereby amplifying the land surface warming [START_REF] Byrne | Understanding decreases in land relative humidity with global warming: conceptual model and {G}{C}{M} simulations[END_REF]. Land surface evapotranspiration can also be suppressed by the soil drying [START_REF] Jung | Recent decline in the global land evapotranspiration trend due to limited moisture supply[END_REF][START_REF] Mueller | Systematic land climate and evapotranspiration biases in CMIP5 simulations[END_REF], which can also lead to a stronger land surface warming [START_REF] Douville | Midlatitude daily summer temperatures reshaped by soil moisture under climate change[END_REF][START_REF] Vogel | Regional amplification of projected changes in extreme temperatures strongly controlled by soil moisture-temperature feedbacks[END_REF].

Based on both in situ observations (https://www.metoffice.gov.uk/hadobs/hadisdh/) and comprehensive European Centre for Medium-Range Weather Forecasts reanalyses (ECMWF) reanalyses, a reduction in RH over low-latitude and midlatitude land areas was detected after 1998 [START_REF] Simmons | Low-frequency variations in surface atmospheric humidity, temperature, and precipitation: Inferences from reanalyses and monthly gridded observational data sets[END_REF]. The various data sets agreed well for both temperature and humidity variations for periods and places of overlap. Near-surface specific humidity was shown to vary similarly over land and sea, suggesting that the recent reduction in RH over land could be due to limited moisture supply from the oceans. This hypothesis was further supported by the simple moisture transport model assuming equal fractional changes in specific humidity over land and ocean [START_REF] Byrne | Trends in continental temperature and humidity directly linked to ocean warming[END_REF]. The predictions of this analytical model were shown to be broadly consistent with the observed trends in land RH given the observed warming over ocean, despite considerable variability about the best-fit trend in land RH that could be related to other factors such as changes in atmospheric circulations and land-surface properties. The detected decline in land surface RH should be however considered with caution at the regional scale given the possible change in measurement methods and the contrasted multi-decadal trends which can be found between raw and homogeneized in situ observations [START_REF] Freychet | Underestimated Change of Wet-Bulb Temperatures Over East and South China[END_REF].

Post-AR5 studies [START_REF] Douville | Midlatitude Summer Drying: An Underestimated Threat in CMIP5 Models?[END_REF][START_REF] Dunn | Comparison of land surface humidity between observations and CMIP5 models[END_REF] suggest that the apparent decrease in land surface RH was underestimated by most models from the fifth phase of the Coupled Model Intercomparison Project (CMIP5). This observed drying was attributed to anthropogenic forcings and used to constrain the projected decline of near surface RH over the boreal summer mid-latitude continents, suggesting a systematic underestimation of the land surface drying by these models [START_REF] Douville | Midlatitude Summer Drying: An Underestimated Threat in CMIP5 Models?[END_REF]. Interestingly, the CNRM-CM6-1 model [START_REF] Voldoire | Evaluation of CMIP6 DECK Experiments With CNRM-CM6-1[END_REF] shows a realistic land surface drying when driven by observed SST (Fig. 1) and an enhanced response of land surface RH to quadrupled atmospheric CO2 compared to the former CNRM-CM5 model [START_REF] Voldoire | The CNRM-CM5.1 global climate model: Description and basic evaluation[END_REF] (Fig. 2). Globally averaged, this enhanced land surface drying is found all year round but is much more pronounced in boreal summer (Fig. 3a). It is related to an enhanced continental warming given the opposite changes in specific humidity which indicate a stronger moistening in CNRM-CM6-1 over both land (Fig. 3c) and sea (Fig. 3d).

The present study is aimed at better understanding the drivers of this boreal summer (June to September, hereafter JJAS) response of land surface RH, mainly using atmosphere-only experiments proposed by the Cloud Feedback Model Intercomparison Project (CFMIP) component of CMIP6 and already tested in a pilot study [START_REF] Chadwick | Timeslice experiments for understanding regional climate projections: applications to the tropical hydrological cycle and European winter circulation[END_REF] with three atmospheric general circulation models (AGCMs) including the atmospheric component of CNRM-CM5. This pilot study revealed that the so-called physiological CO2 effect, not accounted for in CNRM-CM5, may have a significant impact on tropical forest precipitation. There is indeed growing evidence that plants optimize their water use efficiency (WUE, i.e., the amount of CO2 absorbed by photosynthesis per amount of water lost by transpiration) by reducing the opening of their stomata under elevated atmospheric CO2 concentration. This physiological response can lead to decreased land surface RH as a result of reduced plant transpiration [START_REF] Berg | Land-atmosphere feedbacks amplify aridity increase over land under global warming[END_REF][START_REF] Cao | Importance of carbon dioxide physiological forcing to future climate change[END_REF][START_REF] Lemordant | Critical impact of vegetation physiology on the continental hydrologic cycle in response to increasing CO 2[END_REF][START_REF] Swann | Plant responses to increasing CO2 reduce estimates of climate impacts on drought severity[END_REF]. The resulting drying and warming of the atmospheric boundary layer can also lead to a decrease in precipitation, at least over densely vegetated land areas [START_REF] Richardson | Carbon Dioxide Physiological Forcing Dominates Projected Eastern Amazonian Drying[END_REF]). Yet, this stomatal closure effect exhibit strong species specificity and can be partly offset by a CO2 fertilization effect (i.e., increased photosynthesis and vegetation density) so that the net physiological CO2 effect on land surface evapotranspiration and precipitation remains highly model-dependent.

Here, the focus is both on the understanding of the CNRM-CM6-1 near-surface RH response and on the comparison with CNRM-CM5. Note that the new CNRM-CM6-1 model includes a stomatal closure effect of atmospheric CO2 on plant transpiration (not accounted for in CNRM-CM5) but still has a prescribed vegetation phenology (Decharme et al., 2019b). The paper is structured as follows.

Section 2 highlights the contrasted response of CNRM-CM6-1 and CNRM-CM5 to an abrupt quadrupling of atmospheric CO2 and describes the various atmosphere-only experiments which are used in the present study. The breakdown of the JJAS RH response to abrupt-4xCO2 into multiple contributions is shown in Section 3, with a particular focus on the physiological CO2 effect. Section 4 further explores the role of moisture advection in the land surface humidity response to sea surface warming, the potential influence of the mean state, and the linearity of the RH response to increased CO2. The main conclusions are summarized in Section 5 which also discusses the limitations and implications of the study and suggests a few prospects for further research.

Models and experiment design

CNRM-CM6-1 is the standard configuration of the CNRM global climate model which has recently contributed to the core experiments of CMIP6 (Eyring et al., 2016) as well as to many parallel intercomparisons such as the Cloud Feedback Model Intercomparison project (CFMIP, [START_REF] Webb | The Cloud Feedback Model Intercomparison Project (CFMIP) contribution to CMIP6[END_REF]. It is based on the ARPEGE-Climat v6.3 atmospheric general circulation model, the NEMO v3.6 ocean general circulation model, the GELATO sea-ice model, as well as the SURFEX v8.0 landsurface modelling platform associated to the CTRIP v2 river routing scheme (Decharme et al., 2019b). The atmospheric horizontal resolution is about 1.4° and there are 91 vertical levels up to 0.01 hPa, while the oceanic resolution is 1° with a latitudinal refinement to 1/3° at the equator and 75 vertical levels. In the continuation of the study, we will distinguish the coupled model, CNRM-CM6-1, from its land-atmosphere configuration which will be noted CNRM-AM6-1.

While all model components have been updated in CNRM-CM6-1 compared to CNRM-CM5 [START_REF] Voldoire | Evaluation of CMIP6 DECK Experiments With CNRM-CM6-1[END_REF], a particular attention has been paid to the land surface physics. In CNRM-CM5, the land surface was modeled trough the ISBA-TRIP system [START_REF] Decharme | Global validation of the ISBA sub-grid hydrology[END_REF] based on a three-layer soil hydrology and a simple one-layer snow scheme embedded in the SURFEX 5.2 numerical platform. The plant transpiration was simulated via a simple Jarvis-type scheme (Jarvis, 1976) as a function of the LAI (Leaf Area Index), root zone water stress, and meteorological constraints (solar radiation, air temperature, saturation deficit). In CNRM-CM6-1, the new land surface system, ISBA-CTRIP (Decharme et al., 2019a), is embedded in the SURFEX 8.0 numerical platform where ISBA now explicitly solves the one-dimensional Fourier and Darcy laws throughout the soil and a multilayer snow model enables a separate water and energy budgets to be simulated for the soil and the snowpack. Plant transpiration is controlled by the stomatal conductance of leaves, which depends on carbon cycling in vegetation [START_REF] Calvet | An interactive vegetation SVAT model tested against data from six contrasting sites[END_REF] but the LAI phenology remains prescribed. This parameterization allows the model to simulate the stomatal closure effect of atmospheric CO2 on plant transpiration but not the fertilization effect on plant biomass.

A more comprehensive description of the model components and of their coupling can be found in [START_REF] Voldoire | Evaluation of CMIP6 DECK Experiments With CNRM-CM6-1[END_REF], as well as an evaluation of the model performance in present-day climate conditions and a brief overview of the model sensitivity to increasing CO2. A striking feature of CNRM-CM6-1 is the enhanced equilibrium climate sensitivity (ECS=4.9°C) compared to the former CNRM-CM5 model (ECS=3.3°C) involved in CMIP5. A detailed analysis of this enhanced climate sensitivity highlights the key role of the convection scheme through an increase in the longwave tropical cloud feedback (Saint-Martin et al., personal communication). The demonstration partly relies on the implementation of an intermediate model configuration, CNRM-CM6-atm5 (ECS=3.4°C), in which the atmospheric component of CNRM-CM6-1 was replaced by the one from CNRM-CM5 (which also differs from CNRM-CM6-1 by its ocean, sea-ice and land surface components).

The three pairs of piControl and abrupt-4xCO2 experiments achieved with CNRM-CM6-1, CNRM-CM5 and CNRM-CM6-atm5 are also the starting point of the present study. Looking back at the climatological annual cycle of RH, CNRM-CM6-atm5 is very close to CNRM-CM5 over the global ocean (Fig. 3b). In contrast, the hybrid model shows different piControl and abrupt-4xC02 climatologies over land (Fig. 3a) and an intermediate summer drying compared to CNRM-CM5 and CNRM-CM6-1. The results suggest that ocean RH is primarily controlled by the atmospheric component of the coupled model, while land RH is partly controlled by the land surface model and its coupling with the atmosphere. A striking difference between CNRM-CM6-1 and CNRM-CM5 is the enhanced land-sea contrast in RH which is already found under preindustrial climate but is further enhanced in the abrupt-4xCO2 experiment.

Focusing on the JJAS season, Fig. 4a-c compare the geographical patterns of the RH response in the three model configurations. The patterns are scaled by the annual mean global warming to account for the stronger climate sensitivity in CNRM-CM6-1 compared to CNRM-CM5 and CNRM-CM6-atm5. The land surface drying is more than twice in CNRM-CM6-1 (-0.58%/°C) versus CNRM-CM5 (-0.24%/°C), and does not show exactly the same geographical patterns as revealed by their limited spatial correlation (ACC=0.65). The scaled RH response is much stronger in CNRM-CM6-atm5 (-0.50%/°C) compared to CNRM-CM5 despite their similar ECS. The stronger ECS in CNRM-CM6-1 explains why the scaled response is only slightly lower in CNRM-CM6-atm5, unlike the unscaled response shown in Fig. 3a. Regarding the pattern of the JJAS land surface drying, the hybrid model configuration shares common features with both CNRM-CM5 and CNRM-CM6-1, but also regional differences (e.g., stronger Amazonian drying, weaker Mediterranean drying) thereby suggesting a complex interplay between the atmosphere and the other model components. In other words, the enhanced RH decline found in CNRM-CM6-1 is only partially explained by its stronger climate sensitivity compared to CNRM-CM5 and is not dominated by the atmospheric component given the distinct response in CNRM-CM6-atm5 versus CNRM-CM5.

In order to go one step further, the present study takes advantage of CFMIP experiments performed with CNRM-AM6-1 (cf. Table 1). A first pair of time-slice simulations, piSST and a4SSTice-4xCO2, aims at replicating the piControl and abrupt-4xCO2 coupled experiments and was also performed with CNRM-AM5. This is done by prescribing the same atmospheric CO2 concentrations but also monthly mean annually varying oceanic boundary conditions derived from the SST and sea-ice concentration (SIC) simulated from year 111 to 140 in piControl and abrupt-4xCO2 respectively. Such AGCM experiments were shown to capture the main features of the precipitation response to abrupt-4xCO2 in three CMIP5 coupled models including CNRM-CM5 [START_REF] Chadwick | Timeslice experiments for understanding regional climate projections: applications to the tropical hydrological cycle and European winter circulation[END_REF]. Fig. 4d shows that they also allow CNRM-AM6-1 to reproduce the pattern (ACC=0.97) and overall magnitude (-0.56%/°C) of the RH response simulated by CNRM-CM6-1.

The other AGCM experiments are parallel 30-year integrations (after spin-up) aiming at splitting the atmospheric response to abrupt-4xCO2 into separate responses to different aspects of CO2 forcing and related ocean warming (uniform SST warming, patterned SST change, sea-ice change, radiative and/or physiological CO2 effect, cf. Table 1). The total response can be first decomposed as follows : a4SSTice-4xCO2 -piSST = (a4SSTice-4xCO2 -a4SSTice) + (a4SSTice -piSST-pxK) + (piSST-pxK -piSST) (1) where the three right-hand terms represent the direct CO2 effect, the SST anomaly pattern and SIC effects, and the uniform SST warming effect respectively.

It should be here emphasized that the CFMIP experiment design [START_REF] Webb | The Cloud Feedback Model Intercomparison Project (CFMIP) contribution to CMIP6[END_REF] differs slightly from the early pilot study using CNRM-AM5 [START_REF] Chadwick | Timeslice experiments for understanding regional climate projections: applications to the tropical hydrological cycle and European winter circulation[END_REF]. In the pilot study, the sea-ice effect was ignored (in the two other AGCMs) or included in the SST anomaly pattern of CNRM-AM5 in which the monthly mean SST were prescribed over both open and sea-ice covered ocean surfaces.

In other words, the polar amplification of the global ocean warming due to the sea-ice retreat was entirely included in the SST anomaly pattern for CNRM-AM5, while it is also accounted for in the uniform SST warming effect in CNRM-AM6-1 since the polar surface warming is partly due to dynamical effects such as the advection of warmer and moister air masses from the mid-latitudes.

Looking at the annual mean near-surface air temperature (not shown), the so-called SIC and SST anomaly pattern effect leads to a global cooling of -0.62°C in CNRM-AM5 versus a global warming of +0.39°C in CNRM-AM6-1. Such a difference should be kept in mind when interpreting the results of the AGCM experiments. Note also that the uniform and patterned SST warming were rescaled to match a +4°C global and annual mean average in the pilot study, so that a reverse scaling factor is here applied to compare CNRM-AM5 and CNRM-CM5.

In CNRM-AM6-1, the CO2 effect can be further deconstructed into its physiological versus radiative component under preindustrial SST/SIC as follows:

piSST-4xCO2 -piSST = (piSST-4xCO2 -piSST-4xCO2-rad) + (piSST-4xCO2-rad -piSST) [START_REF] Richardson | Carbon Dioxide Physiological Forcing Dominates Projected Eastern Amazonian Drying[END_REF] and it will be shown that the total CO2 effect does not depend too much on the mean climate (cf. section 3.2). While the CFMIP experiment design assumes the additivity of the radiative and physiological CO2 effects, it should be emphasized that this hypothesis cannot be verified since the physiological effect is estimated as a residual in the first right-hand term of Eq. ( 2).

An ultimate time-slice atmospheric simulation was proposed by CFMIP in order to assess the role of the mean state SST on the model response through a comparison with the amip core experiment driven by the 1979-2014 observed monthly mean SST and SIC boundary conditions (section 4.2). This additional experiment (amip-a4SST-4xCO2, cf. Table 1) is the same as amip, but with a quadrupled CO2 and a patterned SST warming applied on top of the amip SSTs. This climatological monthly SST pattern is the same as in the a4SST experiments, but the observed interannual variability from the amip experiment is preserved. Since the projected decline in SIC is not accounted for in the amip-a4SST-4xCO2 experiment, it has been here decided to run an additional a4SST-4xCO2 with preindustrial SIC in order to ensure a clean comparison of the response to CO2 quadrupling and related SST warming on top of present-day versus preindustrial SST boundary conditions. Table 2 summarizes how each component of the total response to abrupt-4xCO2 are calculated. The statistical significance of all contributions will be assessed using a two-sided t-test at the 5% level suitable for multiple testing [START_REF] Wilks | The stippling shows statistically significant grid points": How research results are routinely overstated and overinterpreted, and what to do about it[END_REF].

Breakdown of the JJAS response to abrupt-4xCO2

Comparison between CNRM-AM6-1 and CNRM-AM5

Fig. 5 and 6 show the Eq. ( 1) decomposition of the boreal summer RH response to abrupt-4xCO2 in CNRM-AM6-1 and CNRM-AM5 respectively. Starting with CNRM-AM6-1, the global land averaged decrease in RH (-3.68%) is primarily due to the CO2 effect (-2.26%) and to the uniform SST warming (-1.78%). This uniform warming drives a significant drying in the tropics and subtropics, but an increased RH in the northern latitudes (Fig. 5b). It explains about half of the spatial variability of the total response simulated over land. In contrast, the response to CO2 (Fig. 5d) shows an enhanced drying in the northern extratropics, a less homogeneous response in the tropics, and a limited correlation with the total response. The global contribution of the SIC and SST anomaly pattern is much weaker but also shows significant regional responses.

Moving to CNRM-AM5, the global land averaged decrease in RH is only of -1.65% (Fig. 4a) and the breakdown is different. There is a dominant contribution of the uniform SST warming, which is strengthened by the contribution of the radiative CO2 effect but partly offset by the response to the SST anomaly pattern. As explained in Section 2, this stronger contribution of the SIC and SST anomaly pattern is partly an artifact of the experiment design. Since the polar amplification of the global ocean warming is here included in the SST anomaly pattern effect, this leads to a relative cooling of the tropical and mid-latitude prescribed SST which can partly offset the RH response to the uniform SST warming effect. This explanation is supported by the relative symmetry of the geographical patterns shown in Fig. 6b andc. A more valuable comparison is the sum of the RH responses to the uniform SST warming and to the SIC and SST anomaly pattern. This non-CO2 effect leads to a global land surface drying of -1.42% in CNRM-AM6-1 versus -0.34% in CNRM-AM5, and therefore explains 53% of the enhanced RH decline in CNRM-AM6-1 (-3.68%) compared to CNRM-AM5 (-1.65%).

Changes in low-level atmospheric circulation have been superimposed on JJAS RH anomalies in both Fig. 5 and6. They suggest that atmospheric moisture transport play a key role in regional RH changes. Some features are apparently robust such as the strong drying driven by the uniform SST warming over Sahel in both CNRM-AM6-1 and CNRM-AM5, partly as a response to a weakening of the West African monsoon circulation (Fig. 5b and6b). In contrast, the response to the SST anomaly pattern shows a significant drying signal over India in CNRM-AM6-1, which is associated with a weakening of the South Asian monsoon but is not found in CNRM-AM5 (Fig. 5c and6c).

To sum up, the direct and SST-mediated CO2 effects contribute equally to the enhanced RH decline in CNRM-CM6-1 compared to CNRM-CM5. The stronger drying due to the direct CO2 effect is potentially related to the introduction of the physiological effect on plant transpiration (Section 3.2).

The response to the uniform SST warming will be further discussed in Section 4.1. The SIC and SST pattern leads to a contrasted response between CNRM-AM5 and CNRM-AM6.1, which is partly an artifact of differences in the experiment design. This pattern has a small global signature in CNRM-AM6-1, but significant positive and negative effects at the regional scale. A further decomposition of this pattern effect (not shown) indicates that the SIC effect is negligible and that the RH response is dominated by the tropical SST anomalies and related circulation changes.

Influence of the physiological CO2 effect

The influence of the stomatal closure CO2 effect can be assessed in CNRM-AM6-1 by using Eq. ( 2) and further splitting the fast response to CO2 into a radiative versus physiological component (Fig. 7). Such a decomposition is only possible under preindustrial climate conditions (Fig. 7b) given the CFMIP experiment design. The total CO2 effect is however relatively close between warm (Fig. 7a) and cold (Fig. 7b) climates. The physiological effect (Fig. 7c) drives a widespread decrease in RH over the tropical, temperate and boreal forests. It is associated with reduced evapotranspiration (not shown) and dominates the RH response to the direct CO2 effect over land. Note that the radiativeonly CO2 effect in CNRM-AM6 (Fig. 7d, -0.15%) is however weaker than in CNRM-AM5 (Fig. 6d, -1.30%). The additional physiological CO2 effect therefore contributes to the stronger land surface RH decline found in Fig. 5d compared to Fig. 6d, but this additional contribution is partly offset by a weaker radiative-only effect so that the fast response to CO2 does not explain the whole additional land surface drying found in CNRM-AM6-1 (Fig. 5a) compared to CNRM-AM5 (Fig. 6a). The much weaker radiative-only CO2 effect found in CNRM-AM6-1 may have multiple causes, including a contrasted adjustment of low-level cloudiness and of temperature and humidity profiles in the planetary boundary layer. Yet, it may also raise the question of the potential interaction between the radiative and physiological CO2 effects which is not addressed by the current CFMIP experiment design.

Globally speaking, the land surface drying due to the physiological CO2 effect (-2.22%) accounts for 60% of the total drying (-3.68%) found in CNRM-AM6-1. The model contrast in the total direct CO2 effect represents 47% of the enhanced RH decline in CNRM-AM6-1 versus CNRM-AM5, in line with the previously estimated 53% contribution of the non-CO2 effects. It should be however emphasized that the physiological CO2 effect simulated in CNRM-CM6-1/AM6-1 may represent an extreme scenario in which the CO2 fertilization effect is ignored and the vegetation density (leaf area index) is fixed at a present-day climatology (Decharme et al., 2019b). The fertilization effect is accounted for in the Earth System configuration of the CNRM model [START_REF] Séférian | Evaluation of CNRM Earth-System model, CNRM-ESM 2-1: role of Earth system processes in present-day and future climate[END_REF], which however did not participate to CFMIP and is not considered in the present study.

Discussion

Influence of land-sea contrast in specific humidity

Byrne and O'Gorman (2016) developed a conceptual box model, involving atmospheric moisture transport between the land and ocean, to investigate the overall decline in land RH as the climate warms. The box model was applied to CMIP5 general circulation models and was found to capture many of the features of the projected changes in land humidity. The simplest version of the box model ignores changes in atmospheric circulation and surface evapotranspiration. It just gives equal fractional increases in specific humidity, q, over land and sea:

(ql,2-ql,1)/ql,1 = (q*s,2-q*s,1)/q*s,1 => ql,2 = (q*s,2/q*s,1).ql,1

where subscripts l and s denote land and sea and subscripts 1 and 2 denote control and perturbed climates respectively. Here q* denotes the ocean moisture source for each land grid point and is simply estimated as the seasonal and zonal mean of ocean surface q at the same latitude.

In this model, future q over land is therefore determined by scaling the present-day pattern of land q by the fractional increase in the oceanic moisture source. This simple scaling was shown to agree well with the projected land surface specific humidity, at least much better than the Clausius-Clapeyron scaling which assumes a constant RH and overestimates the projected increase in q over land [START_REF] Chadwick | A simple moisture advection model of specific humidity change over land in response to {S}{S}{T} warming[END_REF]. It implies a global decrease in land RH given the greater warming over land than ocean and the overall modest change in ocean RH. It also implies a stronger increase in q over land in CNRM-CM6-1, given the enhanced q*s,2/q*s,1 ocean ratio suggested by Fig. 3d.

Eq. ( 4) has been here applied to the JJAS mean climatology of specific humidity simulated in our AGCM experiments. The focus is on the response to the uniform SST warming given the neglected role of changes in atmospheric circulation and surface evapotranspiration (which have been shown to contribute to the response to SST anomaly pattern and CO2 effects respectively). Both CNRM-AM5 (Fig. 8a) and CNRM-AM6-1 (Fig. 8b) show a maximum increase in q over the tropical ocean, but with regional differences over land (Fig. 8g). The simple scaling model is able to capture the main features of the AGCM response over land (Fig. 8b ande), but is not accurate enough to understand the regional differences between CNRM-AM5 and CNRM-AM6-1 (Fig. 8h). The only exception is the Sahel region where the weaker increase in q found in CNRM-AM6-1 is partly reproduced by the simple scaling (in line with the difference in moisture advection implied by the contrasted low-level wind climatologies between the two AGCMs). The departures between the AGCM and simple scaling results (Fig. 8c andf) illustrate the limitations of the simple advection model which does not account for changes in atmospheric circulation and land surface evapotranspiration.

Fig. 9 compares again the JJAS RH responses to uniform SST warming in CNRM-AM5 and CNRM-AM6-1 (Fig. 9a andd). Note that the global land average decrease in RH is not the same as in Fig. 5b and Fig. 6b respectively since the focus is here between 60°S and 60°N. The main reason is that the simple advection model does not account for snow-covered surfaces and that we derive RH from the simple model estimate of q (and from the AGCM near-surface temperature and surface pressure outputs) using an empirical formulation of saturation specific humidity [START_REF] Murray | On the Computation of Saturation Vapor Pressure[END_REF] which is less suitable in cold climates. The results show that the simple scaling captures the main features of the RH changes simulated by both CNRM AGCMs when a uniform SST warming is prescribed. The obtained geographical patterns (Fig. 9b ande) show a strong similarity with the AGCM patterns, and so does the predicted difference (Fig. 9h) between the two AGCMs. Yet, the simple moisture advection model (Fig. 9h) does not explain the stronger tropical and mid-latitude drying (-0.81%)

found in CNRM-AM6-1 compared to CNRM-CM5 (Fig. 9g). This result is mostly due to a widespread and overall strong underestimation of the CNRM-AM6-1 drying, which may suggest a stronger role for the land-atmosphere coupling in this model whereby the land surface decrease in RH can be amplified by a more positive soil moisture feedback [START_REF] Berg | Land-atmosphere feedbacks amplify aridity increase over land under global warming[END_REF]. This hypothesis will be further explored in Section 4.3.

Mean state influence on model sensitivity

Another key question of CFMIP [START_REF] Webb | The Cloud Feedback Model Intercomparison Project (CFMIP) contribution to CMIP6[END_REF] is to explore to what extent climate projections depend on coupled model SST biases. While the SST biases of the piControl experiment cannot be assessed accurately given the limited observational record of preindustrial SST, the role of the mean state SST on the CNRM-AM6-1 response to abrupt-4xCO2 can be assessed through the comparison of two pairs of experiments, aSST-4xCO2 versus piSST on the one hand, amip-a4SST-4xCO2 versus amip on the other hand (cf. Table 1). Fig. 10 shows the two pairs of RH anomalies between the perturbed and control climates (Fig. 10c-d), as well as the difference between the two control and two perturbed simulations (Fig. 10a-b). The geographical pattern of the RH response does not much depend on the mean control climate.

Yet, the overall land surface drying is slightly weaker when the CO2 and SST perturbations are applied onto present-day rather than preindustrial conditions. Such a difference may have several plausible explanations. For instance, the stomatal closure effect introduced in CNRM-CM6-1 is quite complex (Decharme et al., 2019b;[START_REF] Joetzjer | Improving the ISBA<sub>CC</sub> land surface model simulation of water and carbon fluxes and stocks over the Amazon forest[END_REF] and is sensitive to the absolute atmospheric CO2 concentration which is higher in amip than in piSST. A more basic explanation is a possible sensitivity to the control climate. Fig. 10a shows that the amip simulation is globally drier than the piSST simulation. This is for instance obvious over North America and Australia which show a weaker drying when the CO2 and SST perturbations are applied onto present-day rather than preindustrial climate conditions (Fig. 10c-d).

A dry bias in the control climate may therefore limit the land surface drying due to climate change.

Such a result is consistent with a soil moisture limitation of land surface evapotranspiration but may not apply where and when surface evapotranspiration is instead limited by the incoming solar radiation. This hypothesis is further supported by the fact that the mean state dependence vanishes in boreal winter (not shown), in line with an overall weaker soil moisture control on land surface evapotranspiration in the Northern Hemisphere. The mean-state dependence can also contribute to the enhanced drying found in CNRM-CM6-1 given its globally improved soil moisture climatology compared to CNRM-CM5 which showed a widespread dry bias over the northern Hemisphere continents (cf. Fig. 9 in Decharme et al., 2019).

Second versus first doubling of atmospheric CO2

Before moving to the conclusion of the study, let's come back to the coupled CNRM-CM6-1 model and explore the linearity of the RH response as a function of the CO2 increase. For this purpose, we make use of the abrupt-2xCO2 coupled experiment proposed within CFMIP [START_REF] Webb | The Cloud Feedback Model Intercomparison Project (CFMIP) contribution to CMIP6[END_REF]. Fig. 11 shows the JJAS RH response to abrupt-2xCO2 (Fig. 11a), abrupt-4xCO2 (Fig. 11b), as well as the difference between the response to the second versus first CO2 doubling (Fig. 11c). The results emphasize a strong non-linearity with the second CO2 doubling leading to a much stronger drying than the first one over many regions including Amazonia, South Africa, Australia, part of the US, Europe and northern Asia. These regions also show an increased land surface warming for the second CO2 doubling (not shown), thereby suggesting a regional effect whereby increased warming is associated with enhanced drying.

Globally averaged over land, the drying due to the second CO2 doubling is almost twice the response to the first one. This result is again consistent with a significant summertime soil moisture feedback in CNRM-CM6-1. This hypothesis is further supported by Fig. 11d-f showing the response of the evaporative fraction (i.e., the ratio of latent heat flux to the sum of latent and sensible heat flux at the surface). A 50% value in the piControl climatology (black solid line) highlights the areas where the latent heat represents less half of the total heat flux at the surface over land. The similar geographical patterns found in Fig. 11c and Fig. 11f suggest that the non-linear additive drying found in CNRM-CM6-1 is associated with a non-linear response of the evaporative fraction (EF). This response implies a strong coupling beween soil moisture and near-surface humidity. If there was no change in soil moisture, the near-surface drying should on the contrary drive an increase in the actual evapotranspiration and, therefore, in EF.

The overall soil drying is evidenced by the relative changes in total soil moisture (Fig. 11g-h). In most subtropical and midlatitude areas, the decrease in EF can be explained by a corresponding decrease in soil moisture. Yet, there are some exceptions in the northern latitudes where EF can increase despite decreasing total soil moisture. In these high-latitude regions, surface evapotranspiration is mostly limited by energy rather than soil moisture and the drying is not necessarily driven by the atmospheric water demand. It could be rather explained by a non-linear deepening of the permafrost summertime active layer associated with a non-linear response of the springtime snow cover (not shown).

Overall, the results suggest a positive land-atmosphere feedback whereby decreased RH over land is likely to increase not only the atmospheric water demand but also the actual land surface evapotranspiration, and thereby deplete soil moisture and further decrease RH. Yet, for a stronger radiative forcing (4 times instead of 2 times CO2) or after a longer period (e.g., the drying is still increasing after year 150 in the abrupt-4xCO2 experiment), the enhanced soil moisture depletion can ultimately lead to a reduced surface evapotranspiration, a decreased evaporative fraction, a stronger increase in land surface temperature, an enhanced land-sea warming contrast and a further decrease in RH through the ocean moisture advection mechanism. This feedback seems to be stronger than the mean-state dependence highlighted in Section 4.2. It deserves further analyses and a careful evaluation, since it may imply an enhanced long-term soil moisture depletion in the absence of climate change mitigation, at least at the regional scale.

Conclusions

The aim of the study was to better understand the contrasted land surface RH response to abrupt-4xCO2 in CNRM-CM6-1 versus CNRM-CM5, as well as to illustrate the interest of the CFMIP Tier While the enhanced drying simulated by CNRM-CM6-1 is therefore not due to its greater ECS (cf.

Fig. 4), the CO2 stomatal closure effect introduced in this model does not necessarily contradict the "warmer is drier" paradigm [START_REF] Sherwood | A drier future?[END_REF] claiming that the evaporative demand increases faster than precipitation over land. The stomatal closure effect increases the near-surface warming and drying over land and, thereby, strengthens the atmospheric demand (in coupled models, but not in off-line land surface models which may therefore overestimate the possible decoupling between the atmosphere and the land surface) and also leads to a soil moisture decline (cf. Fig. 11ghi, although the soil drying is even more obvious when focusing on the top layers, not shown). The response of other agro-ecological aridity indices may however lead to different conclusions [START_REF] Roderick | On the assessment of aridity with changes in atmospheric <scp>CO</scp> 2[END_REF] and could deserve parallel analyses using the same CFMIP framework. Such analyses would also need to consider the CO2 fertilization effect which is not accounted for in CNRM-CM6-1 and could partly offset the CO2 stomatal closure effect.

The present study also suggests that a simple moisture advection model captures many features of the land surface air humidity response to global warming in both CNRM-CM5 and CNRM-CM6-1, as well as some regional contrasts between both model versions. Yet, this simple scaling is not sufficient to explain the overall stronger drying in CNRM-CM6-1. The relevance of changes in land surface evapotranspiration and large-scale atmospheric circulation for understanding regional changes in land RH is therefore confirmed, and is only partly related to the direct CO2 effect and to the SST anomaly pattern effect. The projected land surface drying is sensitive to both land surface biases (given the expected mean state soil moisture impact on the soil moisture response) and land surface feedbacks (that may induce an enhanced drying in many regions in CNRM-CM6-1), which are not accounted for in the simple conceptual model.

A deeper evaluation and understanding of the land surface water budget and of its coupling with the atmosphere is therefore needed for improving the reliability of RH and land-sea warming contrast projections. The participation of other global modelling centers to CFMIP may be also useful to assess the robustness of our results about the relative role of direct versus SST-mediated CO2 effects on projected changes in the atmosphere and at the land surface. The limited role of sea-ice may be here an artifact of the experiment design and should be further investigated in fully coupled experiments.

Similarly, the CFMIP design assumes no land surface influence on SST temperature, which is a severe assumption that deserves further attention, for instance in the framework of LS3MIP [START_REF] Van Den Hurk | LS3MIP (v1.0) contribution to CMIP6: The Land Surface, Snow and Soil moisture Model Intercomparison Project -Aims, setup and expected outcome[END_REF]. Finally, it could be also interesting to replicate the CFMIP experiment design to deconstruct the abrupt-2xCO2 response of the coupled models and, thereby, better understand the drivers of the non-linear response when shifting from 2xCO2 to 4xCO2.

List of Tables :

Table 1: Summary of the AGCM timeslice experiments. All experiments are 30-year integrations after spin-up except amip and amip-a4SST-4xCO2 (36-year long integrations after spin-up). 605

Table 2: Calculation of the different components of the model response to abrupt-4xCO2. Note that the respective contributions of SIC change and SST anomaly pattern will not shown in the present study since their merged contribution is strongly dominated by the SST effect. Stippling highlights areas where the differences are significant at the 5% level. The black solid line is the 50\% isoline of the piControl climatology. GLD denotes the average global land difference and ACC denotes the spatial continental pattern correlation with the differences shown in panel a). experiments, while the AGCM response is averaged using 30-year timeslice simulations aimed at capturing the response of CNRM-CM6-1 (cf. Table 1). Stippling highlights areas where the differences are significant at the 5% level. The black solid line is the 50% isoline of the piControl climatology. GLD denotes the average global land difference and ACC denotes the spatial continental pattern correlation with the differences shown in panel a). Figure 9: Same as Fig. 8 but for changes in JJAS relative humidity (%). In panels b) and c) relative humidity is estimated from specific humidity as shown in Fig. 8, using the empirical formulation by [START_REF] Murray | On the Computation of Saturation Vapor Pressure[END_REF] and the AGCM temperature and surface pressure simulated in piSST and piSST-pxK.

Name of experiment

List of Figures :

2

  AGCM experiments proposed within CFMIP. The results show that such atmosphere-only experiments are able to capture the coupled model response to increased CO2 and are therefore useful for understanding the multiple drivers of this atmospheric response. The breakdown of the RH response indicates that the enhanced summertime land surface drying found in CNRM-CM6-1 compared to CNRM-CM5 is due to both the introduction of a physiological CO2 effect and to a SSTmediated effect.

Figure 1 :

 1 Figure 1: Timeseries of RH anomalies (%) relative to the 1995-2014 climatology: a) Annual mean anomalies averaged over global land areas except Antarctica, b) JJAS mean anomalies averaged over the northern midlatitude land areas. The solid green line represents the ensemble mean of 10 extended AMIP integrations of the atmospheric component of CNRM-CM6-1 driven by observed monthly mean SST and SIC, with the 95% confidence interval in light green shading. The black line represents the anomalies in the ERA-Interim reanalysis. R denotes the time correlation between the ensemble mean and ERA-Interim anomalies.

Figure 2 :

 2 Figure 2: Annual mean RH response (%) to abrupt-4xCO2 in a) CNRM-CM6-1, b) CNRM-CM5. The model responses are averaged over years 111-140 of the piControl and abrupt-4xCO2 experiments.

Figure 3 :

 3 Figure 3: Climatological annual cycle of near-surface a,b) relative humidity (%) and c,d) specific humidity (g/kg) averaged over 30 years (years 111-140) in the piControl and abrupt-4xCO2 simulations with CNRM-CM5, CNRM-CM6-atm5 and CRNM-CM6-1 respectively, averaged over global land areas except Antarctica (a,c), global ocean areas except sea-ice covered surfaces (b,d). CNRM-CM6-atm5 is an intermediate model version in which the atmospheric component of CNRM-CM6-1 has been replaced by the one from CNRM-CM5.

Figure 4 :

 4 Figure 4: JJAS mean scaled RH response (%/°C) to abrupt-4xCO2 in a) CNRM-CM6-1, b) CNRM-CM5, c) the atmospheric component of CNRM-CM6-1, and d) CNRM-CM6-atm5. The response is scaled by the annual and global mean near-surface warming simulated by each model. The AOGCM responses are averaged over years 111-140 of the corresponding piControl and abrupt-4xCO2

Figure 5 :

 5 Figure 5: Breakdown of the JJAS mean RH response (%) to abrupt-4xCO2 in CNRM-CM6-1 using pairs of atmosphere-only timeslice experiments (cf. Tables1 and 2): a) total AGCM response, b) response to uniform SST warming, c) response to SST and SIC anomaly pattern, and d) response to

Figure 6 :

 6 Figure 6: Same as Fig. 5, but for the breakdown of the JJAS mean RH response (%) and 850 hPa wind response (m/s) to abrupt-4xCO2 in CNRM-CM5.

Figure 7 :

 7 Figure 7: Breakdown of the JJAS mean RH response (%) to CO2 in CNRM-CM6-1 using pairs of atmosphere-only timeslice experiments (cf. Tables1 and 2): a) total CO2 effect under warm climate (abrupt-4xCO2 SST), b) total CO2 effect under control climate (piControl SST), c) Physiological CO2 effect under control climate, and d) Radiative CO2 effect under control climate. By experiment design b=c+d. All AGCM responses are averaged over 30 years. Stippling highlights areas where the differences are significant at the 5% level. Vectors represent the wind response (m/s) at 850 hPa. GLD denotes the average global land RH difference and ACC denotes the spatial continental pattern correlation with the RH differences shown in panel a).

Figure 8 :

 8 Figure 8: Changes in JJAS specific humidity (g/kg) due to uniform SST warming a,d) as simulated by the CNRM-CM5 and CNRM-CM6-1 AGCMs respectively (piSST-pxK minus piSST), and b,e) as predicted from a simple advection model based on the AGCM specific response over ocean only (cf. equation 4). Also shown are c) (a)-(b); f) (d)-(e); g) (d)-(a); h) (e)-(b); i) (f)-(c). The piSST JJAS low-level (850 hPa) wind climatology is superimposed in b) and e) and the difference (e)-(b) is superimposed in h).

Figure 10 :

 10 Figure 10: Differences in JJAS relative humidity (%) between: a) amip and piSST, b) amip-a4SST-4xCO2 and a4SST-4xCO2, c) amip-a4SST-4xCO2 and amip, d) a4SST-4xCO2 and piSST. Stippling highlights areas where the differences are significant at the 5% level. GLD denotes the average global land RH difference and ACC denotes the spatial continental pattern correlation between the differences shown in panels c) and d) respectively.

Figure 11 :

 11 Figure11: Differences in JJAS a-c) relative humidity (%), d-f) evaporative fraction (%), g-i) total soil moisture (%) between: a,d,g) abrupt-2xCO2 and piControl, b,e,h) abrupt-4xCO2 and piControl, c,f,i)

  Figure 7

Figure 11

 11 Figure 11

Table 1 :

 1 Summary of the AGCM timeslice experiments. All experiments are 30-year integrations after spin-up except amip and amip-a4SST-4xCO2 (36-year long integrations after spin-up).

		Monthly mean annually varying SST and SIC boundary conditions	CO2 forcing
	piSST	Years 111-140 of piControl	Preindustrial (1xCO2)
		As piSST + uniform SST anomaly computed as the
	piSST-pxK	climatological global mean difference between years	preindustrial
		111-140 of abrupt-4xCO2 and piControl
	piSST-4xCO2	As piSST	4xCO2 seen by radiation and vegetation
	piSST-4xCO2-rad	As piSST	4xCO2 seen by radiation
	a4SSTice	Years 111-140 of abrupt-4xCO2	preindustrial
	a4SSTice-4xCO2	As a4SSTice	4xCO2 seen by radiation and vegetation
	a4SST	As a4SSTice but with sea-ice concentration from years 111-140 of piControl	preindustrial
	a4SST-4xCO2	As a4SST	4xCO2 seen by radiation and vegetation
	amip	Observed 1979-2014 monthly mean SST and SIC	present-day
	amip-a4SST-4xCO2	As amip + climatological patterned SST anomalies calculated as the difference between years 111-140 of abrupt-4xCO2 and piControl	4 x present-day seen by radiation and vegetation
	610

Table 2 :

 2 Calculation of the different components of the model response to abrupt-4xCO2. Note that the respective contributions of SIC change and SST anomaly pattern will not shown in the present study since their merged contribution is strongly dominated by the SST effect.

Acknowledgements :

The authors would like to thank all people at CNRM and CERFACS who were involved in the development of the CNRM-CM5 and CNRM-CM6-1 models. Thanks are also due to Sophie Tyteca and Richard Stchepounoff for the preparation of the SST and SIC boundary conditions 485 of the AGCM experiments with CNRM-CM5 and CNRM-CM6-1 respectively. The CNRM-CM6-1 model outputs from the CMIP6 DECK and CFMIP experiments can be downloaded from the ESGF.


Antoine Etesse

COMPLEX-ANALYTIC INTERMEDIATE HYPERBOLICITY, AND FINITENESS PROPERTIES

Keywords: 2010 Mathematics Subject Classification. 14G99 (11G35, 14G05, 32Q45) Hyperbolicity, Moduli spaces of maps, Ampleness, Positivity

Motivated by the finiteness of the set of automorphisms Aut(X) of a projective manifold X, and by Kobayashi-Ochiai's conjecture that a projective manifold dim(X)-analytically hyperbolic (also known as strongly measure hyperbolic) should be of general type, we investigate the finiteness properties of Aut(X) for a complex manifold satisfying a (pseudo-) intermediate hyperbolicity property. We first show that a complex manifold X which is (dim(X) -1)-analytically hyperbolic has indeed finite automorphisms group. We then obtain a similar statement for a pseudo-(dim(X) -1)-analytically hyperbolic, strongly measure hyperbolic projective manifold X, under an additional hypothesis on the size of the degeneracy set. Some of the properties used during the proofs lead us to introduce a notion of intermediate Picard hyperbolicity, which we last discuss.

Acknowledgements. I would like to thank my supervisor Erwan Rousseau as well as Ariyan Javanpeykar for their useful remarks, advices, and corrections. Specifically, the notion of intermediate Picard hyperbolicity emerged from various discussions we had together.

Introduction

It is a standard result that a hyperbolic (in any sense of the term) compact manifold X satisfies the following strong finiteness result: for any compact manifold Y , the set of surjective holomorphic maps Sur(Y, X) is finite. (See [START_REF] Kobayashi | Hyperbolic complex spaces, volume 318 of Grundlehren der Mathematischen Wissenschaften[END_REF][Chapter 6, Section 6]; the result is actually true for X, Y complex spaces). In particular, the set of automorphisms Aut(X) is finite. In a different direction, a compact complex hyperbolic manifold X satisfies the following remarkable extension property: for any compact complex manifold Y , and any A Y closed complex subspace of Y , every holomorphic map f : Y \ A → X extends holomorphically through A. (See [START_REF] Kobayashi | Hyperbolic complex spaces, volume 318 of Grundlehren der Mathematischen Wissenschaften[END_REF][Chapter 6, Section 3]; the result is actually valid for X a complex space).

In [START_REF] Donald A Eisenman | Intrinsic measures on complex manifolds and holomorphic mappings[END_REF], Eisenmann introduced the so-called Kobayashi-Einsenmann pseudo-metrics, leading to weaker forms of hyperbolicity, namely ℓ-analytic hyperbolicity, 1 ≤ ℓ ≤ dim(X), that we sometimes refer to as "intermediate hyperbolicity". The case ℓ = 1 leads to the usual notion of hyperbolicity, and a special focus was first given to the weakest form (i.e. for ℓ = dim(X)), often called (strong)-measure hyperbolicity (see Section 2.1 for definitions). Restricting ourselves to the realm of projective manifolds, it was proved by Kobayashi and Ochiai that a projective manifold X of general type is (strongly-)measure hyperbolic (see e.g. [START_REF] Kobayashi | Hyperbolic complex spaces, volume 318 of Grundlehren der Mathematischen Wissenschaften[END_REF][Chapter 7, Section 4]), and they further conjectured that the converse should be true.

The finiteness property that we described above, valid for hyperbolic manifolds, turns out to be satisfied by projective manifolds of general type as well (See [START_REF] Kobayashi | Hyperbolic complex spaces, volume 318 of Grundlehren der Mathematischen Wissenschaften[END_REF][Chapter 7, Section 6]); as for the property of extension, the following weaker form is valid for projective manifolds of general type:

(*) if dim(Y ) ≥ dim(X), then any non-degenerate holomorphic map f : Y \ A → X extends meromorphically through A, where f is said to be non-degenerate if its differential is of a maximal rank at some point (see Section 2.4 where we recall the definition of a meromorphic map).

In light of the conjecture we mentioned above, it is natural to expect that such finiteness and extension properties should be satisfied by measure hyperbolic projective manifold. As such kind of properties are actually satisfied by hyperbolic manifold, not necessarily projective, we may even be tempted to drop the projective hypothesis. In any case, this leads naturally to the following two conjectures. (See also [START_REF] Graham | Some remarks on the intrinsic measures of eisenman[END_REF]).

Conjecture 1.1 (Finiteness property). Let X be a projective manifold dim(X)-analytically hyperbolic. Then Aut(X) is finite.

Conjecture 1.2 (Extension property). Let X be a projective dim(X)-analytically hyperbolic manifold, let Y be a projective manifold of dimension dim(Y ) ≥ dim(X), and let H ⊂ Y be an hypersurface. Then any non-degenerate holomorphic map

f : Y \ H → X extends to a meromorphic map f : Y X.
It is worth noting here that the usual notion of (pseudo-)hyperbolicity, defined for complex spaces, has its analogues in the algebraic and arithmetic worlds, and that such finiteness properties were investigated in these settings: see [START_REF] Bogomolov | Algebraically hyperbolic manifolds have finite automorphism groups[END_REF] and [JK] for algebraic hyperbolicity, and [Java, JX] for pseudo algebraic hyperbolicity and arithmetic hyperbolicity.

In this paper, we investigate the finiteness properties for intermediate (analytic) hyperbolicities, and prove first the following theorem.

Theorem 1.3. Let X be a compact complex manifold which is (dim(X)-1)-analytically hyperbolic. Then Aut(X) is finite.

Note that X is not supposed projective, and that unfortunately we were not able to obtain the suspected optimal hypothesis, i.e. dim(X)-analytic hyperbolicity.

Following Lang's terminology, each intermediate hyperbolicity has its pseudo-version, where a closed degeneracy set is allowed for the Kobayashi-Eisenmann (pseudo-)metrics (see Section 2.1 for definitions). Depending on whether we want the degeneracy set to be analytically closed, or simply topologically closed, we will write pseudo an or pseudo top . In view of Lang's conjecture that a projective manifold of general type should be pseudo an -hyperbolic, it is expected that, in the realm of projective manifolds, the intermediate pseudo an -hyperbolicities should be equivalent. For more details on Lang's conjectures, and their analogues in the algebraic and arithmetic settings, see e.g. [Javb].

We were then naturally lead to investigate the question of finiteness of Aut(X) under the weaker assumptions of dim(X)-analytic hyperbolicity and pseudo top / pseudo an -(dim(X) -1)-analytic hyperbolicity, and were able to prove the following Theorem 1.4. Let X be projective manifold, dim(X)-analytically hyperbolic and (dim(X) -1)analytically hyperbolic modulo a topologically closed subset ∆ = ∆ X,ℓ of zero (2 dim(X)-3) Haussdorf dimension. Then Aut(X) is finite.

Note that this time, the manifold is supposed to be projective, and that any Zariski closed proper subset ∆ of codimension ≥ 2 satisfies the assumptions of the theorem. The hypothesis on the degeneracy set may seem odd: this is a technical hypothesis necessary to apply a very general extension theorem due to Siu [START_REF] Siu | Extension of meromorphic maps into kahler manifolds[END_REF].

In the case of pseudo an -(dim(X) -1)-analytic hyperbolicity, i.e. when ∆ is algebraic, the extension property needed in the course of the proof is in fact exactly the extension property (*) we saw above. Accordingly, and interestingly enough, the questions on finiteness are connected to the ones on extension. Whereas this seemed to us to be a much harder task to relate intermediate hyperbolicities to this extension property, in view of the recent papers [START_REF] Javanpeykar | Algebraicity of analytic maps to a hyperbolic variety[END_REF] and [START_REF] Deng | Big picard theorem and algebraic hyperbolicity for varieties admitting a variation of hodge structures[END_REF], we realized that there is a natural notion of intermediate Picard hyperbolicity that precisely encapsulates these extension properties. In this framework, the scheme of proof of 1.4 allowed us to obtain the following theorem (see Section 3 for the intermediate notions of ℓ-Picard hyperbolicity) Theorem 1.5. Let X be a projective manifold, dim(X)-Picard hyperbolic, dim(X)-analytically hyperbolic and pseudo an -(dim(X) -1)-analytically hyperbolic. Then Aut(X) is finite.

The paper is organized as follows. In the first part, we recall the notions of pseudo top/anintermediate hyperbolicities, and prove some basic results regarding them. In the second part, we start by proving the finiteness result of Aut(X) in the absolute case (Theorem 1.3), and then move onto the proof in the pseudo case (Theorem 1.4). In the third and last part, we introduce the notions of intermediate Picard hyperbolicities, give some of their basic properties, and prove Theorem 1.5.

Automorphisms of ℓ-analytically hyperbolic manifolds

This section, motivated by Conjecture 1.1, is devoted to the proof of Theorem 1.3 and its pseudoversion Theorem 1.4. We start by proving some basic properties on intermediate hyperbolicity, in particular that ℓ-analytic hyperbolicity modulo ∆ implies (ℓ + 1)-analytic hyperbolicity modulo ∆ (See also [START_REF] Graham | Some remarks on the intrinsic measures of eisenman[END_REF], where it is proved in the absolute case), and then move to the two proofs in question. We end this section by some conditions ensuring intermediate hyperbolicity, and hence the finiteness of Aut(X) in suitable cases.

2.1. Basics on ℓ-analytic hyperbolicity. Throughout this section, we let X be a compact complex manifold of dimension p and 1 ≤ ℓ ≤ p be an integer. Following [START_REF] Demailly | Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials[END_REF] and [START_REF] Graham | Some remarks on the intrinsic measures of eisenman[END_REF], define the Kobayashi-Eisenmann pseudo-metric e ℓ on pure multi-vectors of ℓ T X by setting for

ξ = v 1 ∧ . . . ∧ v ℓ ∈ ℓ T x X, x ∈ X: e ℓ (x, ξ) = {inf 1 R | ∃ f : B ℓ → X holomorphic, f (0) = x, f * ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z ℓ = Rξ}.
For ∆ X a closed subset of X (of empty interior), the compact complex manifold X is said ℓ-analytically hyperbolic modulo ∆ if and only if there exists a continuous Finsler pseudo-metric ω defined on pure multi-vectors of ℓ T X, which is a metric on X \ ∆ and such that e ℓ ≥ ω on X. We will denote ∆ X,ℓ the smallest closed subset satisfying such a property, commonly referred to as the exceptional locus (with respect to the pseudo-metric e ℓ ).

Remark 2.1. Such a set is well defined since if X is ℓ-hyperbolic modulo ∆ 1 and ℓ-hyperbolic modulo ∆ 2 , then X is ℓ-hyperbolic modulo ∆ 1 ∩ ∆ 2 : for i = 1, 2, take ω i a continuous Finsler pseudo-metric on (pure multi-vectors of) ℓ T X, which is a metric on X \ ∆ i , and such that e ℓ ≥ ω i on X, and consider

ω = ω 1 + ω 2 2 ,
which is a continuous Finsler pseudo-metric on (pure multi-vectors of) ℓ T X, a metric on X \ (∆ 1 ∩ ∆ 2 ), and satisfies e ℓ ≥ ω on X.

We will say that X pseudo top (resp. pseudo an ) ℓ-analytically hyperbolic if ∆ X,ℓ is of empty interior (resp. if ∆ X,ℓ is a proper analytically closed subset). Note that if ℓ = dim X and ∆ := ∅, such a manifold is called strongly measure hyperbolic in [START_REF] Kobayashi | Hyperbolic complex spaces, volume 318 of Grundlehren der Mathematischen Wissenschaften[END_REF] and more generally strongly ℓ-measure hyperbolic in [START_REF] Graham | Some remarks on the intrinsic measures of eisenman[END_REF] for 1 ≤ ℓ ≤ dim X and ∆ := ∅.

A first basic observation is that, for f : C × B ℓ-1 → X a holomorphic map, the pseudo-metric e ℓ degenerates along the image of every non-degenerate point of f . Note that we say that f is non-degenerate if there exists at least one point (and thus a non-empty Zariski open subset as being degenerate for an holomorphic map is an analytically closed condition) at which f is nondegenerate. If ℓ = 1, by Brody's reparametrization lemma (see e.g. [START_REF] Kobayashi | Hyperbolic complex spaces, volume 318 of Grundlehren der Mathematischen Wissenschaften[END_REF][Lemma 3.6.2]), the existence of a degeneracy locus for e 1 implies the existence of a non-constant holomorphic map f : C → X, commonly referred to as an entire curve in X. Let us make two comments on that. First, the degeneracy locus of e 1 need not be equal to the union of the closure of the image of all entire curves. The reason for this lies in the way one constructs an entire curve via Brody's lemma: we have no control on the image of the entire curve, and in particular, we can not ensure that there exists an entire curve passing arbitrarly close to a point where e 1 degenerates. Second, we see that it implies that a compact complex manifold X is 1-analytically hyperbolic if and only if X contains no entire curves (X is commonly called Brody-hyperbolic). However, in the case where we allow a degeneracy set for e 1 , there is to our knowledge no clean relationship between the geometry of entire curve and 1-analytic hyperbolicity modulo ∆. Even worse, in the case p > 1, the non-existence of non-degenerate holomorphic maps f : C × B ℓ-1 → X does not imply, a priori, that X is ℓ-analytically hyperbolic. Also, note that, in the case where p = 1, the notion of 1analytic hyperbolicity modulo ∆ is the same as the notion of hyperbolicity modulo ∆ (in the sense of Kobayashi), where we recall that X is hyperbolic modulo ∆ if and only if the Kobayashi metric d X satisfies d X (p, q) = 0 ⇐⇒ p = q, or p, q ∈ ∆.

Indeed, one implication follows from a result of Royden ([Roy71]), stating the Kobayashi pseudodistance d X is obtained as the integration of the infinitesimal pseudo-metric e 1 , whereas the reverse implication follows for instance from [START_REF] Kobayashi | Hyperbolic complex spaces, volume 318 of Grundlehren der Mathematischen Wissenschaften[END_REF][Cor. (3.5.41)].

2.2. From ℓ to ℓ + 1. Equip X with any smooth hermitian metric with its induced Riemaniann metric on X. Moreover, for 1 ≤ ℓ ≤ p, equip the vector bundles ℓ T X with smooth (Finsler) metrics defined on pure multi-vectors that we denote F ℓ . Here, F 1 is the hermitian metric. Let U ϕ ≃ B p be any trivializing open set, and denote ( ∂ ∂z i ) 1≤i≤p the standard vector fields on

B p . Let 1 ≤ ℓ ≤ p be an integer. For v 1 ∧ . . . ∧ v ℓ ∈ ℓ T x X and x ∈ U , denote A(x; v 1 , . . . , v ℓ ) the (ℓ × p)-matrix which represents (ϕ) * (v 1 ), . . . , (ϕ) * (v ℓ ) in the basis ( ∂ ∂z 1 , . . . , ∂ ∂zp ).
We define a smooth Finsler metric on pure multi-vectors of ℓ T X |U by setting

h ℓ,U (x; v 1 ∧ . . . ∧ v ℓ ) := max{| det(M )| | M is a ℓ × ℓ extracted matrix of A(x; v 1 , . . . , v ℓ )}. Note that this metric is well-defined. Indeed, if v ′ 1 ∧ . . . ∧ v ′ ℓ = v 1 ∧ . . . ∧ v ℓ , then there exists a matrix Q in GL ℓ (C) with det(Q) = 1 such that QA(x; v ′ 1 , . . . , v ′ ℓ ) = A(x; v 1 , . . . , v ℓ ).
Since both F ℓ and h ℓ,U are continuous on U , for any compact subset K ⊂ U , there exist real numbers 0 < c ≤ C such that, for all x ∈ K and all ξ ∈ ℓ T x X pure multi-vector, the inequalities

c • ξ h ℓ,U (x) ≤ ξ F ℓ (x) ≤ C • ξ h ℓ,U (x)
hold. Accordingly, when dealing with questions of convergence towards 0 or divergence to ∞ with respect to the metric F ℓ of sequences of pure multi-vectors, we can always use the local metric h ℓ,U .

To prove Proposition 2.4 below, we will use the following basic lemma, which is merely a reformulation of what ℓ-analytic hyperbolicity modulo ∆ means.

Lemma 2.2. Let ∆ X be a closed set and let 1 ≤ ℓ ≤ p. If X is not ℓ-analytically hyperbolic modulo ∆, then there exists h n : B ℓ → X holomorphic maps such that

(h n ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z ℓ ) F ℓ (xn) → ∞, with x n = h n (0) → x / ∈ ∆.
Reciprocally, if there exists h n : B ℓ → X holomorphic maps, and z n → 0 such that

(h n ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z ℓ ) F ℓ (xn) → ∞, with x n = h n (z n ) → x / ∈ ∆, then X is not ℓ-analytically hyperbolic modulo ∆.
Proof. Suppose first that X is not ℓ-analytically hyperbolic modulo ∆. Consider (U n ) n∈N an exhaustion of relatively compact domains of X \ ∆, i.e. U n ⊂ U n+1 ⊂ X \ ∆ for all n ∈ N, and

n∈N U n = X \ ∆. Observe then that the following property is not satisfied (2.1) ∀n ∈ N, ∃δ n > 0 such that e ℓ ≥ δ n F ℓ on U n .
Indeed, otherwise by constructing a nonnegative continuous function ϕ on X, which is equal to 0 on ∆, and satisfies 0 < ϕ ≤ δ n on U n , we deduce that ϕF ℓ ≤ e l on X. As ϕF ℓ is a continuous Finsler pseudo-metric on (pure multi-vectors of) ℓ T X, which is a metric on X \ ∆, this contradicts the fact is not ℓ-analytically hyperbolic modulo ∆. Now, as (2.1) is not satisfied, we deduce in particular the existence of a sequence of points x n → x ∈ X \ ∆, aswell as a sequence of pure multi-vectors

ξ n ∈ ℓ T xn X, |ξ n | F ℓ (xn) = 1, such that e ℓ (x n ; ξ n ) → 0.
By definition of e ℓ , this in turn gives a sequence of holomorphic maps h n :

B ℓ → X, h n (0) = x n , satisfying (h n ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z ℓ ) = R n ξ n ,
with R n → ∞, so that the first implication is proved.

In the other direction, observe first that we can always suppose that z n = 0 for all n ∈ N: indeed, for all n, pick τ n ∈ Aut(B ℓ ) such that τ n (0) = z n , and consider instead hn = h n • τ n ; the announced fact then follows from the equality

( hn ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z ℓ ) F ℓ (xn) = det(dτ n ) 0 • (h n ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z ℓ ) F ℓ (xn) ,
and the fact that det(dτ n ) 0 remains bounded (as z n stays away of ∂B ℓ ). If X were ℓ-analytically hyperbolic modulo ∆, then by definition there would exist a continuous Finsler metric ω on (pure multi-vectors of) l T X, which is a metric on X \ ∆, such that e ℓ ≥ ω on X. In particular, letting U be a relatively compact neighborhood of x in X \ ∆, there exists δ > 0 such that for any x ∈ U and ξ

= v 1 ∧ . . . ∧ v ℓ ∈ ℓ T x X, we have the inequality (2.2) |ξ| e ℓ (x) ≥ |ξ| ω(x) ≥ δ|ξ| F ℓ (x) .
Now, the sequence

ξ n = (h n ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z ℓ ) (h n ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z ℓ ) F ℓ (xn) ∈ ℓ T xn X satisfies |ξ n | F ℓ (zn)
= 1 by construction, and by the very definition of the metric e ℓ , we have:

ξ n e ℓ (xn) ≤ 1 (h n ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z ℓ ) F ℓ (xn)
-→ 0, which contradicts the inequality (2.2), as for all n large enough, x n lies in U . Therefore X is not ℓ-analytically hyperbolic modulo ∆.

We will also use the following elementary lemma in linear algebra.

Lemma 2.3. Let A 1 , A 2 , . . . be a sequence of matrices in GL ℓ (C) with | det(A n )| -→ n→∞ ∞. Then, the sequence (A n ) has a subsequence (A ϕ(n) )
such that the norm of some fixed minor of (A ϕ(n) ) tends to infinity.

Proof. This follows immediately from the equality

| det(A n )| ℓ-1 = | det(Com(A n ))| → ∞,
where Com(A n ) is the co-matrix of A n , i.e., the matrix whose entries are the minors of A n .

We now come to the following proposition (see also [START_REF] Graham | Some remarks on the intrinsic measures of eisenman[END_REF], Proposition 2.17)

Proposition 2.4. If X is ℓ-analytically hyperbolic modulo ∆, then X is (ℓ + 1)-analytically hyperbolic modulo at least ∆.

Proof. Assume that X is not (ℓ + 1)-analytically hyperbolic modulo ∆. We show that it is not ℓ-analytically hyperbolic modulo ∆. By our assumption and the Lemma 2.2, we can find a sequence of points

x n ∈ X, with x n → x ∈ X \ ∆, a sequence of holomorphic maps f n : B ℓ+1 → X with f n (0) = x n , and a sequence of pure multi-vectors ξ n ∈ ℓ+1 T xn X with |ξ n | F ℓ+1 (xn) = 1 such that (f n ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z ℓ+1 ) = R n ξ n ,
where R n is a sequence of real numbers such that R n → ∞.

Let U ≃ B p be a trivializing open set around x, relatively compact in X \∆. We now use the local Finsler metrics h i,U for 1 ≤ i ≤ p. For 1 ≤ j ≤ ℓ + 1, define v j,n := (df n ) 0 ( ∂ ∂z j ); using the notations introduced earlier, we deduce (up to passing to a subsequence) that the determinant of a fixed extracted square matrix of maximal size, i.e. size (ℓ + 1), of A(x n ; v 1,n , . . . , v ℓ+1,n ) ∈ M p,ℓ+1 (C), tends to infinity in norm. By Lemma 2.3, we deduce (up to passing to a subsequence) that the determinant of a fixed square matrix of size ℓ of A(x n ; v 1,n , . . . , v ℓ+1,n ) also tends to infinity in norm; we can always suppose (up to re-ordering the variables) that this matrix is obtained by keeping the first ℓ columns fixed. Now, consider the sequence of maps

h n : B ℓ → X, (z 1 , . . . , z ℓ ) → f n (z 1 , . . . , z ℓ , 0).
Note that, by construction, this sequence satisfies

(h n ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z ℓ ) h ℓ,U (xn) → ∞.
In particular, it follows that

(h n ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z ℓ ) F ℓ (xn) → ∞.
It now follows from Lemma 2.2 that X is not ℓ-analytically hyperbolic modulo ∆. This concludes the proof.

2.3. Finiteness of automorphism groups. In this section we prove that, if X is a compact complex manifold which is (dim(X) -1)-analytically hyperbolic, then the automorphism group Aut(X) of X is finite. As before, we start with a basic lemma in linear algebra.

Lemma 2.5. Let |.| be a norm on C p , let ε > 0 be a real number, and let A ∈ M p (C) be such that

inf |v|=1 |Av| ≥ ε.
Then, there exists a real number K > 0 independant of A such that

Kε p-1 ||A|| ≤ | det(A)|,
where the norm ||A|| is computed with respect to |.|.

Proof. Let us first consider the case of the usual hermitian norm on C p . Note that, by our assumption, the matrix A is invertible. Now, consider the polar decomposition

A = U S of A,
|Sv| = p k=1 |α k | 2 λ 2 k ≤ λ p ≤ λ p λ p-1 ε . . . λ 1 ε = 1 ε p-1 det(S).
This shows that the statement of the lemma holds when |.| is the usual hermitian norm on C p with K = 1.

To conclude the proof, let |.| be a norm on C p . Then, there exist real numbers 0

< c ≤ C such that c.|.| eucl ≤ |.| ≤ C|.
| eucl , where |.| eucl is the usual hermitian norm. Then, by the above, it follows readily that the lemma holds with K := ( c C ) p . To prove the desired finiteness of Aut(X) when X is (dim X -1)-analytically hyperbolic, we will show that Aut(X) is discrete and compact. The discreteness is well-known, and we record it in the following lemma.

Lemma 2.6. Let X be a compact complex manifold. If X is pseudo top -dim X-analytically hyperbolic, then Aut(X) is discrete.

Proof. As X is a compact complex variety (of dimension p), it is known that Aut(X) is a complex Lie-group by a theorem of Bochner-Montgomery ( [START_REF] Bochner | Groups on analytic manifolds[END_REF]). Accordingly, it must be discrete since otherwise, by considering a non-zero complex vector field on X, one can construct a non-degenerate holomorphic map

F : C × B p-1 → X.
But this contradicts the pseudo dim(X)-analytic hyperbolicity of X as the image of such a map would have a non-empty interior and would be included in the degeneracy set of e p .

We now prove the finiteness of Aut(X), assuming X is (dim X -1)-analytically hyperbolic, by showing that it is compact.

Proof of Theorem 1.3. To prove the theorem, by Lemma 2.6, it is enough to prove that Aut(X) is compact. To show this compactness, let (f n ) n∈N be a sequence in Aut(X). By classic arguments involving Arzela-Ascoli theorem and convergence of holomorphic maps, in order to prove that (f n ) n∈N has an adherence value in Aut(X) (i.e. a converging subsequence) it suffices to prove that lim inf

n∈N max x∈X ||(df n ) x || < ∞.
Arguing by contradiction, up to passing to a suitable subsequence, we can suppose that there exists

x n → x, with y n = f n (x n ) → y, satisfying ||(df n ) xn || → ∞, where the norm is computed with respect to F 1 . Let U ϕ ≃ B p be
a trivializing open set around x with ϕ(x) = 0, and let V ψ ≃ B p be a trivializing open set around y with ψ(y) = 0. Replacing x n and y n by subsequences if necessary, we may assume that, for every n, x n ∈ U and y n ∈ V . Define z n = ϕ(x n ) and define g n := ψ • f n • ϕ -1 ; note that this function g n is well-defined in a neighborhood of z n (but maybe not on the whole set B p ).

From now on, we are going to use the metrics h ℓ,U and h ℓ,V defined via the trivializations ϕ and ψ, as described in the beginning of this section. Let us show that there exist real numbers 0 < m ≤ M such that, for every n ∈ N,

m ≤ (f n • ϕ -1 ) * ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z p h p,V (yn) ≤ M.
Indeed, the upper bound follows immediately from the Lemma 2.2. To prove the lower bound, let A n ∈ GL p (C) be the matrix representing the linear map (dg n ) zn with respect to the canonical basis of C p , and observe that, by the definition of h p,V , the equality

(f n • ϕ -1 ) * ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z p h p,V (yn) = | det(A n )| holds. Consider f -1 n and g -1 n = ϕ • f -1 n • ψ -1
. Now, the matrix representing the linear map (dg -1 n ) gn(zn) with respect to the canonical basis of C p is A -1 n , and similarly we have that

(f -1 n • ψ -1 ) * ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z p h p,U (xn) = | det(A -1 n )| = 1 | det(A n )| .
Thus, by applying once again Lemma 2.2, we obtain the desired lower bound. Now, we show that, up to extracting and re-ordering the variables, there exists

v n = (α 1 n , . . . , α p n ) → v = (α 1 , . . . , α p ) ∈ C p with |v n | = |α p n | = 1 such that A n v n → 0,
where we consider the sup norm on C p , i.e., |(v 1 , . . . , v ℓ )| = max

1≤i≤ℓ |v i |. Indeed, recall that ||(df n ) xn || → ∞, so that ||A n || → ∞.
Arguing by contradiction, there must exist ε > 0 such that, for every v ∈ C p with |v| = 1 and every n ∈ N, |A n v| ≥ ε.

In this case, by Lemma 2.5, there is a real number

K > 0 such that | det(A n )| ≥ Kε p-1 ||A n ||.
However, as the latter diverges to infinity, this implies that | det(A n )| → ∞. This contradicts the (dim X)-analytic hyperbolicity by Lemma 2.2, as

(f n • ϕ -1 ) * ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z p h p,V (yn) = | det(A n )|.
To conclude the proof, note that

(f n • ϕ -1 ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z p ) h p,V (yn) = 1 |α p n | (f n • ϕ -1 ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z p-1 ∧ v n ) h p,V (yn) = | det(B n )|,
where B n is defined to be the matrix A n except that its last column (equal to A n ∂ ∂zp ) has been replaced by A n v n . Now, developing the determinant with respect to the last column, since A n v n → 0 and | det(A n )| is bounded from below by m > 0, we deduce that at least one minor of the matrix representing A n ∂ ∂z 1 , . . . , A n ∂ ∂z p-1 has its determinant tending to infinity in norm. This shows that

(f n • ϕ -1 ) * ( ∂ ∂z 1 ∧ . . . ∧ ∂ ∂z p-1 ) h p-1,V (yn) → ∞.
By Lemma 2.2, we conclude that X is not (p -1)-analytically hyperbolic. This contradiction concludes the proof.

2.4. Pseudo version. This goal of this section is to obtain the same finiteness result as in the previous part, but by allowing an exceptional locus ∆ = ∆ X,ℓ for the (dim(X) -1) analytic hyperbolicity assumption, and adding accordingly the dim(X)-analytic hyperbolicity condition. Some technical assumptions are to be added on the exceptional locus ∆, and we aim at proving the following slightly more general theorem than Theorem 1.4.

Theorem 2.7. Let X be a compact Kähler manifold of dimension dim(X) = p, which is panalytically hyperbolic, and (p -1)-analytically hyperbolic modulo ∆ = ∆ X,ℓ . Suppose furthermore that • the (2p -3)-Haussdorf measure of ∆ is zero,

• ∆ has a projective neighborhood in X (e.g. X, if X is supposed projective). Then Aut(X) is finite.

For the notion of Haussdorf measure, we refer for instance to [START_REF] Donald A Eisenman | Intrinsic measures on complex manifolds and holomorphic mappings[END_REF]. Let us start with the following elementary proposition: Proposition 2.8. Let X, Y be compact complex manifolds of same dimension, with X ℓ-analytically hyperbolic modulo ∆ = ∆ X,ℓ . Suppose there exists a (non-degenerate) holomorphic map f : Y → X. Then Y is ℓ-analytically hyperbolic modulo f -1 (∆) ∪ Degen(f ), where Degen(f ) is the set of points where the differential df is not an isomorphism.

Proof. By hypothesis, there exists ω a continuous pseudo-metric on (pure multi-vectors of) ℓ T X, which is a metric on X \ ∆ and such that e ℓ,X ≥ ω on X. It is straightforward to check that, as f is holomorphic, almost by definition, f * e ℓ,X ≤ e ℓ,Y .

Consider then ω ′ := f * ω, which is a continuous pseudo-metric on (pure multi-vectors of) ℓ T X, a metric on X \ Degen(f ) ∪ f -1 (∆) , and satisfies

ω ′ ≤ e ℓ,Y
on X, so that Y is indeed ℓ-analytically hyperbolic modulo f -1 (∆) ∪ Degen(f ).

Applying this in particular to f and f -1 in Aut(X) yields the following corollary Corollary 2.9. Let X be a compact complex variety which is ℓ-analytically hyperbolic modulo ∆ = ∆ X,ℓ . Then any automorphism f ∈ Aut(X) induces an automorphism f ∈ Aut(X \ ∆), i.e. f (∆) = ∆.

Proof. By the Lemma 2.8 applied to f , X is ℓ-analytically hyperbolic modulo f -1 (∆), so that necessarily ∆ ⊂ f -1 (∆), as ∆ = ∆ X,ℓ is the smallest closed set modulo which X is p-analytically hyperbolic. Applying it to f -1 yields the reverse inclusion, and proves the corollary.

Let us recall the notion of meromorphic map (in the sense of Remmert). A meromorphic map f : X Y between two complex manifolds X, Y is a correspondance satisfying the following conditions:

• ∀ x ∈ X, f (x) is a non-empty compact complex subspace of Y ,

• the graph

G f = {(x, y) ∈ X × Y | y ∈ f (x)} is a connected complex subspace of X × Y , with dim G f = dim X,
• there exists a dense subset X * of X such that for all x ∈ X * , f (x) is a single point. Let π : X × Y → X be the first projection, E ⊂ G f the set of points where π degenerates, i.e.

E := {(x, y) ∈ G f | dim f (x) > 0},
and S = π(E), called the singular locus of f . All we need to know here is that S is a closed complex subspace of X of codimension ≥ 2, and f : X \ S → Y is holomorphic (see e.g. [START_REF] Remmert | Holomorphe und meromorphe abbildungen komplexer räume[END_REF]).

Letting ∆ X be a closed subset (of empty interior), a key property that we will need is the following:

(P ∆ ) Every holomorphic map f : X \ ∆ → X extends to a meromorphic map f : X X. Now, an important result to prove Theorem 2.7 is the following lemma, whose idea can essentially by found in a paper of Yau ([Yau75]) Lemma 2.10. Let X be a compact complex manifold which is dim(X)-analytically hyperbolic, (dim(X) -1)-analytically hyperbolic modulo ∆ = ∆ X,ℓ , and satisfies the property P ∆ . Suppose furthermore that ∆ has a projective neighborhood in X. Then from any sequence (f n ) n ∈ Aut(X) N , we can extract a subsequence that converges uniformly on any compact of X \ ∆ to an element of Aut(X \ ∆), which extends to an element of Aut(X).

Proof. The reasoning of the proof of 1.3 works verbatim, and allows to show that (f n ) n∈N admits a converging subsequence to an element f of Hol(X \ ∆, X). Similarly, up to extraction, we can suppose that (f -1 n ) n∈N converges to g ∈ Hol(X \ ∆, X). Observe that there exists m > 0 such that for all x ∈ X \ ∆, the following inequality is satisfied

| det(df x )| ≥ m > 0,
where |.| is a fixed continuous metric on p Ω X (p = dim(X)). Indeed, it is enough to prove that such an inequality is satisfied for f n for all n ∈ N and all x ∈ X. Arguing by contradiction, by compacity of X, we can construct

x n k → x, with y n k = f n k (x n k ) → y, such that det (df n k ) xn k → 0. But this implies that det (df -1 n k ) yn k → ∞
, which allows to contradict the dim(X)-analytic hyperbolicity via Lemma 2.2 in the same fashion that it was done in the course of the proof of Theorem 1.3. Obviously, we can suppose that the same inequality is satisfied for g.

By the property (P ∆ ), f and g extends meromorphically to X. Let then I f (resp. I g ) be the singular locus of the meromorphic map f (resp. g), which is of codimension greater than 2, and observe that (df ) x (resp. (dg) x ) is invertible for all x ∈ X \ I f (resp. X \ I g ): indeed, this follows from the continuity of df on X \ I f combined with the density of X \ ∆ in X \ I f and the previous inequality we have just shown. Accordingly, f has a local inverse everywhere on X \I f . But observe that g is inverse to f on the dense open set f (X \ ∆) ∩ (X \ ∆) so that g must actually extend holomorphically to f (X \ I f ). Therefore, we deduce that f (X \ I f ) ⊂ X \ I g , and by doing the same reasoning for g, we obtain aswell that g(X \ I g ) ⊂ X \ I f . But as we necessarily have g • f = Id X\I f and f • g = Id X\Ig , we actually deduce the following equalities

f (X \ I f ) = X \ I g and g(X \ I g ) = X \ I f .
Recall that for S any set in X, the image of S by the meromorphic map f is defined as

π 2 (G f ∩ (S × X)),
where π 2 is the projection onto the second factor. Since f (X) = X (G f ⊂ X × X is closed and so is its image by π 2 as it is a proper map; the equality follows as this projection contains a dense open set of X), by writing

X = f (X) = f (X \ I f ) ∪ f (I f ) = (X \ I g ) ∪ f (I f ),
we deduce that I g ⊂ f (I f ), and similarly, we obtain that I f ⊂ g(I g ).

The goal is to prove that

I f = I g = ∅.
Arguing by contradiction, we suppose that it is not empty. Let then N be a projective neighborhood of ∆ (which exists by hypothesis), and H an irreducible hypersurface of N such that N \ H is Stein, and H ∩ I f = I f . Observe that f (N ) is an open set, and that f (H) is a subvariety of f (N ). Furthermore, defining

A := I g ∩ f (N ) \ f (H) ,
we prove that A is not the empty set. Indeed, otherwise we have the inclusion I g ⊂ f (H), so that, using the above, we deduce that

I f ⊂ g(I g ) ⊂ g(f (H)).
But g(f (H)) is a closed irreducible analytic set, which contains the irreducible open set H \ I f , so that necessarily g(f (H)) = H, and thus I f ⊂ H, which was excluded by construction.

To conclude, observe that g maps the open set f (N ) \ f (H) \ A into the Stein manifold N \ H, with codim(A) ≥ 2, so that g extends to a holomorphic map through A = ∅ (see e.g.[AS60][Theorem 2, p.316]), which yields a contradiction to the very definition of I g . Therefore, we deduce that I g = ∅.

Similarly, we obtain that I f = ∅. Accordingly, f and g are defined on X, and as

f • g = Id X\Ig = Id X = Id X\I f = g • f,
we deduce that f is indeed in Aut(X). And by Proposition 2.8, we know that f induces aswell an element in Aut(X \ ∆).

We are now in position to prove the announced theorem.

Proof of Theorem 2.7. We equip X with an hermitian metric, with which we define the topology of uniform convergence on C 0 (X), and the topology of uniform convergence on any compact of X \ ∆ on C 0 (X \ ∆). Note that these two topologies are metrizable. As the closed set ∆ is of (2p -3)-Haussdorf measure 0, it is in particular of empty interior. Furthermore, as X is compact and Kähler, a theorem of Siu ([Siu75], Theorem 1) ensures that the property P ∆ is indeed satisfied for X. Now, by Lemma 2.10, there is a continuous injective restriction map

r : Aut(X) → Aut(X \ ∆),
whose image G = r(Aut(X)) is compact. But it is also easily seen that G is a subgroup of Aut(X \ ∆), therefore acting on X \ ∆. By a theorem of Bochner-Montgomery [START_REF] Bochner | Locally compact groups of differntiable transformations[END_REF], G has a structure of a compact real Lie group. But G is at most countable (because Aut(X) is, see Lemma 2.6), so that it must be of dimension zero, and as it is compact, it must be finite. Accordingly, we deduce that Aut(X) is finite, which finishes the proof.

As closed analytic subset of codimension ≥ 2 in a compact complex variety X of dimension p has zero (2p -3)-Haussdorf measure, and as projective varieties are Kähler, we immediately deduce the following corollary Corollary 2.11. Let X be a complex projective manifold of dimension dim(X) = p, which is panalytically hyperbolic, and (p -1)-analytically hyperbolic modulo ∆, where ∆ = ∆ X,ℓ is an analytic subset of X of codimension ≥ 2. Then Aut(X) is finite.

Conditions for intermediate (analytic) hyperbolicity.

2.5.1. Algebraic condition. Starting from the classic fact that a projective manifold X, with Ω X ample, is hyperbolic (and hence has Aut(X) finite), Demailly proved in [START_REF] Demailly | Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials[END_REF] the following generalization:

Theorem 2.12 (Demailly, [START_REF] Demailly | Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials[END_REF]). Let X be a projective manifold of dimension p. Let 1 ≤ ℓ ≤ p, and suppose that ℓ Ω X is ample. Then X is ℓ-analytically hyperbolic.

In [START_REF] Etesse | Ampleness of schur powers of cotangent bundles and p-hyperbolicity[END_REF], the author studied complete intersections in projective spaces satisfying such a positivity property, and proved that a general complete intersection X ⊂ P N of c ≥ N ℓ+1 hypersurfaces of large enough degrees (with an explicit very large bound) has its ℓth exterior power of its cotangent bundle ℓ Ω X ample. 2.5.2. Curvature condition. It is well known that negative curvature properties imply hyperbolic properties, which therefore should imply finiteness results. For example, a compact complex manifold endowed with a smooth hermitian metric of negative holomorphic sectional curvature is known to be hyperbolic (see [START_REF] Kobayashi | Hyperbolic complex spaces, volume 318 of Grundlehren der Mathematischen Wissenschaften[END_REF], and therefore has finite automorphisms group. A natural question is to extend this result to the setting of a singular metric. Regarding this question, a recent work of Guenancia [Gue] generalizing results of Wu-Yau [START_REF] Wu | Negative holomorphic curvature and positive canonical bundle[END_REF] provides the following corollary.

Theorem 2.13 (Guenancia [Gue]). Let X be complex projective manifold endowed with a smooth closed semipositive (1, 1)-form ω such that there exists a Zariski open subset where ω defines a Kähler metric which has uniformly negative holomorphic sectional curvature . Then X is of general type (and therefore has finite automorphisms group).

It is natural to try to extend these kind of results to intermediate hyperbolicity. Graham and Wu have introduced in [GW85] curvature conditions which imply ℓ-analytic hyperbolicity. Namely, Theorem 4.5 in [START_REF] Graham | Some remarks on the intrinsic measures of eisenman[END_REF] show that strongly negative ℓ-th Ricci curvature (see loc. it. for a definition) imply ℓ-analytic hyperbolicity.

An immediate corollary of this statement and Theorem 1.3 is the following.

Corollary 2.14. Let X be complex compact hermitian manifold with strongly negative ℓ-th Ricci curvature for some 1 ≤ ℓ ≤ dim X. Then Aut(X) is finite.

Proof. If ℓ = dim X then X is of general type and we are done. If ℓ < dim X then Theorem 4.5 of [START_REF] Graham | Some remarks on the intrinsic measures of eisenman[END_REF] implies that X is ℓ-analytically hyperbolic, which implies that X is (dim X -1)-analytically hyperbolic by Proposition 2.4. Then Theorem 1.3 implies that Aut(X) is finite.

Remark 2.15. As noted by Graham and Wu ([GW85] p. 638), it is unknown if the existence of a Hermitian metric of strongly negative ℓ-th Ricci curvature implies the existence of one with strongly negative (ℓ + 1)-th Ricci curvature.

Remark 2.16. It would be interesting to generalize this statement to the setting of singular metric such as in Theorem 2.13 above. It does not seem obvious how to use Theorem 1.4 in this context. More precisely, the following question seems interesting to us. Suppose that X is a complex compact hermitian manifold with pseudo-strongly negative ℓ-th Ricci curvature for some 1 ≤ ℓ ≤ dim X. Then is it true that Aut(X) is finite?

Intermediate Picard hyperbolicity

Motivated by the extension property (P ∆ ) introduced during the proof of Theorem 1.4, we generalize the notion of Picard hyperbolicity introduced in [JK20] and [START_REF] Deng | Big picard theorem and algebraic hyperbolicity for varieties admitting a variation of hodge structures[END_REF] to the context of intermediate hyperbolicity.

Definition 3.1. Let X be a compact complex manifold and ∆ a proper closed subset of empty interior. X is said ℓ-Picard hyperbolic modulo ∆ if for every p, q ∈ N, p+q = ℓ, every non-degenerate holomorphic map f : B p × (B * ) q → X whose image is not contained in ∆ extends meromorphically to a map f : B ℓ X.

Remark 3.2. Note that if X is ℓ-Picard hyperbolic, then so is any compact complex manifold X ′ bimeromorphic to X, as we are interested in meromorphic extensions.

If X is moreover supposed to be Kähler, then in order to obtain the pseudo ℓ-Picard hyperbolicity, it is enough to check the conditions only for q = 1, i.e. for non-degenerate holomorphic maps f : B ℓ-1 × B * → X whose image is not contained in the exceptional locus: Lemma 3.3. Let X is a compact Kähler manifold, and let 1 ≤ ℓ ≤ dim(X). Then X is ℓ-Picard hyperbolic modulo ∆ if and only if any non-degenerate holomorphic map f : B ℓ-1 × B * → X whose image is not included in ∆ extends meromorphically to B ℓ . Proof. This is an application of the extension theorem of Siu [START_REF] Siu | Extension of meromorphic maps into kahler manifolds[END_REF][Theorem1], which implies in particular that any meromorphic map (without any further condition) from B ℓ \ E to X extends meromorphically as soon as E is a closed subset of codimension ≥ 2. Indeed, let q ≥ 1, and let f : B p × (B * ) q → X, p + q = ℓ, be a non-degenerate holomorphic map whose image is not included in ∆. Extending f meromorphically is a local question, and observe that around a point z 0 ∈ B p × (B * ) q-j × {0} j , 0 ≤ j ≤ q, up to a renormalization, f can be interpreted as a holomorphic map from B ℓ-j × (B * ) j to X. If j = 1, we can extend f meromorphically by hypothesis. If j > 1, an obvious induction shows that we can extend meromorphically f to B ℓ-j × B j \ (0, . . . , 0). Now, as j > 1, we can apply Siu's extension theorem, so that f actually extends meromorphically to B ℓ , and we are done.

One also sees from [START_REF] Siu | Extension of meromorphic maps into kahler manifolds[END_REF] that if X is 1-Picard hyperbolic then X is ℓ-Picard hyperbolic for all 1 ≤ ℓ ≤ dim X (see [START_REF] Deng | Big picard theorem and algebraic hyperbolicity for varieties admitting a variation of hodge structures[END_REF][Prop. 3.4]). In fact, one has the more general property that if X is pseudo-ℓ-Picard hyperbolic then X is pseudo-(ℓ + 1)-Picard hyperbolic as shown in the next proposition.

Proposition 3.4. Let X be a compact Kähler manifold. If X is ℓ-Picard hyperbolic modulo ∆ then X is ℓ + 1-Picard hyperbolic modulo ∆.

Proof. By Lemma 3.3, in order to prove the (ℓ + 1)-Picard hyperbolicity modulo ∆, it is enough to show that any non-degenerate holomorphic map f : B ℓ × B * → X whose image is not included in ∆ extends to a meromorphic map. Since f is non-degenerate, there exists a dense Zariski open set U on which the differential df is of maximal rank. Denoting π : B × (B ℓ-1 × B * ) → B the projection on the first factor, it is easily seen that for any z in the open set π(U ), the restriction map f (z, .) : B ℓ-1 × B * → X remains non-degenerate. Furthermore, one can find O ⊂ π(U ) open set such that f (z, .) is not included in ∆ for every z ∈ O. In particular, since X is ℓ-Picard hyperbolic modulo ∆, this map extends to a meromorphic map, which à fortiori implies that for any w in the open set p(π -1 (O)), where p is the projection p : B ℓ × B * → B ℓ , the map f (w, .) : B * → X extends to a meromorphic map. Now, by invoking the extension result of Siu [START_REF] Siu | Extension of meromorphic maps into kahler manifolds[END_REF][result (*), p.442], this actually implies that the holomorphic map f : B ℓ × B * → X extends to a meromorphic map, which indeed shows that X is (ℓ + 1)-Picard hyperbolic modulo ∆.

Examples of Picard hyperbolic varieties are provided by projective manifolds whose cotangent bundles satisfy some positivity properties as shown by the following result of Noguchi [START_REF] Noguchi | Meromorphic mappings into a compact complex space[END_REF] which generalizes the classical fact that varieties of general type are dim X-Picard hyperbolic. Theorem 3.5 (Noguchi). Let X be a smooth projective variety over C and let ∆ ⊂ X be a proper Zariski-closed subset. If p Ω 1 X is ample modulo ∆, then X is p-Picard hyperbolic modulo ∆. It is natural to try to give positivity conditions which will still ensure some intermediate Picard hyperbolicity. In this direction, one can state the following corollary of a recent result of [START_REF] Campana | Foliations with positive slopes and birational stability of orbifold cotangent bundles[END_REF].

Theorem 3.6 (Campana, Pȃun). Let X be a smooth projective variety over C such that some tensor power of Ω 1 X is big. Then X is of general type, in particular dim X-Picard hyperbolic. Remark 3.7. A natural question is to provide negative curvature conditions (such as negative holomorphic sectional curvature for ℓ = 1) which will guarantee Picard hyperbolicity. Graham and Wu have introduced in [GW85] curvature conditions which imply ℓ-analytic hyperbolicity. It seems unknown whether such conditions guarantee ℓ-Picard hyperbolicity. In the singular setting, it is shown recently in [DLSZ] that a projective manifold equipped with a singular metric of negative holomorphic sectional curvature (in the sense of currents) is 1-Picard hyperbolic modulo the singular set of the metric.

We now justify why the notion of intermediate Picard hyperbolicity is suitable for dealing with extension problems such as Conjecture 1.2. (See also [START_REF] Deng | Big picard theorem and algebraic hyperbolicity for varieties admitting a variation of hodge structures[END_REF][Proposition 3.4]). Proposition 3.8. Let X be a projective manifold which is ℓ-Picard hyperbolic, 1 ≤ ℓ ≤ dim X. Let Y be a projective manifold of dimension dim(Y ) ≥ ℓ, and let H ⊂ Y be a proper closed subset. Then any non-degenerate holomorphic map f : Y \ H → X extends meromorphically through H.

Proof. As we can always take an hypersurface containing the proper closed subset H, it is enough to prove the result when H is an hypersurface. Having in mind Remark 3.2, one sees that, by using a resolution of singularities (see e.g. [START_REF] Lazarsfeld | Positivity in algebraic geometry. I[END_REF][Theorem 4.3.1]), it is enough to prove the result when H is a simple normal crossing divisor. But in this situation, the local picture around a point h ∈ H is clear: there exists a neighborhood U h ⊂ Y of h such that

U h \ (H ∩ U h ) ≃ B p × (B * ) q
for some p, q such that p + q = dim(Y ) ≥ ℓ. Since X is ℓ-Picard hyperbolic, we know by Proposition 3.4 that it is also dim(Y ) ≥ ℓ-Picard hyperbolic, so that any non-degenerate holomorphic map

U h \ (H ∩ U h ) → X

  where U is a unitary matrix and S is a positive definite hermitian matrix. As | det(A)| = | det(S)|, and |Av| = |U Sv| = |Sv| for any v ∈ C N , it suffices to prove the lemma for S, i.e., we may and do assume that A is a positive-definite hermitian matrix. Let ε ≤ λ 1 ≤ . . . ≤ λ p be the (real) eigen values of A, where the first inequality comes from the hypothesis inf |v|=1 |Av| ≥ ε. Let (v 1 , . . . , v p ) be an orthonormal basis of C p such that A is diagonal with respect to this basis, and let v ∈ C p with |v| = 1. Then, we may write v = p k=1 α k v k , where α k ∈ C and p k=1 |α k | 2 = 1. Therefore, we have that

extends meromorphically through H ∩ U h . As this is true for any h ∈ H, this shows in particular the announced result.

We now end this section by proving the announced Theorem 1.5.

Proof of Theorem 1.5. In view of the scheme of proof of Theorem 1.4, it is enough to check that under the assumption of dim(X)-Picard hyperbolicity, Conjecture 1.2 is satisfied. But this is in particular the content of Proposition 3.8, so that the result follows.

From all of this, it is obviously tempting to make the following conjecture: Conjecture 3.9. A projective manifold X is ℓ-Picard hyperbolic if and only if it is ℓ-analytically hyperbolic.

However, none of the direction seems easy at first. A first step that seems feasible would be to prove that (ℓ -1)-analytic hyperbolicity implies ℓ-Picard hyperbolicity.