Eberhard J Gischler
email: gischler@em.uni-frankfurt.de

Harold Hudson

Marc Humblet

Juan Carlos Braga

Dominik Schmitt

Anja Isaack

Anton Eisenhauer

Gilbert F Camoin

J Harold Hudson

Juan Carlos Braga

Giovanna Della Porta

Holocene and Pleistocene fringing reef growth and the role of accommodation space and exposure to waves and currents (Bora Bora, Society Islands, French Polynesia)

Keywords: Holocene, Pacific, Pleistocene, reef, U-series dating

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Fringing reefs are the most common tropical, shallow-water reef type [START_REF] Smithers | Fringing reefs[END_REF]. In the subsidence model of reef growth (Darwin, 1842), fringing reefs represent the basic type in the genetic sequence to barrier reefs and atolls. In uplift areas such as Barbados and the Huon Peninsula, fringing reefs have been used as important long-term gauges of late Quaternary sea-level change (Mesolella et al., 1969;Chappell et al., 1996). Fringing reefs are usually attached to the shoreline and their geomorphology and history appear to be comparatively simple. Despite this, Kennedy & Woodroffe (2002) have shown that fringing reef growth and internal architecture may be rather complex. In their review, these authors discussed various growth models including vertical accretion, lateral accretion, episodic progradation, offshore accretion with lagoon formation and offshore rubble formation.

Accommodation space appears to be of high significance as it dictates reef architecture.

Changes in relative sea-level and antecedent topography, for example, slope angle and relief, in turn, control accommodation space.

Progradation due to a decrease in accommodation space appears to be a very common feature of fringing reef development. As relative sea-level rise both in the Indo-Pacific (transgressive-regressive) and the western Atlantic (transgressive) exhibits mid to late Holocene deceleration [START_REF] Woodroffe | Coral reefs and sea-level change[END_REF]Camoin and Webster, 2015;and references therein), accommodation space has been decreasing in time and causing lateral reef accretion. Fringing reef development on the Queensland shelf behind the Great Barrier Reef has received significant attention (Kennedy and Woodroffe, 2002;[START_REF] Smithers | Fringing and nearshore coral reefs of the Great Barrier Reef: episodic Holocene development and future prospects[END_REF]and references therein). Prograding types predominate (e.g. Hopley and Barnes, 1985;Johnson and Risk, 1987;Lewis et al., 2012) over those that started out in somewhat deeper water and caught up with sea-level (Chappell et al., 1983). Some of these reefs persisted during terrigenous input (e.g. [START_REF] Ryan | Chronostratigraphy of Bramston Reef reveals a long-term record of fringing reef growth under muddy conditions in the central Great Barrier Reef[END_REF]. [START_REF] Smithers | Fringing and nearshore coral reefs of the Great Barrier Reef: episodic Holocene development and future prospects[END_REF]Hopley et al. (2007) have provided a structural classification of fringing reefs in the Great Barrier Reef province in which progradation plays a crucial role. In the Ryukyu Islands, Japan, examples of fringing reefs have been documented in detail, which exhibit progradation and downstepping due to tectonic uplift (Kan et al., 1997;[START_REF] Webster | Model of fringing reef development in response to progressive sea level fall over the last 7000 years -(Kikai-jima, Ryukyu Islands, Japan)[END_REF]. In the northern Red Sea, [START_REF] Shaked | Evolution of fringing reefs: space and time constraints from the Gulf of Aqaba[END_REF] detected aggradation and subsequent lateral growth in Holocene fringing reefs. In the Indian Ocean, Mauritius fringing reefs first aggraded and then prograded (Montaggioni and Faure, 1997). Fringing reefs in the isolated barrier reef system of Mayotte (Indian Ocean) have apparently also prograded during the Holocene [START_REF] Zinke | -0.75 m Acropora 96.6 Holocene Puhia 2, -4.94 m Acropora ? 99.1 Holocene Puhia 2, -7.5 m Acropora 99.8 Holocene Puhia 2, -8.0 m Acropora 87.2 Pleistocene Puhia 2, -8.5 m Porites 83.9 Pleistocene Puhia 2, -9.5 m Acropora 92.8 Pleistocene Puhia 2, -11.0 m Acropora 21.3 Pleistocene Puhia 3, -6.5 m Acropora 99.3 Holocene Puhia 3, -9.75 m Acropora 97.2 Holocene Puhia 3, -11.5 m Acropora? 99.7 Holocene Puhia 3, -12.75 m Acropora 99.1 Holocene Puhia 3, -13.5 m Acropora 55.9 Holocene Ome 1, -0.1 m Acropora 99.5 Holocene Ome 1, -5.2 m Acropora? 98.8 Holocene Ome 1, -10.25 m Porites 99.7 Holocene Ome 2, -0.1 m Pocillopora 98 Holocene Ome 2, -8.26 m Acropora 99[END_REF]. [START_REF] Braithwaite | Origins and development of Holocene coral reefs: a revisited model based on reef boreholes in the Seychelles, Indian Ocean[END_REF] have stressed the importance of storm redeposition during fringing reef accretion in the Seychelles. Likewise, [START_REF] Blanchon | Anatomy of a fringing reef around Grand Cayman island: storm rubble, not coral framework[END_REF][START_REF] Blanchon | Retrograde accretion of a Caribbean fringing reef controlled by hurricanes and sea-level rise[END_REF] have shown in Grand Cayman and eastern Yucatan fringing reefs (Caribbean Sea) that coral storm rubble has been transported landward over back reef areas and played an important role in reef accretion. In these Caribbean examples, fringing reefs have been retrograding significantly during the Holocene. In post-glacial fringing reef development that experienced high rates of sea-level rise following the last glacial maximum, retrogradation has been common also until modern level was reached (e.g. Cabioch et al., 1998).

The classic study of [START_REF] Walther | Die Korallenriffe der Sinaihalbinsel[END_REF]English translation by Ginsburg et al., 1994) along the north-western coast of the Red Sea has shown that antecedent topography may play a crucial role in fringing reef formation. Bedrock or terrigenous clastics form the basement of many fringing reefs as seen in examples on the Queensland shelf (e.g. Hopley et al., 2007;p. 191-232). In other instances, Pleistocene fringing reefs may act as Holocene fringing reef pedestals (Montaggioni and Faure, 1997;Kennedy and Woodroffe, 2002;and references therein). On the other hand, antecedent topography may also be largely masked by Holocene fringing reefs, as in the examples of Hanauma Bay, Oahu, Hawaii, and Galeta Point, Caribbean, Panama. Both of these studies represent the first time that core traverses were drilled across the reefs thereby allowing two-dimensional reconstructions. In Hanauma Bay, the fringing reef has been prograding during the Holocene (Easton and Olson, 1976).

The fringing reef of Galeta Point exhibits aggradation and slight progradation of the marginal Acropora facies over the deeper water massive coral facies during the Holocene (Macintyre and Glynn, 1976).

Exposure to waves and currents is another potential environmental factor of fringing reef growth (e.g. Geister, 1977). However, there are only few studies in which windward and leeward differences in fringing reefs have been elaborated on. Cabioch et al. (1995) found that reef framework was more common in exposed than in protected sites on New Caledonia. Curiously, accretion rates in the fringing reefs studied were higher in more protected sites as compared to exposed sites. Hongo and Kayanne (2009) showed that a windward, robust Holocene reef framework contrasted with a leeward, detrital reef facies fringing Ishigaki Island in the Ryukyu Islands and found that the accretion rate was higher in the windward fringing reef.

This study presents results of a core study in which windward fringing reef growth on Bora Bora (Society Islands) is detailed and then compared with data from the leeward fringing reef of the same reef system in order to test to which extent variations in accommodation space and exposure to waves and currents exert control on reef development and architecture. Whereas the late Quaternary development of barrier and atoll reefs in French Polynesia has been the subject of detailed studies (e.g. Cabioch et al., 1999a;Camoin et al., 2001Camoin et al., , 2012)), fringing reefs have been largely neglected. The work by Montaggioni (1988) on fringing reefs of Tahiti and Moorea is based on one core each and revealed largely detrital facies and both keep-up and catch-up growth patterns, respectively.

Fringing reef development on the leeward side of Bora Bora is characterized by both This article is protected by copyright. All rights reserved.

coralgal-microbial and detrital facies, based on a core traverse (Gischler et al., 2016). The former study may be characterized as preliminary and the latter included only the leeward island side. In this study, fringing reef growth around Bora Bora will be compared with barrier reef accretion in the same reef system to obtain a comprehensive picture of the development of this isolated barrier reef system.

STUDY AREA

Bora Bora is located in the Society Archipelago (French Polynesia) in the central south Pacific Ocean. This archipelago comprises nine islands and five atolls (Fig. 1). Ages of the volcanic islands (4.3 to 0.3 Ma) increase from south-east to north-west and indicate a plate movement of 11 cm/year over the Society hotspot [START_REF] Blais | Géologie, géochimie et géochronologie de l'île de Bora Bora (Société, Polynésie Française)[END_REF]Guillou et al., 2005).

The volcanic islands of Bora Bora are 3.45 to 3.10 Ma old based on radiometric (K/Ar) dating of basalts [START_REF] Blais | Géologie, géochimie et géochronologie de l'île de Bora Bora (Société, Polynésie Française)[END_REF]. The islands are composed of alkali basalt, rare hawaiites, intrusive gabbros and a volcanic breccia. The Baie de Povai is supposed to outline the former caldera (Fig. 2).

The climate of Bora Bora is characterized by a hot and wet season during the summer (November to April), and a colder and drier period in the winter (May to October) (Gabrié and Salvat, 1985). Trade winds blow from the north-eastern to south-eastern directions (Pirazzoli et al., 1985a). Eleven major cyclones have passed over the Society Islands from 1901to 1968. Major storms Lisa, Reva and Veena passed over during 1982and 1983(Pirazzoli et al., 1985a). In February 2010, category 4 cyclone Oli made landfall on the Society Islands; tropical cyclone Niko passed over French Polynesia in January 2015.

Annual average air temperature in French Polynesia ranges from 27°C in the north to 21°C in the south. Monthly air-temperature extremes in the Society Islands range from 24 to 28°C (Gabrié and Salvat, 1985). Annual precipitation in Bora Bora, measured from 1951to 1961(Guilcher et al., 1969), averages 2000 mm/year. Measurements of sea-surface temperatures in the lagoon ranged from 23.8 to 26.7°C in August 1963; salinity ranged from 36.7 to 36.9‰ during the same time period (Guilcher et al., 1969). The spring tidal range is up to 40 cm (Pirazzoli et al., 1985a).

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The volcanic island of Bora Bora covers ca 30 km 2 . The 32 km long coastline creates extensive bays and long peninsulas (Fig. 2). The highest point, Mount Otemanu, rises 727 m above sea-level. One ephemeral water course drains into Faanui Bay. The shoreline is almost completely lined by fringing reefs. The relief of the lagoon floor is high and there are six basins up to 40 m deep (Guilcher et al., 1969;Gabrié et al., 1994). Lagoonal patch reefs are largely lacking. One interruption in the barrier reef is up to 48 m deep and connects the lagoon with the open ocean. The barrier reef, including the reef crest and the extensive sand apron, is 1 to 2 km wide and covers some 70 km 2 . The reef crest consists largely of coralline algae (Porolithon) and the brown algae Turbinaria (Gabrié et al., 1994). Water depth on sand aprons does not exceed 3.5 m. Coral patch reefs on the sand apron occur in the northeastern and eastern parts of Bora Bora, usually in the lee of narrow, shallow waterways (hoa) through elongated sand cays (motu). On the eastern, northern and north-western sides of Bora Bora, long and continuous motus are developed; they are interrupted by a few very shallow hoa. Lagoonal circulation is sustained by water entering the lagoon on the eastern reef and leaving through the western channel (Gabrié et al., 1994). On the ocean sides of the motus, coral rubble conglomerate, beachrock and elevated fossil reef terraces occur (Pirazzoli et al., 1985b;Pirazzoli and Montaggioni 1988a,b). The terraces are Holocene in age and are evidence of a higher than present Holocene sea-level (Rashid et al., 2014;Hallmann et al., 2018). Seaward of the reef crest, a well-developed spur and groove system can be found (Gabrié et al., 1994) which is characterized by a pavement of crustose coralline algae and few corals including Acropora and Pocillopora.

METHODS

Eight rotary cores were drilled on the windward island side of Bora Bora during May 2017 using a hydraulic drill attached to a tripod with wireline core barrel (Fig. 2). Barrel length was 1.5 m. Geographic coordinates were recorded with GPS. Elevation was measured to mean sea-level (m.s.l.). The fringing reef traverse of three cores was drilled on the ca 80 m wide eastern fringing reef of Bora Bora with inter-core distances of ca 40 m. Fringing reef cores include Puhia 1 (16°28.850'S; 151°43.234'W; drilled 0.3 m below m.s.l.), Puhia 2 (16°28.839'S; 151°43.217'W; drilled 0.6 m below m.s.l.) and Puhia 3 (16°28.82.5'S; 151°43.197'W; drilled 0.6 m below m.s.l.) (Fig. 3). Fringing reef core recovery ranged from 0 to 89% and averaged 22 to 63%; recovery decreased from the shore towards the lagoon (Table 1). In addition, two core traverses of three and two cores, respectively, were drilled on the barrier reef of Motu Tofari and Motu Ome (Fig. 2). Cores include Tofari 1 (16°29.636'S; 151°41.812'W; drilled 1 m above m.s.l.), Tofari 2 (16°29.652'S; 151°41.797'W; drilled 0.45 This article is protected by copyright. All rights reserved. below m.s.l.) and Tofari 3 (16°29.652'S: 151°41.849'W; drilled 0.4 m above m.s.l.), Ome 1 (16°27.084'S; 151°43.685'W; drilled at m.s.l.) and Ome 2 (16°27.091'S; 151°43.698'W; drilled at m.s.l.). Barrier reef core recovery ranges from 21 to 43% in individual cores and averages 27.3% (Table 1). Depths of samples in cores including error ranges were calculated based on recovery in individual core barrels.

In the laboratory, cores were cut with an angle grinder and analysed with regard to sedimentology including diagenesis, systematics and geochronology. Thirty-five thin sections from samples taken from cores (16 fringing reef; 19 barrier reef samples) were qualitatively studied under a Leica DM 2500 M polarizing microscope (Leica, Wetzlar, Germany). Subsamples were powdered and relative amounts of carbonate minerals measured by X-ray diffraction (XRD) using a PANalytical X'Pert Pro diffractometer (Malvern Panalytical, Malvern, UK), following the method Milliman (1974). The same method was used to assess the aragonite content of corals selected for age dating (see below).

Corals in cores were identified using the standard publications of [START_REF] Wallace | Staghorn corals of the world[END_REF], [START_REF] Veron | Corals of the World[END_REF] and the guide of Humblet et al. (2015). The taxonomic nomenclature of scleractinian corals include recent modifications by Huang et al. (2014) in the family Merulinidae. Corals were identified at the lowest taxonomic level possible. Morphogroups were defined when several species in the same genus could not be distinguished (for example, Acropora gr. humilis). Coral colonies described as massive are 5 cm in thickness or more. The width of coral branches was measured and they were categorized as fine (<1.0 cm), medium (1.0 to 1.5 cm), or robust (>1.5 cm).

Coralline algae were identified in thin sections relying on standard descriptions. This study used the generic classification scheme proposed by [START_REF] Rösler | Phylogenetic relationships of corallinaceae (Corallinales, Rhodophyta): taxonomic implications for reef-building corallines[END_REF]Cargnano et al. (2018) for the order Corallinales. The palaeodepths inferred from coralline algal assemblages were derived from the published distributions of living members of the identified taxa in the Pacific Ocean [START_REF] Adey | The crustose coralline algae (Rhodophyta: Corallinaceae) of the Hawaiian Islands[END_REF]Cabioch et al., 1999b;Payri et al., 2000;Dechnik et al., 2017). The thickness of coralline algal crusts was measured. The occurrences of vermetids and the encrusting foraminifer Homotrema rubrum were noted.

Twenty-eight uranium-series measurements of coral ages (13 fringing reef; 15 barrier reef samples) were made following standard procedures. For U/Th dating, corals with no indications of early diagenesis and an aragonite content of >97% were selected. The three most aragonite-rich Pleistocene corals had aragonite contents ranging from 87 to 93% and were also included. Separation of uranium and thorium from the sample matrix was done using Eichrom-UTEVA resin following the methods of Fietzke et al. (2005). Determination of uranium and thorium isotope ratios were done at GEOMAR, Kiel, using a Thermo Fisher Neptune multi-ion-counting inductively coupled plasma mass spectrometer (MC-ICP-MS) (Thermo Fisher Scientific, Waltham, MA, USA) as described by Fietzke et al. (2005). For isotope dilution measurements, a combined 233 U/ 236 U/ 229 Th spike was used. Stock solutions were calibrated for concentration using NIST-SRM 3164 (U) and NIST-SRM 3159 (Th) as combi-spike, calibrated against CRM-145 uranium standard solution (formerly NBL-112A) for uranium isotope composition and against a secular equilibrium standard (HU-1, uranium ore solution) for the precise determination of 230 Th/ 234 U activity ratios. In clean room laboratories at GEOMAR, Kiel, usually whole-procedure blank values of this kind of samples were measured between 0.5 pg and 1.0 pg for thorium and between 10 pg and 20 pg for uranium.

Both values are in the range typical of this method and the laboratory (Fietzke et al., 2005).

Based on the 230 Th/ 232 Th and 234 U/ 238 U ratios, ages were calculated using the uranium and thorium half-lives of Cheng et al. (2000).

RESULTS

Fringing reef

The three cores recovered reefal successions ranging in thickness from 9.25 to 13.1 m in a lagoonward direction (Fig. 4). The reef limestones are underlain by basalt. In cores Puhia 1 and Puhia 2, a Holocene reef superimposes a Pleistocene reef. In core Puhia 3, the entire reef limestone succession is Holocene in age (Fig. 4).

Holocene

The Holocene fringing reef is 3.6 to 13.1 m thick. Thickness significantly increases lagoonward. It consists of coral-rich sections, crusts of coralline algae, coralgal sections, microbialite and unconsolidated sand and rubble (Fig. 4). Coralgal sections are very common in all cores and reach thicknesses of up to 4 m (Fig. 5A to D). Microbialite sections contain both structureless and laminated textures; they are as thick as 20 cm, with an average thickness of 3.5 cm (Fig. 5E). Coral successions get as thick as several decimetres.

Fourteen coral taxa have been identified (Table S1), and two coralgal assemblages may be distinguished, namely Acropora and Pocillopora assemblages (Fig. 4). The Acropora assemblage contains medium to robust-sized branches of acroporids and thick crusts of the algae Porolithon onkodes. The second assemblage is dominated by Pocillopora and also comprises thick crusts of P. onkodes. Crust thickness amounts to several centimetres. At the base of core Puhia 3, rhodoliths with large coral nuclei and thin coralline algal covers mainly composed of Hydrolithon boergesenii are found (Fig. 5F). Recovered coralline algal crusts may get as long as 1.5 m (for example, in core Puhia 2) because they grew vertically, presumably on the sides of coral framework (Fig. 5G). The vertical crusts are dominated by laminar, contorted growths of Mesophyllum, Lithophyllum and Harveylithon, as well as Peyssonneliaceans (Table S2). Some 20 algal taxa have been identified (Table S2). In core Puhia 3, a 6 m thick section at the top of the core consists of unconsolidated sand and rubble. Small amounts of millimetre-sized to centimetre-sized fragments of basalt occur at the middle and basal sections of Puhia 2 and at the base of core Puhia 3. Carbonate phases encountered in XRD of bulk samples include 55.6% aragonite, 44.1% high-magnesium calcite and 0.3% low-magnesium calcite on average (Table 2). The variation is large; however, aragonite and high-magnesium calcite predominate and low-magnesium calcite is rare. Marine cements are not abundant and include aragonite needle (acicular), botryoidal and peloidal high-magnesium calcite cements (Fig. 6A to C). Acicular aragonite crystals reach 150 µm in length, and botryoidal crystals are up to 400 µm long. Peloids in the peloidal cement are 30 to 45 µm and surrounded by clear equant crystals <10 µm in diameter. All Holocene U-series ages obtained are reliable based on aragonite content and 234/238 U init.-values, and range from 4661 to 8697 yr BP (Table 3). Reef accretion rates calculated between dated reef intervals in core range from 0.74 to 5.82 m/kyr with an average value of 3.12 m/kyr (Table 4). Accretion rates appear to generally decrease up core.

Pleistocene

The underlying Pleistocene fringing reef is 3.7 to 5.6 m thick (Fig. 4). In contrast to the Holocene reef, thickness of the Pleistocene reef decreases lagoonward. The Pleistocene reef consists of sections of coral, coralline algal crusts, coralgal sections and microbialite (Fig. 7A to D). At the top of the Pleistocene reef, reddish-brown stains are visible. In core Puhia 2, a 50 cm thick breccia with fragments of coral, coralline algae including rhodoliths and small (coarse sand to fine pebble) basalt grains occur (Fig. 7E). Coral sections get as thick as 0.5 m. Five coral taxa have been identified (Table S1) and there are two coral assemblages. The Pocillopora assemblage with thick P. onkodes crusts, and a Porites assemblage with encrusting to massive Porites colonies and P. onkodes crusts (Fig. 4).

Recovered sections with coralline algal crusts may exceed 1 m in thickness, and like in the Holocene, layering is vertical. These crusts are mainly composed of laminar, contorted thalli of Mesophyllum, Lithophyllum and Harveylithon (Fig. 7D). Twelve algal taxa have been identified (Table S2). Microbialite crusts are usually structureless and a few centimetres thick. In cores Puhia 1 and 2, coarse sand-sized to small pebble-sized fragments of basalt occur throughout the Pleistocene reef (Fig. 7A to C). In general, the Pleistocene core sections are somewhat darker as compared with the Holocene sections and have a greyish hue. Carbonate phases identified in bulk samples using XRD comprise 60.3% aragonite, 6.4% high-magnesium calcite and 33.3% low-magnesium calcite on average (Table 2). In contrast to the Holocene reef section, low-magnesium calcite contents are high and highmagnesium calcite contents are low. Blocky low-magnesium calcite cement with crystal sizes of 80 to 120 µm is encountered both in inter-particle porosity and within coral and coralline algal components (Fig. 6D to F). Aragonite and high-magnesium calcite components are in many cases neomorphosed and dissolved portions replaced by blocky low-magnesium calcite crystals. Based on the comparatively low aragonite contents and the elevated 234/238 U init. -values, the U-series ages from Pleistocene corals are not considered reliable. In core Puhia 1, the dates are moderately reliable and range from 141.67 to 144.02 kyr BP (Table 3). The calculated accretion rate, which has to be treated with caution, would amount to 2.41 m/kyr (Table 4), and is in the same range as the Holocene rates. The dates from core Puhia 2 range from 167.12 to 169.02 kyr BP, but are characterized as being unreliable.

Barrier reef

Unlike the fringing reef cores, none of the windward barrier reef cores reached the Holocene reef base due to drill problems caused by abundant unconsolidated sand (Fig. 8). Core thickness ranges from 6 m to a little more than 20 m. Species of Acropora with mediumsized to robust-sized branches predominate. Eighteen coral taxa have been identified (Table S3). Thick crusts of Porolithon onkodes are common, indicating shallow (<10 m), highenergy depositional environments. At the tops of three seaward cores, vermetids and Homotrema rubrum are found in addition to robust acroporids and thick Porolithon crusts, suggesting a shallower palaeo-water depth of <6 m. Eighteen coralline algal taxa were identified (Table S4). Among coralline algal crusts, the encrusting foraminifer Carpentaria sp. occurs rarely. Other than in the fringing reef cores, microbialite crusts are very rare and thin, This article is protected by copyright. All rights reserved.

and were found only in the middle part of Tofari 1 and at the base of Tofari 2. At the top of core Ome 2, a 3 m thick section with very well-cemented rudstone was recovered, most probably massive beachrock. Marine cements are not abundant, and include aragonite needles, high-magnesium calcite peloidal and microcrystalline cements. Reliable U-series age data in barrier reef cores range from 6849 to 5035 yr BP at core bases and from 5168 to 2620 yr BP at core tops (Table 3; Fig. 8). Barrier reef accretion rates are from 2.9 to 20.4 m/kyr and average 7.02 m/kyr (Table 4). However, most values are in the range of 4 to 5 m/kyr; the high value of >20 m/kyr has to be treated with caution because it comes from a core section with relatively low recovery (Ome 2).

DISCUSSION

Holocene fringing reef development

The Holocene Puhia fringing reef exhibits a zonation with a landward Pocillopora and lagoonward Acropora coralgal assemblage. While both assemblages presumably occurred in shallow water (0 to 10 m), as additionally indicated by the commonly found thick crusts of Porolithon onkodes, vermetids and Homotrema; the acroporid corals are characteristic of a more open setting and the pocilloporids of a somewhat more protected environment [START_REF] Abbey | Variation in deglacial coralgal assemblages and their paleoenvironmental significance: IODP Expedition 310, "Tahiti Sea Level[END_REF]. The curious thick, vertically oriented coralline algal crusts in the Puhia cores are dominated by taxa that typically occur in poorly illuminated settings, which usually occur in deeper reef areas or in shaded habitats, such as crevices or caves. In Puhia, they probably grew on the shaded sides of coral colonies and reef framework. In the leeward fringing reef, the landward core Faanui 1 was also characterized by a Pocillopora assemblage in the upper core section. The lower portion (>7.8 kyr BP) of the core consisted of a deeper water (10 to 20 m) assemblage dominated by laminar Montipora and a basal sandy facies (Gischler et al., 2016). Microbialite crusts usually occur in core sections older than approximately 6 kyr BP. This is in accordance with the findings in the other cores in Bora Bora (Gischler et al., 2016) and in Tahiti, and has been explained by changes in environmental parameters such as light and energy, nutrients and alkalinity [START_REF] Cabioch | Postglacial growth history of a French Polynesian barrier reef tract, Tahiti, central Pacific[END_REF][START_REF] Seard | Microbialite development patterns in the last deglacial reefs from Tahiti (French Polynesia; IODP Expedition #310): implications on reef framework architecture[END_REF]Heindel et al., 2012;Riding et al., 2014). The youngest, progradational section of the windward Puhia fringing reef is detrital and largely composed of unconsolidated sand and rubble. Likewise, the upper prograding section of the leeward Faanui fringing reef consists of unconsolidated material.

The Holocene Puhia fringing reef has largely kept up with sea-level rise, which provided ample accommodation space. Both the new age data from the windward and that from the leeward fringing reefs plot within the upper part of the data obtained from Bora Bora (Fig. 9). Whereas the windward Puhia fringing reef exhibits retrogradation followed by progradation, the leeward Faanui fringing reef apparently solely prograded throughout the Holocene. The reason for this difference is not entirely clear, however, because of the existing Pleistocene reef topography, Holocene reef growth may have been forced to initiate more lagoonward and migrated onto the fossil reef as sea-level was rising. On the leeward island side, the early nearshore reef initiation 8.7 kyr BP in the leeward Faanui fringing reef was probably due to the greater accommodation space. An older portion of the Faanui fringing reef might exist at the base of the most seaward core FAA 2, but unfortunately this section was not recovered (Fig. 10). The fringing reefs in the geomorphologically similar Mayotte barrier reef system are also composed of massive and branched corals and bioclastic sand and gravel, established ca 8 kyr BP, and reach thicknesses of up to 10 m, but it is not entirely clear whether they accreted in keep-up or catch-up modes [START_REF] Zinke | -0.75 m Acropora 96.6 Holocene Puhia 2, -4.94 m Acropora ? 99.1 Holocene Puhia 2, -7.5 m Acropora 99.8 Holocene Puhia 2, -8.0 m Acropora 87.2 Pleistocene Puhia 2, -8.5 m Porites 83.9 Pleistocene Puhia 2, -9.5 m Acropora 92.8 Pleistocene Puhia 2, -11.0 m Acropora 21.3 Pleistocene Puhia 3, -6.5 m Acropora 99.3 Holocene Puhia 3, -9.75 m Acropora 97.2 Holocene Puhia 3, -11.5 m Acropora? 99.7 Holocene Puhia 3, -12.75 m Acropora 99.1 Holocene Puhia 3, -13.5 m Acropora 55.9 Holocene Ome 1, -0.1 m Acropora 99.5 Holocene Ome 1, -5.2 m Acropora? 98.8 Holocene Ome 1, -10.25 m Porites 99.7 Holocene Ome 2, -0.1 m Pocillopora 98 Holocene Ome 2, -8.26 m Acropora 99[END_REF]. Many fringing reefs in other regions such as the Great Barrier Reef province are dominated by progradation (Kennedy and Woodroffe, 2002;[START_REF] Smithers | Fringing and nearshore coral reefs of the Great Barrier Reef: episodic Holocene development and future prospects[END_REF]and references therein). These reefs initiated usually 1.5 to 2.0 kyr earlier and their overall thickness of 5 to 10 m [START_REF] Smithers | Fringing and nearshore coral reefs of the Great Barrier Reef: episodic Holocene development and future prospects[END_REF] 2007) also show initial retrogradation and subsequent progradation, like the Puhia fringing reef in Bora Bora. Reef type A occurs in the narrow, rocky foreshore, is rather rare in the Great Barrier Reef province, and the history has not been detailed by drilling as yet according to these authors. The fringing reefs along the north-western coast of Australia (western Indian Ocean) analysed by [START_REF] Twiggs | Development and demise of a fringing coral reef during Holocene environmental change, eastern Ningaloo Reef, western Australia[END_REF] also exhibit initial aggradation, retrogradation and then progradation during the Holocene. During the transition from retrogradation to progradation, reef growth supposedly declined likely due to enhanced run-off and turbidity. The retrograding to prograding reef architecture may be interpreted as a direct response to the mid to late Holocene decreasing rate of rise of sea-level and the coeval decrease in accommodation space. The vertical accretion rate of the windward fringing reef in Bora Bora amounts to 3.12 m/kyr on average and is somewhat lower than the reported rate of the leeward fringing reef of 5.65 m/kyr (Gischler et al., 2016). This observation is similar to the data of Cabioch et al. (1998) in New Caledonia where accretion rates of the leeward fringing reef exceed those of the windward fringing reef. The lateral accretion of both windward and leeward fringing reefs in Bora Bora is estimated to some 80 m in 6 kyr, i.e. 13.3 m/kyr. Recovery in the windward fringing reef (42.9%) is significantly higher as compared to the leeward fringing reef (18.1%), which is This article is protected by copyright. All rights reserved.

interpreted to be a consequence of both the larger amount of reefal framework facies and the stronger early submarine cementation, which is commonly higher in more exposed reefs.

Both Cabioch et al. (1995) and Hongo and Kayanne (2009) in their windward-leeward comparisons of New Caledonia and Ishigaki Island, Japan, respectively, found framework facies to be more common on windward fringing reefs as compared to leeward fringing reefs where detrital facies were more abundant. Higher exposure to waves and currents at windward positions leads to stronger flushing of pore spaces with seawater that is supersaturated with regard to calcium carbonate and, hence, results in more pervasive cementation in these settings (Marshall and Davies, 1981;[START_REF] Aissaoui | Reef diagenesis: cementation at Mururoa Atoll (French Polynesia)[END_REF]Gischler and Lomando, 1997).

Pleistocene fringing reef development

The geochronology data, the occurrence and the elevation of the underlying Pleistocene fringing reef recovered in cores Puhia 1 and 2 makes formation during Marine Isotope Stage (MIS) 5e very likely. The age data ranging from 141 to 169 kyr BP are not reliable; however, they may be explained by the loss of uranium during light meteoric diagenesis. Using an average age of 125 kyr BP for MIS 5e and a 6 m higher than present sea-level (Hearty et al., 2007;Dutton et al., 2015), the subsidence rate of the island of Bora Bora may be estimated to 0 to 0.08 m/kyr (core Puhia 1) and to 0.03 to 0.11 m/kyr (core Puhia 2). These subsidence ranges compare well with the estimated range of 0.05 to 0.14 m/kyr based on the Bora Bora barrier reef core TEV 1 (Gischler et al., 2016). Based on the value of 0.11 m/kyr, maximum subsidence during Holocene fringing reef accretion since 8.8 kyr BP amounts to 0.97 m. Thus, in contrast to sea-level rise, subsidence has provided only a minor portion of accommodation space for reef accretion.

Like its Holocene counterpart, the Pleistocene fringing reef shows a zonation with a landward Pocillopora assemblage (core Puhia 1) and a lagoonward Porites coralgal assemblage (core Puhia 2). Both of these assemblages contain thick crusts of P. onkodes and vermetids, and are considered to have been deposited in shallow water (0 to 10 m). It is not entirely clear as to why there is a Porites-dominated assemblage seaward of the Pocillopora-dominated assemblage, which is a zonation different from that observed in the overlying Holocene fringing reef. Porites is considered as being indicative of either deeper water or more turbid conditions [START_REF] Abbey | Variation in deglacial coralgal assemblages and their paleoenvironmental significance: IODP Expedition 310, "Tahiti Sea Level[END_REF]. Siliciclastic input was presumably somewhat greater in the more landward setting of core Puhia 1. However,small, This article is protected by copyright. All rights reserved.

basalt fragments occur throughout both Pleistocene core sections of Puhia 1 and 2. This is an interesting difference compared to the Holocene core sections where terrigenous input is largely missing. Temperatures are considered higher during MIS 5e in the tropics in general and in the south Pacific (e.g. Cortese et al., 2013), and it could be speculated that at the same time somewhat wetter conditions during MIS 5e as compared to the Holocene led to more erosion and terrigenous runoff during precipitation events. The complete lack of a Pleistocene fringing reef at the leeward island location might also be a consequence of stronger runoff. The only watercourse on the island drains into Faanui Bay and might have rendered conditions too turbid for reef development during MIS 5e around this location.

Based on the much weaker Holocene cementation on the leeward side, it could also be speculated that a leeward Pleistocene reef has been eroded during subaerial exposure after MIS 5e. Apart from P. onkodes crusts, thick vertically oriented coralline algal crusts are found also in the Pleistocene fringing reef. Again, taxa indicative of shaded environments dominate and suggest growth on the sides of corals and the reef framework. Microbialite crusts occur in the Pleistocene core sections as well, but for unknown reasons they are not as abundant as in the Holocene. This observation suggests that apart from nutrient input and runoff from a basaltic hinterland additional environmental factors must be responsible for abundant microbialite formation. In accordance with the data compiled by Riding et al.

(2014), it could possibly be explained by low alkalinity at the peak of MIS 5e.

Fringing versus barrier reef growth

For Darwin (1842), fringing reefs were the initial reef type in the genetic sequence of fringing, barrier and atoll reefs, controlled by subsidence of volcanic islands. As elegant as this model is, changes in relative sea-level, especially during the Quaternary high-amplitude sea-level variability, as well as other environmental factors were not considered (e.g. Purdy, 1974;Montaggioni, 2005;Purdy and Winterer, 2006;[START_REF] Toomey | Profiles of ocean island coral reefs controlled by sea-level history and carbonate accumulation rates[END_REF]. 'Fringing to barrier' reef transitions as predicted by the subsidence model have not been documented convincingly as yet. [START_REF] Blanchon | Postglacial fringing-reef to barrier-reef conversion on Tahiti links Darwin's reef types[END_REF] suggested for Tahiti a fringing to barrier reef transformation during 14.o to 12.3 kyr BP. These authors proposed that a transition to fastgrowing acroporids boosted barrier-reef accretion when the fringing reef was isolated from the coast and from terrigenous influx. However, the fringing to barrier transition was not recovered and could not be dated exactly. Likewise, there are no age data of the top of the underlying Pleistocene platform. In general, it is largely unknown when and how the modern fringing reefs around the island of Tahiti came into existence and developed, because they have not been investigated systematically by drilling thus far. Previous drilling in Bora Bora This article is protected by copyright. All rights reserved.

has shown that the leeward Holocene barrier reef likely grew on top of a late Pleistocene barrier reef, not on a fringing reef (Gischler et al., 2016). There, the Pleistocene reef was encountered 30 m below modern sea-level, and the adjacent Pleistocene lagoon deposits at -40 m. A comparable Pleistocene barrier reef geomorphology is likely for the windward barrier reef even though its top has not been reached during the present study. The maximum lagoon depths behind the windward Tofari and Ome drill sites reach 31 to 34 m, and adding another 10 m of Holocene lagoonal sediment thickness (Gischler et al., 2016;Isaack et al., 2016) places the Pleistocene lagoon floor some 40 m below modern sea-level, as on the leeward island side. It remains to be seen whether or not the windward barrier reef thickness also amounts to 30 m. This study and preceding drill projects on Bora Bora (Gischler et al., 2016;Isaack et al., 2016) have shown that Holocene fringing and barrier reefs developed contemporaneously in the same reef system. Comparable results were obtained elsewhere, for example, in the Mayotte [START_REF] Zinke | -0.75 m Acropora 96.6 Holocene Puhia 2, -4.94 m Acropora ? 99.1 Holocene Puhia 2, -7.5 m Acropora 99.8 Holocene Puhia 2, -8.0 m Acropora 87.2 Pleistocene Puhia 2, -8.5 m Porites 83.9 Pleistocene Puhia 2, -9.5 m Acropora 92.8 Pleistocene Puhia 2, -11.0 m Acropora 21.3 Pleistocene Puhia 3, -6.5 m Acropora 99.3 Holocene Puhia 3, -9.75 m Acropora 97.2 Holocene Puhia 3, -11.5 m Acropora? 99.7 Holocene Puhia 3, -12.75 m Acropora 99.1 Holocene Puhia 3, -13.5 m Acropora 55.9 Holocene Ome 1, -0.1 m Acropora 99.5 Holocene Ome 1, -5.2 m Acropora? 98.8 Holocene Ome 1, -10.25 m Porites 99.7 Holocene Ome 2, -0.1 m Pocillopora 98 Holocene Ome 2, -8.26 m Acropora 99[END_REF], Chuuk (Truk) and the New Caldonian barrier reef systems (Purdy and Winterer, 2006, and references therein). It is noteworthy in this context that the fringing reef of Bora Bora apparently developed in water depths somewhat shallower as compared to the nearby barrier reef during ca 9 to 5 kyr BP. Both windward and leeward fringing reef age data plot consistently several metres above the barrier reef data of Bora Bora and Tahiti before 5 kyr BP (Fig. 9). However, it remains somewhat enigmatic as to why this palaeobathymetrical difference developed between fringing and barrier reefs during the early to mid Holocene. Both the leeward and the windward barrier reef of Bora Bora is dominated by an Acropora coral assemblage with medium-sized to robust-sized branches and thick crusts of Porolithon onkodes. Both the leeward and the windward barrier reef exhibit progradation (Gischler et al., 2016). A limitation of the palaeobathymetrical interpretation is the fact that the resolution of the palaeo-depth does not go below 10 m, and is similar in the Acropora assemblage, which dominates at the barrier reef, and the Pocillopora assemblages, that is common at the fringing reef. Only in the late Holocene, the abundant occurrences of vermetids and Homotrema together with robust acroporids allow for a somewhat more precise palaeobathymetrical interpretation at the barrier reef locations.

CONCLUSIONS

Holocene windward and leeward fringing reefs, located in the isolated barrier-reef system of Bora Bora, are more or less of the same age but show considerable differences with regard to composition and architecture. The windward fringing reef retrograded and then prograded during the Holocene. The reef exhibits zonation with a landward Pocillopora and a lagoonward Acropora coralgal assemblage. Sand and rubble is common in the prograding phase only. The leeward fringing reef simply prograded during the Holocene. Pocillopora and Montipora coralgal assemblages occur; sand and rubble are very common. Fringing reef consolidation is much stronger on the windward as compared to the leeward side of Bora Bora. These differences are most likely to be the result of differences in available accommodation space and exposure to waves and currents. Windward-leeward differences must have been even greater during the late Pleistocene as reef framework occurs only on the windward island side. Environmental conditions, i.e. elevated input of terrigenous material due to stronger runoff during Marine Isotope Stage 5e, were supposedly deleterious for reef development at the leeward location. For reasons which are not entirely clear, fringing reefs probably grew in somewhat shallower water during the early-mid Holocene as compared to the nearby barrier reef. Comparisons with fringing reefs in other regions, especially the Great Barrier Reef region, where this reef type has been studied intensively, reveals both similarities and differences. While progradation is a common architectural element in many fringing reefs as it is in Bora Bora, initial retrogradation and subsequent progradation has only been observed and studied in a limited number of examples and requires further studies. 2018) are from elevated, late Holocene reef terraces on Bora Bora. 14 C age data from [START_REF] Bard | Deglacial sea-level record from Tahiti corals and the timing of global meltwater discharge[END_REF] have been calibrated (Reimer et al., 2013).

FIGURE CAPTIONS

Bora Bora sea-level curve is from Gischler et al. (2016).

  is significantly lower than the south Pacific examples studied here. Still, in the Great Barrier Reef, fringing reefs of type A of Smithers et al. (2006) and Hopley et al. (

Fig. 1 .

 1 Fig. 1. Map of the Society Archipelago, French Polynesia, including Bora Bora. After Gabriéand Salvat (1985).

Fig. 2 .

 2 Fig. 2. Map of Bora Bora including drill sites of this study (Puhia 1 to 3; Tofari 1 to 3; Ome 1to 2), and other drill locations of previous studies(Gischler et al., 2016).

Fig. 3 .

 3 Fig. 3. Satellite image (from Google Earth) that shows the exact position of the fringing reef cores Puhia 1 to 3 on the eastern island side.

Fig. 4 .

 4 Fig. 4. Core logs across the Puhia fringing reef, eastern side of Bora Bora; including Useries age data and indicating the distribution of coralgal assemblages. Note that coralline algal crusts and microbialite crusts occur throughout the cores, and are therefore not indicated on the logs.

Fig. 5 .

 5 Fig. 5. Selected core photographs from the Holocene sections. (A) Merulinid coral (Astrea curta?) and coralline algal crust; the latter associated with Homotrema and vermetids. Puhia 1, -0.75 m. (B) Branched corals (Pocillopora), coralline algae and microbialite infill. Puhia 1, -2.5 m. (C) Thick coralline algal crust with vermetids; coral (Porites?) at top. Puhia 1, -1.75 m. (D) Acropora humilis, thin coralline algal crust, and boring mollusc. Puhia 2, -2.5 m. (E) Thick coralline algal crust with Homotrema and microbialite. Puhia 2, -4.25 m. (F) Rhodoliths at the base of Holocene section. Puhia 3, -13.5 m. (G) Coralline algal crust with vertical layering. Puhia 2, -5.0 m.

Fig. 8 .

 8 Fig. 8. Core log transects across the windward barrier reef; including U-series age data and indicating the distribution of coralgal assemblages. (A) Motu Tofari core traverse. (B) Motu Ome core traverse. Note that coralline algal crusts occur throughout the cores, and therefore are not indicated. Microbialite crusts are very rare.

Fig. 9 .

 9 Fig. 9. Uranium-series age data from windward Puhia fringing reef plotted together with existing Holocene sea-level data from Bora Bora and Tahiti, Society Islands. Age data from leeward Faanui fringing reef are plotted separately. Age data from Gischler (2011), Rashid et al. (2014) and Hallmann et al. (2018) are from elevated, late Holocene reef terraces on

Fig. 10 .

 10 Fig. 10. Reconstruction of reef development in the Holocene including occurrences of coral assemblages and reef history (isochrones). Upper diagram; Puhia fringing reef on the windward island side originated in somewhat deeper water and then retrograded at first, before prograding later on. Lower diagram; Faanui fringing reef on the leeward island side shows only progradation throughout the Holocene (based on data from Gischler et al., 2016). Numbers at isochrones are ages in kyr BP. Sections are not vertically exaggerated.

TABLE CAPTIONS Table 1 .

 CAPTIONS1 Recovery in the fringing and barrier reef cores.

Table 2 .

 2 Results of X-ray diffraction (XRD) analyses of bulk samples as well as of corals and mollusc shells in cores.

Table 3 .

 3 Uranium-series dates from fringing and barrier reef cores.

Table 4 .

 4 Corrected elevations of samples including calculated accretion rates.

	Core, barrel recovery recovery	Core, barrel recovery recovery
		(m)	(%)		(m)	(%)
	fringing reef			barrier reef
	Puhia 1			Tofari 1
	barrel 1	0.78	52.0	barrel 1	0.60	40.0
	barrel 2	1.33	88.7	barrel 2	0.60	40.0
	barrel 3	0.82	54.7	barrel 3	0.27	18.0
	barrel 4	1.10	73.3	barrel 4	0.33	22.0
	barrel 5	0.78	52.0	barrel 5	0.27	18.0
	barrel 6	0.83	55.3	barrel 6	0.23	15.3
	barrel 7	0.93	62.0	barrel 7	0.11	7.3
	mean	0.94	62.6	mean	0.34	23.0
	Puhia 2			Tofari 2
	barrel 1	0.25	16.7	barrel 1	0.80	53.3
	barrel 2	0.95	63.3	barrel 2	0.92	61.3
	barrel 3	0.95	63.3	barrel 3	0.43	28.7
	barrel 4	1.12	74.7	barrel 4	0.54	36.0
	barrel 5	0.57	38.0	barrel 5	0.33	22.0
	barrel 6	0.98	65.3	barrel 6	0.22	14.7
	barrel 7	0.77	51.3	barrel 7	0.38	25.3
	barrel 8	0.68	45.3	barrel 8	0.22	14.7
	mean	0.78	52.2	barrel 9	0.23	15.3
				barrel 10	0.47	31.3
	Puhia 3			barrel 11	0.14	9.3
	barrel 1	0.00	0.0	mean	0.43	28.4
	barrel 2	0.00	0.0
	barrel 3	0.00	0.0	Tofari 3
	barrel 4	0.00	0.0	barrel 1	0.55	36.7
	barrel 5	0.36	24.0	barrel 2	0.36	24.0
	barrel 6	0.57	38.0	barrel 3	0.42	28.0
	barrel 7	0.50	33.3	barrel 4	0.59	39.3
	barrel 8	0.73	48.7	barrel 5	0.13	8.7
	barrel 9	0.42	28.0	mean	0.41	27.3
	barrel 10	0.65	43.3
	mean	0.32	21.5	Ome 1
				barrel 1	1.42	94.7
	total mean	0.64	42.9	barrel 2	0.65	43.3
				barrel 3	0.68	45.3
				barrel 4	0.70	46.7
				barrel 5	0.26	17.3
				barrel 6	0.41	27.3
				barrel 7	0.41	27.3
				mean	0.65	43.1
				Ome 2
				barrel 1	0.65	43.3
				barrel 2	0.95	63.3
				barrel 3	0.52	34.7
				barrel 4	0.50	33.3
				barrel 5	0.22	14.7
				barrel 6	0.48	32.0
				barrel 7	0.00	0.0
				barrel 8	0.22	14.7
				barrel 9	0.17	11.3
				barrel 10	0.00	0.0
				barrel 11	0.20	13.3
				barrel 12	0.04	2.7
				barrel 13	0.27	18.0
				barrel 14	0.27	18.0
				mean	0.32	21.4
				total mean	0.41	27.3

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dpm dpm/dpm dpm/dpm dpm/dpm dpm/dpm dpm/dpm dpm/dpm dpm/dpm initial dpm/dpm dpm/dpm

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	sample	material	Age	± depth in core recovery corr. coreerrorwater depth depth below SLerror	min. corr. depth below SL accretion
		coral	ky	ky	m	in barrel (m)depth (m) ± m		m	m	± m subsidence min. subsidence	rate
																m	0,05 m/kyr	m/kyr
	HOLOCENE FRINGING REEF
	Puhia 1, -0.10 m Acropora Puhia 1, barrel 1 Acropora 5.116 ### 99.0 5.116 0.027 4.8605 0.0078 1.232 0.002 4.187 -0.1 0.78 -0.55	0.4	0	639.03 -0.3	2.48 -0.85	12127.04 0.4	37.34 -0.26	0.052	0.0001 -1.11	1.148 0.74	0.004	1.15	0.004
	Puhia 1, -2.10 m Acropora Puhia 1, barrel 2 Acropora 7.356 ### 99.6 7.356 0.037 3.0764 0.0037 5.243 0.018 3.7673 0.0005 -2 1.33 -2.09 0.1	134.33 -0.3	0.49 -2.39	1793.1 0.1	6.68 -0.37	0.074	0.0001 -2.76	1.142 5.82	0.004	1.145	0.004
	Puhia 1, -3.40 mPocillopora 99.2 Puhia 1, barrel 3 Pocillopora 7.607 ### 7.607 0.050 2.1802 0.0033 7.46 0.017 2.772 -3.4 0.82 -3.54	0.3	0	69.54 -0.3	0.26 -3.84	893.9 0.3	2.59 -0.38	0.0768	0.0001 -4.22	1.143	0.004	1.146	0.004
	Puhia 1, -3.75 m Porites	97.7 141.673 1.432 2.7259 0.0034 14.873 0.168 36.648 0.001		460.95	5.38	560.42	6.38	0.8121	0.001	1.11	0.004	1.164	0.005
	Puhia 1, -9.20 m Porites Puhia 2, barrel 1 Acropora 4.661 ### 92.2 144.025 1.655 2.3392 0.0032 33.58 0.722 31.854 0.001 -0.75 0.25 -0.81 0.6	177.39 -0.6	3.85 -1.41	212.94 0.6	4.59 -0.23	0.8225	0.0011 -1.64	1.112 1.48	0.004	1.169	0.005
	Puhia 2, barrel 4 Acropora? 7.554 ###	-4.75	1.12		-4.94	0.2		-0.6	-5.54	0.2	-0.38	-5.92	3.67
	Puhia 2, -0.75 m Acropora Puhia 2, barrel 6 Acropora? 8.374 ### 96.6 4.661 0.026 5.4036 0.0082 7.249 0.038 4.322 -7.5 0.98 -7.91	0.3	0	111.57 -0.6	0.68 -8.51	2280.09 0.3	12.67 -0.42	0.0483	0.0001 -8.93	1.162	0.004	1.164	0.004
	Puhia 2, -4.95 mAcropora? 99.1	7.554 0.043 2.8399 0.0058 2.468 0.006 3.5613 0.0005		269.78	0.8	3516.52	12.24	0.0757	0.0002	1.142	0.004	1.145	0.004
	Puhia 2, -7.50 m Acropora Puhia 3, barrel 5 Acropora 6.829 ### 99.8 8.374 0.042 3.0324 0.0051 2.524 0.003 4.208 -6.5 0.36 -6.84	0.6	0	312.52 -0.6	1.05 -7.44	3681.59 0.6	9.03 -0.34	0.0838	0.0001 -7.78	1.145 2.44	0.004	1.148	0.004
	Puhia 2, -8.00 m Acropora Puhia 3, barrel 7 Acropora 8.046 ### 87.2 169.019 1.892 2.7992 0.0039 4.912 0.016 41.646 0.001 -9.25 0.5 -9.75 0.5	1587.53 -0.6	7.14 -10.35	1744.19 0.5	6.54 -0.40	0.8987	0.0013 -10.75	1.128 4.59	0.003	1.207	0.004
	Puhia 2, -9.50 m Acropora Puhia 3, barrel 9 Acropora 8.697 ### 92.8 167.131 2.058 3.4239 0.0047 2.013 0.004 50.8 -12.75 0.42 -12.71	0.003 0.5	4735.24 -0.6	17.96 -13.31	5216.81 0.5	14.9 -0.43	0.8962	0.0012 -13.74	1.131	0.004	1.21	0.005
	Puhia 3, -6.50 m Acropora	99.3	6.829 0.036 3.0164 0.005 3.937 0.013 3.439		0		163.56	0.74	2345.24	9.19	0.0689	0.0001 mean	1.143 3.12	0.004	1.146	0.004
	Puhia 3, -9.75 m Acropora PLEISTOCENE FRINGING REEF 97.2 8.046 0.117 2.8343 0.005 41.259 0.829 3.9019 0.0005 Puhia 3, -12.75 mAcropora 99.1 8.697 0.051 2.9804 0.0019 12.875 0.116 4.327 0 Puhia 1, barrel 3 Porites ##### ### -3.75 0.82 -3.81 0.3 Puhia 1, barrel 7 coral ##### ### -9.2 0.93 -9.47 0.3	17.68 62.87 -0.3 -0.3	0.36 0.6 -4.11 -9.77	209.95 707.88 0.3 0.3	4.24 6.43	0.0832 0.0877	0.0001 0.0001	1.142 1.146 2.41	0.004 0.003	1.146 1.149	0.004 0.003
	Ome 1, -0.10 m Acropora	99.5	2.620 0.01 2.6417 0.0028 1.085 0.001 1.179		0		204.69	0.77	7493.99	20.22	0.027	0	1.147	0.003	1.148	0.003
	Ome 1, -5.20 m Acropora? 98.8 Puhia 2, barrel 6 Acropora ##### ### 4.118 0.03 2.9616 0.0071 0.307 0.005 2.054 0.0005 -8 0.98 -8.14 0.3	1249.02 -0.6	20.95 -8.74	29437.72 497.72 0.3	0.0419	0.0001	1.144	0.004	1.145	0.004
	Ome 1, -10.25 m Porites Puhia 2, barrel 7 Acropora ##### ### 99.7 5.035 0.03 2.5139 0.0044 0.95 0.001 2.131 -9.5 0.77 -9.77	0.4	0	422.27 -0.6	1.56 -10.37	8141.75 0.4	23.96	0.0512	0.0001	1.147	0.004	1.149	0.004
																	mean	2.41
	Ome 2, -0.10 mPocillopora BARRIER REEF CORES	98	3.459 0.01 2.8329 0.003 0.311	0	1.659		0		1019.5	7.31	28449.34	189.7	0.0354	0	1.146	0.003	1.148	0.003
	Ome 2, -8.25 m Acropora Ome 1, barrel 1 Acropora 2.620 ### 99.4 5.434 0.04 2.8249 0.009 0.292 0.005 2.5693 0.0004 -0.1 1.42 -0.23 0.1	1645.18 0	27.39 -0.23	29566.73 0.1	500.3 -0.13	0.0549	0.0002 -0.36	1.143 3.32	0.005	1.145	0.005
	Ome 2, -20.75 mAcropora Ome 1, barrel 4 Acropora? 4.118 ### 99.1 5.993 0.02 4.9707 0.0042 1.004 0.001 4.994 -4.8 0.7 -5.2 Ome 1, barrel 7 Porites 5.035 ### -10.25 0.41 -9.85	0.4 0.6	0	0 0	936.49	3.5 -5.2 -9.85	15235.19 0.4 0.6	39.09 -0.21 -0.25	0.0607	0.0001 -5.41 -10.10	1.147 5.07	0.003	1.15	0.003
	Tofari 1, ±0 m Acropora Tofari 1, -5.30 m Acropora Ome 2, barrel 1 Pocillopora 3.459 ### 97.5 3.671 0.02 2.7444 0.0042 0.233 100 5.357 0.03 2.6807 0.0037 0.073 0.005 2.4017 0.0005 0 1.707 0 -0.1 0.65 -0.68 0.4 Tofari 1, -9.75 m Acropora 94.9 6.849 0.03 3.2537 0.004 1.721 0.002 3.726 0 Ome 2, barrel 6 Acropora 5.434 ### -7.75 0.48 -8.26 0.5 Ome 2, barrel 13-14 Acropora 5.993 ### -20.75 0.27 -19.64 1.4	1412.15 6129.22 0 406.36 0 0	13.71 421.13 111826.98 7684.3 37120.05 349.1 -0.68 0.4 -0.17 1.4 5800.98 13.25 -8.26 0.5 -0.27 -19.64 1.4 -0.30	0.0376 0.0541 0.0692	0.0001 0.0001 -0.85 0.0001 -8.53 -19.94	1.148 1.142 3.84 1.148 20.36	0.003 0.004 0.003	1.15 1.144 1.15	0.003 0.004 0.003
	Tofari 2, ±0 m Acropora Tofari 2, -8.25 m Acropora Tofari 1, barrel 1 Acropora 3.671 ### 99.4 4.693 0.02 2.8058 0.0036 0.259 98.9 6.104 0.04 3.0803 0.0064 0.785 0.005 3.1513 0.0005 0 2.221 0 0 0.6 -0.45 0.5 Tofari 2, -15.40 mAcropora 95.6 5.910 0.03 4.438 0.0061 3.81 0.007 4.407 0 Tofari 1, barrel 4 Acropora 5.357 ### -4.65 0.33 -5.29 0.5 Tofari 1, barrel 7 Acropora 6.849 ### -9.75 0.11 -9.75 0.6	1647.25 750.88 1 216.67 1 1	15.2 5.14 0.55 0.77 -4.29 -8.75	34010.38 301.66 11997.37 84.99 0.5 -0.18 3566.83 8.93 0.5 -0.27 0.6 -0.34	0.0478 0.0618 0.06	0.0001 0.0001 0.37 0.0001 -4.56 -9.09	1.148 1.148 2.87 1.147 2.99	0.003 0.004 0.004	1.15 1.15 1.15	0.003 0.004 0.004
	Tofari 3, ±0m Acropora Tofari 3, -3.75 m Acropora Tofari 2, barrel 1 Acropora 4.693 ### 99.6 5.168 0.02 3.8057 0.0042 0.206 99.2 5.368 0.04 2.8779 0.009 0.123 0.005 2.5916 0.0005 0 3.315 0 0 0.8 -0.43 0.4 Tofari 3, -7.20 m Acropora 97.3 6.332 0.03 2.7989 0.0037 0.664 0 2.966 0 Tofari 2, barrel 6 Acropora 6.104 ### -7.62 0.22 -8.26 0.6 Tofari 2, barrel 11 Acropora 5.910 ### -15.4 0.14 -15.75 0.7	3109.11 3946.27 -0.45 843.46 -0.45 -0.45	34.45 166.27 71631.62 3025.42 58349.65 628.19 -0.88 0.4 -0.23 3.66 13010.03 45.86 -8.71 0.6 -0.31 -16.2 0.7 -0.30	0.0526 0.0544 0.064	0.0001 0.0002 -1.11 0.0001 -9.02 -16.50	1.15 1.146 5.31 1.147	0.003 0.005 0.004	1.152 1.148 1.15	0.003 0.005 0.004
	Tofari 3, barrel 1 Acropora 5.168 ###	0	0.55		-0.48	0.5		0.4	-0.08	0.5	-0.26	-0.34	16.3
	Tofari 3, barrel 3 Acropora 5.368 ###	-3.2	0.42		-3.74	0.5		0.4	-3.34	0.5	-0.27	-3.61	3.12
	Tofari 3, barrel 5 Acropora 6.332 ###	-7.2	0.13		-6.75	0.7		0.4	-6.35	0.7	-0.32	-6.67
																	mean	7.02

ACKNOWLEDGMENTS

We thank the Deutsche Forschungsgemeinschaft (DFG) who has been funding the research in Bora Bora (Gi 222/23-2). Maiarii Teriipaia, Rick Hunter, and Philip Teriipaia assisted us during field-work. Gabriela Meyer also helped during drilling, operated the GPS, and acted as French interpreter. Gérard Bion supported us in various ways. Dr. Marie-Joseph Juen tended to repeatedly occurring minor to moderate injuries of the drill team. The mayor of Bora Bora and the Ministère du Logement, des Affaires Foncières, de l'Économie Numérique et de l'Artisanat (Papeete) kindly issued research permits. Two anonymous journal reviewers and the associate editor made very helpful comments that improved this paper.

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All rights reserved. This article is protected by copyright. All rights reserved. This article is protected by copyright. All rights reserved. This article is protected by copyright. All rights reserved. This article is protected by copyright. All rights reserved. This article is protected by copyright. All rights reserved.

This article is protected by copyright. All rights reserved.

SUPPLEMENTARY TABLE CAPTIONS

Table S1. Corals identified in Puhia cores, including information on coralline algal crust thickness and occurrence of vermetids and Homotrema. BR = branching; COR = corymbose; DIG = digitate; EN = encrusting; F-BR = fine branching; M = massive; M-BR = medium branching; PL = platy; R-BR = robust branching; STAG = staghorn; TAB = tabular; IS = in situ; ISX = not in situ.

Table S2. Occurrence of coralline algae and Halimeda in thin-section along the fringing reef cores.

Table S3. Corals identified in barrier reef cores, including information on coralline algal crust


Sergei Kuksin
email: sergei.kuksin@imj-prg.fr

Asymptotic expansions for some integrals of quotients with degenerated divisors

We study asymptotic expansion as ν → 0 for integrals over R 2d = {(x, y)} of quotients F (x, y) (x • y) 2 + (νΓ(x, y)) 2 , where Γ is strictly positive and F decays at infinity sufficiently fast. Integrals of this kind appear in description of the four-waves interactions.

Introduction s2

Our concern is the integrals

I ν = R d ×R d
dx dy F (x, y) (x • y) 2 + (νΓ(x, y)) 2 , d ≥ 2 , 0 < ν 1.

(1.1) I_s Such integrals and their singular limits ν → 0 appear in physical works on the four-waves interaction, where the latter is suggested as a mechanism, dictating the long-time behaviour of solutions for nonlinear Hamiltonian PDEs with cubic nonlinearities and large values of the space-period. Usually the integrals I ν appear there in an implicit form, and become visible as a result of rigorous mathematical analysis of the objects and constructions, involved in the heuristic physical argument (see below in this section). We denote (x, y) = z ∈ R 2d , ω(z) = x • y, and assume that F and Γ are C 2 -smooth real functions, satisfying 1

|∂ α z F (z)| ≤ K z -M -|α| ∀ z, ∀ |α| ≤ 2 ; (1.2) F_1 |Γ(z)| ≥ K -1 z r * ∀ z , |∂ α z Γ(z)| ≤ K z r * -|α| ∀ z, ∀ |α| ≤ 2 . (1.3) Ga_1
Here r * , M, K are any real constants such that

M + r * > 2d -2, M > 2d -4, K > 1 .
(1.4) hr

As usual we denote z = |z|2 + 1.

The main difficulty in the study of I ν comes from the vicinity of the quadric Σ = {ω(z) = 0}. The latter has a locus at 0 ∈ R 2d and is smooth outside it. Firstly we will study I ν near 0, next -near the smooth part of the quadric, Σ \ {0} =: Σ * , and finally will combine the results obtained to get the main result of this work: t_singint Theorem 1.1. As ν → 0, the integral I ν has the following asymptotic:

I ν = πν -1 Σ * F (z) |z| Γ(z) d Σ * z + I ∆ . (1.5) p22
Here d Σ * z is the volume element on Σ * 2 and |I ∆ | ≤ C χ d (ν), where

χ d (ν) = 1, d ≥ 3 , max(1, ln(ν -1 )), d = 2 .
(1.6) chi_d

The integral in (1.5) converges absolutely, and the constant C depends on d, K, M and r * .

The integral in (1.5) may be regarded as integrating of the function F/Γ against a measure in R 2d , supported by Σ, which we will denote |z| -1 δ Σ * (here δ Σ * is the delta-function of the hypersurface Σ * ). For any real number m let C m (R 2d ) be the space of continuous functions on R 2d with the finite norm |f | m = sup z |f (z)| z m . By (1.2) and (1.3), F/Γ ∈ C M +r * (R 2d ), where M +r * > 2d-2.

prop Proposition 1.2. The measure |z| -1 δ Σ * is an atomlesss σ-finite Borel measure on R 2d . The integrating over it defines a continuous linear functional on the space

C m (R 2d ) if m > 2d -2. Since any function F ∈ C ∞ 0 (R 2d ) satisfies (1.
2) for every M , then we have Corollary 1.3. Let a C 2 -function Γ meets (1.3) with some r * . Then the function ν/ (x • y) 2 + (νΓ(x, y)) 2 converges to the measure |z| -1 δ Σ * as ν → 0, in the space of distributions.

The theorem and the proposition are proved below in Sections 2-4.

In the mentioned above works from the non-linear physics, to describe the long-time behaviour of solutions for nonlinear Hamiltonian PDEs with cubic nonlinearities, physicists derived nonlinear kinetic equations, called the (four-) wave kinetic equations. The k-th component of the kinetic kernel K (k ∈ R d ) for such equation is given by an integral of the following form:

K k = R d R d R d F k (k 1 , k 2 , k 3 )δ kk3 k1k2 δ(ω kk3 k1k2 ) dk 1 dk 2 dk 3 . (1.7) heur
Here δ kk3 k1k2 is the delta-function {k

+ k 3 = k 1 + k 2 } and δ(ω kk3 k1k2 ) is the delta- function {ω k + ω k3 = ω k1 + ω k2 },
where {ω k } is the spectrum of oscillations for the linearised at zero equation. If the corresponding nonlinear PDE is the cubic NLS equation, then ω k = |k| 2 . In this case the two delta-functions define the following algebraic set:

{(k 1 , k 2 , k 3 ) ∈ R 3d : k + k 3 = k 1 + k 2 , |k| 2 + |k 3 | 2 = |k 1 | 2 + |k 2 | 2 } ,
see [START_REF] Nazarenko | Wave Turbulence[END_REF], p.91, and [START_REF] Kuksin | Derivation of a wave kinetic equation from the resonant-averaged stochastic NLS equation[END_REF]. Excluding s 3 using the first relation we write the second as

-2(k 1 -k) • (k 2 -k) = 0. Or -2x • y = 0, if we denote x = k 1 -k, y = k 2 -k. That is, K k is
given by an integral over the set Σ * ⊂ R 2d as in (1.5). In a work in progress (see [START_REF] Kuksin | The Zakharov-Lvov stochastic model for the wave turbulence[END_REF]) we make an attempt to derive rigorously a wave kinetic equation for NLS with added small dissipation and small random force (see [START_REF] Kuksin | Derivation of a wave kinetic equation from the resonant-averaged stochastic NLS equation[END_REF][START_REF] Kuksin | The Zakharov-Lvov stochastic model for the wave turbulence[END_REF] for a discussion of this model). On this way nonlinearities of the form (1.7) appear naturally as limits for ν → 0 of certain integrals of the form (1.1), where, again, x = k 1 -k, y = k 2 -k. 3 We strongly believe that more asymptotical expansions of integrals, similar to (1.1), will appear when more works on rigorous justification of physical methods to treat nonlinear waves will come out.

Proof of Theotem 1.1, given below in Sections 2-4, is rather general and applies to other integrals with singular divisors. Some of these applications are discussed in Section 5.

Notation. By χ A we denote the characteristic function of a set A. For an integral

I = R 2d f (z) dz and a submanifold M ⊂ R 2d , dim M = m ≤ 2d, compact or not (if m = 2d, then M is an open domain in R 2d ) we write I, M = M f (z) d M (z),
where d M (z) is the volume-element on M , induced from R 2d . Acknowledgments. We acknowledge the support from the Centre National de la Recherche Scientifique (France) through the grant PRC CNRS/RFBR 2017-2019 No 1556 "Multi-dimensional semi-classical problems of condensed matter physics and quantum dynamics", and thank Johannes Sjöstrand for explaining the way to estimate singular integrals (5.1), presented in Appendix to this work.

2 Integral over the vicinity of 0. s_71 For 0 < δ ≤ 1 consider the domain

K δ = {|x| ≤ δ, |y| ≤ δ} ⊂ R d × R d ,
and the integral We write Īν (δ) as

K δ |F (x, y)| dx dy (x • y) 2 + (νΓ(x, y)) 2 . ( 2
Īν (δ) = |x|≤δ J x dx , J x = |y|≤δ dy (x • y) 2 + (Cν) 2 .
Let us introduce in the y-space a coordinate system (y 1 , . . . , y d ) with the first basis vector e 1 = x/r, where r = |x|. Since the volume of the layer, lying in the ball {|y| ≤ δ} above an infinitesimal segment [y 1 , y 1 + dy 1 ] is ≤ C d δ d-1 dy 1 and since (x • y) = ry 1 , then

J x ≤ C d r -2 δ 0 dy 1 δ d-1 y 1 2 + (Cν/r) 2 = C d δ d-1 tan -1 (rδ/Cν) Crν ≤ π 2 C d δ d-1 Crν . So Īν (δ) = |x|≤δ J x dx ≤ C d δ d-1 Cν δ 0 r d-2 dr ≤ C d δ 2d-2 ν -1 .
Thus we have proved l_nearsing Lemma 2.1. The integral (2.1) is bounded by Cν -1 δ 2d-2 .

Now we pass to the global study of the integral (1.1) and begin with studying the geometry of the manifold Σ * and its vicinity in R 2d .

3 The manifold Σ * and its vicinity. The set Σ * = Σ \ (0, 0) is a smooth submanifold of R 2d of dimension 2d -1. Let ξ ∈ R 2d-1 be a local coordinate on Σ * with the coordinate mapping ξ → (x ξ , y ξ ) = z ξ ∈ Σ * . Abusing notation we write |ξ| = |(x ξ , y ξ )|. The vector N (ξ) = (y ξ , x ξ ) is a normal to Σ * at ξ of length |ξ|, and

N (ξ) • (x ξ , y ξ ) = 2 x ξ • y ξ = 0 . (3.1) orth For any 0 ≤ R 1 < R 2 we denote S R1 = {z ∈ R 2d : |z| = R 1 } , Σ R1 = Σ ∩ S R1 , S R2 R1 = {z : R 1 < |z| < R 2 } , Σ R2 R1 = Σ ∩ S R2 R1 , (3.
2) nota and for t > 0 denote by D t the dilation operator

D t : R 2d → R 2d , z → tz .
It preserves Σ * , and for any ξ ∈ Σ * we denote by tξ the point D t (x ξ , y ξ ). l_p1 Lemma 3.1. 1) There exists θ * 0 ∈ (0, 1] such that for any 0

< θ 0 ≤ θ * 0 a suitable neighbourhood Σ nbh = Σ nbh (θ 0 ) of Σ * in R 2d \ {0} may be uniquely parametrised as Σ nbh = {π(ξ, θ) : ξ ∈ Σ * , |θ| < θ 0 } , (3.3) par where π(ξ, θ) = (x ξ , y ξ ) + θN ξ = (x ξ , y ξ ) + θ(y ξ , x ξ ).
2) For any vector π = π(ξ, θ) ∈ Σ nbh its length equals

|π| = |ξ| 1 + θ 2 . (3.4) length
The distance from π to Σ equals |ξ||θ|, and the shortest path from

π to Σ is the segment [ξ, π] = {π(ξ, tθ) : 0 ≤ t ≤ 1} =: S. 3) If z = (x, y) ∈ S R is such that dist(z, Σ) ≤ 1 2 Rθ 0 , then z = π(ξ, θ) ∈ Σ nbh , where |θ| < θ 0 and |ξ| ≤ R ≤ |ξ| 1 + θ 2 0 . 4) If π(ξ, θ) ∈ Σ nbh , then ω π(ξ, θ) = |ξ| 2 θ. (3.5) p0 5) If (x, y) ∈ S R ∩ (Σ nbh ) c , then |x • y| ≥ cR 2 for some c = c(θ 0 ) > 0.
The coordinates (3.3) are known as the normal coordinates, and their existence follows easily from the implicit function theorem. The assertion 1) is a bit more precise than the general result since it specifies the size of the neighbourhood Σ nbh .

Proof. 1) Fix any positive κ < 1. Then for θ * 0 small enough it is well known that the points π(ξ, θ) with ξ ∈ Σ 1+κ 1-κ and |θ| < θ 0 ≤ θ * 0 form a neighbourhood of Σ 1+κ 1-κ in R 2d and parametrise it in a unique and smooth way. Besides, any point π ∈ R 2d such that dist (π , Σ 1 ) ≤ 1 2 θ 0 , may be represented as

π = π(ξ , θ ), ξ ∈ Σ 1+κ 1-κ , |θ | < θ 0 , (3.6) p3 and π(ξ 1 , θ 1 ) = π(ξ 2 , θ 2 ), ξ 1 , ξ 2 ∈ Σ 1+κ 1-κ ⇒ |θ 1 |, |θ 2 | ≥ 2θ * 0 . (3.7) p39
We may assume that θ * 0 < 1 2 κ. The mapping D t sends Σ 1+κ 1-κ to Σ t+tκ t-tκ and sends π Σ 1+κ 1-κ × (-θ 0 , θ 0 ) to π Σ t+tκ t-tκ × (-θ 0 , θ 0 ) . This implies that the set Σ nbh , defined as a collection of all points π(ξ, θ) as in (3.3), makes a neighbourhood of Σ * . To prove that the parametrisation is unique assume that it is not. Then there exist

t 1 > t 2 > 0, θ 1 , θ 2 ∈ (-θ 0 , θ 0 ) and ξ 1 ∈ Σ t1 , ξ 2 ∈ Σ t2 such that π(ξ 1 , θ 1 ) = π(ξ 2 , θ 2 ). So π 1 = π 2 , where π 1 = π(t -1 1 ξ 1 , θ 1 ), π 2 = π(t -1 1 ξ 2 , θ 2 ),
and

1 = |t -1 1 ξ 1 | > |t -1 1 ξ 2 |. Let us write |t -1 1 ξ 2 | as 1 -κ , κ > 0. If κ < κ, then (ξ 1 , θ 1 ) = (ξ 2 , θ 2 ) by what was said above. If κ > κ, then |t -1 1 ξ 1 -t -1 1 ξ 2 | ≥ |t -1 1 ξ 1 | -|t -1 1 ξ 2 | ≥ κ. Since |π 1 -t -1 1 ξ 1 | = θ 1 , |π 2 -t -1 2 ξ 2 | = |θ 2 N t -1 1 ξ2 | ≤ θ 2 , then |π 1 -π 2 | ≥ κ -θ 1 -θ 2 ≥ κ -2θ 0 . Decreasing θ * 0 if needed, we achieve that κ > 2θ 0 , so |π 1 -π 2 | > 0. Contradiction.
2) The first assertion holds since by (3.1) the vector N ξ is orthogonal to (x ξ , y ξ ) and since its norm equals |ξ|. The second assertion holds since the segment S is a geodesic from π to Σ * , orthogonal to Σ * . Any other geodesic from π to Σ * must be a segment S = [π, ξ ], ξ ∈ Σ * , orthogonal to Σ * . It is longer than |θ||ξ|. To prove this, by scaling (i.e. by applying a delation operator), we reduce the problem to the case

|ξ| = 1. Now, if ξ ∈ Σ 1+κ 1-κ , then π(ξ , θ ) = π = π(ξ, θ) for some real number θ . So by (3.7), |θ| ≥ 2|θ * 0 |, which is a contradiction. While if ξ ∈ Σ 1+κ 1-κ , then the distance from π to ξ is bigger than κ -θ 0 > θ 0 . Indeed, if |ξ | ≥ 1 + κ, then the distance is bigger than 1 + κ -|π| ≥ 1 + κ -1 -θ 0 = κ -θ 0 . The case |ξ | ≤ 1 -κ is similar.
3) If R = 1, then the assertion follows from (3.6) and (3.4). If R = 1, we apply the operator D R -1 and use the result with R = 1.

4) Follows immediately from (3.1). 5) If R = 1, then the assertion with some c > 0 follows from the compactness of S 1 ∩ (Σ nbh ) c . If R = 1, then again we apply D R -1 and use the result with R = 1.

Let as fix any 0 < θ 0 ≤ θ * 0 , and consider the manifold Σ nbh = Σ nbh (θ 0 ). Below we provide it with some additional structures and during the corresponding constructions decrease θ * 0 , if needed. Consider the set Σ 1 . It equals

Σ 1 = {(x, y) : x • y = 0, x 2 + y 2 = 1} .
Since the differentials of the two relations, defining Σ 1 , are independent on Σ 1 , then this set is a smooth compact submanifold of R 2d of codimension 2. Let us cover it by some finite system of charts N 1 , . . . , N ñ, N j = {η j = (η j 1 , . . . , η j 2d-2 )}. Denote by m(dη) the volume element on Σ 1 , induced from R 2d , and denote the coordinate maps as N j η j → (x η j , y η j ) ∈ Σ 1 . We will write points of Σ 1 both as η and (x η , y η ).

The mapping

Σ 1 × R + → Σ * , ((x η , y η ), t) → D t (x η , y η ),
is 1-1 and is a local diffeomorphism; so this is a global diffeomorphism. Accordingly, we can cover Σ * by the ñ charts N j × R + , with the coordinate maps

(η j , t) → D t (x η j , y η j ), η j ∈ N j , t > 0 ,
and can apply Lemma 3.1, taking (η, t) for the coordinates ξ. In these coordinates the volume element on Σ t is t 2d-2 m(dη). Since ∂/∂t ∈ T η,t Σ * is a vector of unit length, perpendicular to Σ t ,4 then the volume element on Σ * is

d Σ * = t 2d-2 m(dη) dt . (3.8) vol_on_*
The coordinates (η, t, θ) with η ∈ N j , t > 0, |θ| < θ 0 , where 1 ≤ j ≤ ñ, make coordinate systems on the open set Σ nbh . Since the vectors ∂/∂t and t -1 ∂/∂θ form an orthonormal base of the orthogonal complement in R 2d to T (η,t,0) Σ t ,5 then in Σ nbh the volume element dx dy may be written as dx dy = t 2d-1 µ(η, t, θ)m(dη)dt dθ , where µ(η, t, 0) = 1 .

(3.9) p4

For r > 0 the transformation D r multiplies the form in the l.h.s. by r 2d , preserves dη and dθ, and multiplies dt by r. Hence, µ does not depend on t, and we have got

l_p2
Lemma 3.2. The coordinates (η j , t, θ), where η j ∈ N j , t > 0, |θ| < θ 0 , (3.10) chj and 1 ≤ j ≤ ñ, define on Σ nbh coordinate systems, jointly covering Σ nbh . In these coordinates the dilations D r , r > 0, reed as

D r : (η, t, θ) → (η, rt, θ) ,
and the volume element has the form (3.9), where µ does not depend on t.

Besides, since at a point z = (x, y) = π(ξ, θ) ∈ Σ nbh we have (∂/∂θ) = ∇ z • (y, x), then in view of (1.2), (1.3)

∂ k ∂θ k F (η, t, θ) ≤ K (1 + t) -M , ∂ k ∂θ k Γ(η, t, θ) ≤ K (1 + t) r * (3.11) new_est
for 0 ≤ k ≤ 2 and for all (η, t, θ).

For 0 ≤ R 1 < R 2 we denote Σ nbh R2 R1 = π Σ R2 R1 × (-θ 0 , θ 0 ) .
In a chart (3.10) this domain is {(η j , t, θ) :

η j ∈ N j , R 1 < t < R 2 , |θ| < θ 0 }.

Global study of the integral (1.1) s4

To study I ν and the integrals I ν , Σ nbh R2 R1 , 0 ≤ R 1 < R 2 , we pass to the coordinates (η, t, θ) and write the former using (3.9). First we will examine in details the integrals over µ(η, θ) dθ, and next integrate them over t 2d-1 m(dη) dt.

Desintegration of I ν and integrating over µ dθ.

Using (3.9) we write the integral I ν , Σ nbh R2 R1 as where by (3.5)

Σ 1 m(dη) R2 R1 dt t 2d-1 θ0 -θ0 dθ F (η, t, θ)µ(η, θ) (x • y) 2 + (νΓ(η, t, θ)) 2 = Σ 1 m(dη) R2 R1 dt t 2d-1 J ν (η, t) ,
J ν (η, t) = θ0 -θ0 dθ F (η, t, θ)µ(η, θ) t 4 θ 2 + (νΓ(η, t, θ)) 2 .
To study J ν (η, t) =: J ν we write Γ as Γ(η, t, θ) = h η,t (θ)Γ(η, t, 0) .

The function h(θ)

:= h η,t (θ) is C 2 -smooth
, and in view of (3.11) and (1.3) it satisfies

|h(θ)| ≥ C -1 0 , ∂ k ∂θ k h(θ) ≤ C k ∀ η, t, θ, 0 ≤ k ≤ 2 . (4.2) p11 Denoting ε = νt -2 Γ(η, t, 0),
we write J ν as

J ν = t -4 θ0 -θ0 F (η, t, θ)µ(η, θ)h -2 (θ) dθ θ 2 h -2 (θ) + ε 2 .
Since h(0) = 1, then in view of (4.2) the mapping

f = f η,t : [-θ 0 , θ 0 ] θ → θ = θ/h(θ)
is a C 2 -diffeomorphism on its image such that f (0) = 0, f (0) = 1 and the C 2 -norms of f and f -1 are bounded by a constant, independent from η, t (to achieve that, if needed, we decrease θ * 0 ). Denote

θ + 0 = f (θ 0 ), θ - 0 = -f (-θ 0 ), θ0 = min(θ + 0 , θ - 0 ) . Then 2 -1 θ 0 ≤ θ ± 0 ≤ 2θ 0 if θ * 0 is small, and
J ν = t -4 θ + 0 θ - 0 F (η, t, θ)µ(η, θ)h -2 (θ)(f -1 (θ)) d θ θ2 + ε 2 .
Denote the nominator of the integrand as Φ(η, t, θ). This is a C 2 -smooth function, and by (3.11) and (4.2) it satisfies

| ∂ k ∂θ k Φ| ≤ C(1 + t) -M for 0 ≤ k ≤ 2 .
Moreover, since h(0) = 1 and (f -1 (0)) = f (0) = 1, then in view of (3.9) we have that Φ(η, t, 0) = F (η, t, 0) . (4.3) p13

Consider the interval ∆ η,t = f -1 η,t (-θ0 , θ0 ). Then (-θ 0 /2, θ 0 /2) ⊂ ∆ η,t ⊂ (-θ 0 , θ 0 ) for all η and t. Now we modify the neighbourhood Σ nbh (θ 0 ) to

Σ nbhm = Σ nbhm (θ 0 ) = {π(η, t, θ) : η ∈ Σ 1 , t > 0, θ ∈ ∆ η,t } . Then Σ nbh ( 1 2 θ 0 ) ⊂ Σ nbhm (θ 0 ) ⊂ Σ nbh (θ 0 ) . (4.4) Smod
The modified analogy J m ν of the integral J ν has the same form as J ν , but the domain of integrating becomes not (-θ 0 , θ 0 ), but ∆ η,t . Then

J m ν = t -4 θ0 -θ0 Φ(η, t, θ) d θ θ2 + ε 2 .
To estimate J m ν , consider first the integral J 0m ν , obtained from J m ν by frozening Φ at θ = 0:

J 0m ν = t -4 θ0 -θ0 Φ(η, t, 0) d θ θ2 + ε 2 = 2t -4 F (η, t, 0)ε -1 tan -1 θ0 ε (we use (4.3)).
From here

|J 0m ν | ≤ πε -1 t -4 |F (η, t, 0)| . (4.5) p_triv As 0 < π 2 -tan -1 1 ε < ε for 0 < ε ≤ 1 2 , then also 0 < πν -1 t -2 (F/Γ) | θ=0 -J 0m ν < 2 θ0 t -4 F (η, t, 0) , (4.6) p17 if νt -2 Γ(η, t, 0) ≤ 1 2 θ0 . (4.7) p18
Now we estimate the difference between J m ν and J 0m ν . We have:

J m ν -J 0m ν = t -4 θ0 -θ0 Φ(η, t, θ) -Φ(η, t, 0) θ2 + ε 2 d θ . Since each C k -norm of Φ , k ≤ 2, is bounded by C(1 + t) -M , then Φ(η, t, θ) -Φ(η, t, 0) = A(η, t) θ + B(η, t, θ) θ2 , where |A|, |B| ≤ C(1 + t) -M . From here |J m ν -J 0m ν | ≤ C 1 (1 + t) -M t -4 θ0 0 θ2 d θ θ2 + ε 2 ≤ C 1 (1 + t) -M t -4 θ0 . Denote J ν (η, t) = πt -2 F Γ -1 (η, t, 0) . (4.8) p20
Then, jointly with (4.6), the last estimate tell us that

|J m ν -ν -1 J ν (η, t)| ≤ C(1 + t) -M t -4 θ-1 0 if (4.7) p19
holds. (4.9) p19

Proof of Theorem 1.1 s_76

To estimate I ν we write it as

I ν = I ν , (Σ nbhm ) C -1 √ ν 0 + I ν , (Σ nbhm ) ∞ C -1 √ ν .
We will show that the first term is small by Lemma 2.1, and that the second may be approximately calculated using the representation (4.1) and the estimate (4.9). Doing that we have to distinguish the cases r * ≤ 2 and r * > 2.

Let r * ≤ 2. Then by (1.3) assumption (4.7) holds if t ≥ C -1 √ ν (where C depends on θ0 ). Integrating J m ν (η, t) and ν -1 J ν (η, t) with respect to the measure t 2d-1 m(dη) dt and using (4.9), (4.1) and (1.4) we get that

| I ν , (Σ nbhm ) ∞ C -1 √ ν -ν -1 Σ 1 m(dη) ∞ C -1 √ ν dt t 2d-1 J ν (η, t)| ≤ C Σ 1 m(dη) ∞ C -1 √ ν dt t 2d-1 t -4 (1 + t) -M ≤ Cχ d (ν) (4.10) p41
(for the quantity χ d (ν) see (1.6)). In view of (4.8) and Lemma 2.

1 with δ = 2C -1 √ ν, | I ν , (Σ nbhm ) C -1 √ ν 0 -ν -1 Σ 1 m(dη) C -1 √ ν 0 dt t 2d-1 J ν (η, t)| ≤ Cν -1 ν d-1 + Cν -1 C -1 √ ν 0 t 2d-1 t -2 dt ≤ C 1 (4.11) p42
as d ≥ 2. Next, by (4.8) and (1.4)

Σ 1 m(dη) ∞ 0 dt t 2d-1 |J ν (η, t)| ≤ C ∞ 0 t 2d-1-2 (1 + t) -M -r * ≤ C 1 , (4.12) p43
and by (3.8)

Σ 1 m(dη) ∞ 0 dt t 2d-1 J ν =π Σ 1 m(dη) ∞ 0 dt t 2d-3 (F/Γ) | θ=0 =π Σ * |z| -1 (F/Γ)(z)d Σ * z . (4.13) p44
This gives us asymptotic description as ν → 0 of the integral (1.1), calculated over the vicinity Σ nbhm of Σ * . It remains to estimate the integral over the complement to Σ nbhm . But this is easy: by (4.4),

| I ν , R 2d \ Σ nbhm | ≤ | I ν , {|(x, y)| ≤ 2ν} | + C d ∞ ν dr r 2d-1 S r \Σ nbh (θ0/2) F (x, y) d S r (x • y) 2 + (νΓ((x, y)) 2 .
By item 5) of Lemma 3.1 the divisor of the integrand is ≥ C -2 r 4 . Due to this and (1.2), the second term in the r.h.s. is bounded by

C ∞ ν (1 + r) -M r 2d-5 dr ≤ C 1 χ d (ν) .
This estimate and Lemma 2.1 with δ = 2ν imply that

| I ν , R 2d \ Σ nbhm | ≤ Cχ d (ν) . (4.14) p45
Now relations (4.10), (4.11), (4.13), (4.14) imply (1.5), while (3.8) and (4.12) imply that the integral in (1.5) converges absolutely. Let r * > 2. Then condition (4.6) holds if

C β ν -β ≥ t ≥ C -1 √ ν , β = 1 r * -2 .
Accordingly, the term in the l.h.s. of (4.10) should be split in two. The first corresponds to the integrating from C -1 √ ν to C β ν -β and estimates exactly as before. The second is

| I ν , (Σ nbhm ) ∞ C β ν -β -ν -1 Σ 1 m(dη) ∞ C β ν -β dt t 2d-1 J ν (η, t)| . ( 4

.15) p02

To bound it we estimate the norm of the difference of the two integrals via the sum of their norms. In view of (4.5) and (4.8) both of them are bounded by

Cν -1 Σ1 m(dη) ∞ C β ν -β dt t 2d-3 (F/Γ)(η, t, 0) . So (4.15) ≤ C β ν -1+β(M +r * +2-2d) ≤ C β since M > 2d -4.
Adding this relation to (4.10), applied to the integrating from C -1 √ ν to C β ν -β , and -as before -using this jointly with (4.11), (4.13), (4.14), we again get (1.5) (while the absolute convergence of the integral still follows from (4.12)).

Proof of Proposition 1.2

For any R > r > 0 let us denote by B R and B R r the ball {|z| ≤ R} and the spherical layer {r ≤ |z| ≤ R}, and consider the measure µ R r = χ B R r (z)|z| -1 δ Σ * . This is a well defined Borel measure on the manifold Σ * and on the space R 2d , supported by B R . By the Riesz theorem the family of measures {µ R ε , 0 < ε ≤ 1} is weakly compact in the space of measures on B R , and it is easy to see that for any bounded continuos function f (z) the curve ε → f (z) dµ R ε (dz) satisfies the Cauchy condition as ε → 0. So the measures µ R ε weakly converge to a limit as ε → 0. This is the restriction of the measure |z| -1 δ Σ * to B R , and its further restriction to B R r equals µ R r . So |z| -1 δ Σ * is a σ-additive Borel measure on R 2d , and it has no atoms outside the origin. Let us abbreviate |z| -1 δ Σ * = µ. By (3.8), for any 0 < r ≤ R and ε ≤ r,

B R r d µ = B R r d µ R ε = R r t 2d-2 t -1 dt = 1 2d -2 (R 2d-2 -r 2d-2 ).
From here and the weak convergence of the measures µ R ε to µ | B R we get that µ{|z| < ρ} ≤ ρ 2d-2 /(2d -2) if ρ < R, so µ has no atom in the origin and is atomless. Next, for any function f ∈ C m (R 2d ) we have

f (z) µ(dz) ≤ |f | m ∞ R=0 B R+1 R z -m µ(dz) ≤ C 1 |f | m ∞ R=0 R -m (R + 1) 2d-2 -R 2d-2 2d -2 ≤ C 2 |f | m ∞ R=0 (R + 1) 2d-1 R -m = C 3 |f | m , if m > 2d -2.
This proves the proposition.

Other integrals s_other_integrals

The geometrical approach to treat integrals (1.1), developed above, applies to various modifications of these integrals. Below we briefly discuss three more examples.

Integrals (1.1) with d = 1 s_d1

The restriction d ≥ 2 was imposed in the previous sections since in the onedimensional case some integrals, involved in the construction, strongly diverge at the locus of the quadric Σ. This problem disappears if the function F vanishes near the locus. Indeed, consider

I ν = R 2
F (x, y) x 2 y 2 + (νΓ(x, y)) 2 dxdy , where F ∈ C 2 0 (R 2 ) vanishes near the origin. The quadric Σ = {xy = 0} ⊂ R 2 is one dimensional, has a singularity at the origin, and its smooth part Σ * = Σ \0 has four connected components. Consider one of them: C 1 = {(x, y) : y = 0, x > 0}. Now the coordinate ξ is a point in R + with (x ξ , y ξ ) = (ξ, 0) and with the normal N (ξ) = (0, ξ), the set Σ 1 ∩ C 1 is the single point (1, 0) and the coordinate (η, t, θ) in the vicinity of C 1 degenerates to (t, θ), t > 0, |θ| < θ 0 , with the coordinate-map (t, θ) → (t, tθ). The relations (3.8) and (3.9) are now obvious, and the integral (2.1) vanishes if δ > 0 is sufficiently small. Interpreting z = (x, y) as a complex number, we write the assertion of Theorem 1.1 as

J ν -πν -1 Σ F (z) |z| Γ(z) dz ≤ C ,
where the integral is a contour integral in the complex plane.

Integrals of quotients with divisors, linear in ω.

Consider

I ν = R 2d F (x, y) dxdy x • y + iνΓ(x, y) . ( 5

.1) another_integral

Now there is no need to separate the integral over the vicinity of the origin, and we just split I ν to an integral over Σ nbhm and over its complement.

To calculate I ν , Σ nbhm we observe that an analogy of

J 0m ν is the integral θ0 -θ0 F (η, t, 0) d θ t 2 θ + iνΓ(η, t, 0) = t -2 F (η, t, 0) ln θ + 0 + iνt -2 Γ(η, t, 0) -θ0 + iνt -2 Γ(η, t, 0) , which equals πt -2 F (η, t, 0) + O(ν). So I ν , Σ nbhm = π Σ * F (z) |z| d Σ * z + O(ν) . The integral over the complement to Σ nbhm is R 2d \Σ nbhm F (x, y) dxdy x • y + iΓ(x, y) = R 2d \Σ nbhm F (x, y) dxdy x • y + o(1)
as ν → 0 (the integral in the r.h.s. is regular). In difference with (1.1) the last integral is of the same order as the integral over Σ nbhm . So we have that

I ν = π Σ * F (z) |z| d Σ * z + R 2d \Σ nbh F (x, y) dxdy x • y + o(1) ,
in agreement with the estimate (6.4), applied to (5.1).

Integrals, coming from the three-waves interaction

The three-waves interacting systems lead to integrals, similar to (1.7), where R 3d is replaced by R 2d and the δ-factor is replaced by δ k k1k2 δ(ω k k1k2 ), which gives rise to the algebraic set

{(k 1 , k 2 ) : k 1 + k 2 = k, |k 1 | 2 + |k 2 | 2 = |k| 2 } (see [5], Section 6). I.e., k 2 = k -k 1 , where k 1 ∈ {r ∈ R d : |r -1 2 k| 2 = 1 4 |k| 2 }.
Accordingly, in the variable z = r -1 2 k some constructions from the study of the three-waves interaction lead to the integrals

I ν = R d dz F (z) ω(z) 2 + (νΓ(z)) 2 , d ≥ 2,
with ω(z) = |z| 2 -1 4 |k| 2 . Now the quadric Σ = {ω = 0} is a sphere, i.e. a smooth compact manifold. Denoting by η a local coordinate on Σ with a coordinate mapping η → z(η) ∈ Σ and the volume form m(dη) we see that, similar to Section 3, the local coordinate in the vicinity Σ nbh of Σ is (η, θ), |θ| < θ 0 , with the coordinate mapping (η, θ) → z(η)(1 + θ) and the volume form Assume that g(x) has a unique critical point x 0 , which is non-degenerate. Then, by the stationary phase method, I(λ) = 2π λ n/2 | det g xx (x 0 )| -1/2 ϕ(x 0 )e iλg(x0)+(iπ/4) sgn gxx(x0) + Rλ -n/2-1 (6.1) stph_thm for λ ≥ 1, where R depends on ϕ C 2 , g C 3 , the measure of the support of ϕ and on sup x∈supp ϕ (|x -x 0 |/|∇g(x)|) =: C # (g). See Section 7.7 of [START_REF] Hörmander | The Analysis of Linear Partial Differential Equations[END_REF] and Section 5 of [START_REF] Dimassi | Spectral Asymptotics in the Semi-Classical Limit[END_REF].

If the functions ϕ and g are not C ∞ -smooth, but ϕ ∈ C 2 0 (R n ) and g ∈ C 3 (R n ), then, approximating ϕ and g by smooth functions and applying the result above we get from (6.1) that

|I(λ)| ≤ C λ -n/2 ∀ λ ≥ 1 , (6.2)
I(ν) = R d f (x) Γ + iν -1 g(x) dx = ν R d f (x) νΓ + ig(x) dx , 0 < ν ≤ 1 ,
where Γ is a positive constant. Let us write it as

I(ν) = R d 0 -∞ f (
x)e t(Γ+iν -1 g(x)) dt dx =:

R d 0 -∞ F ν (t, x) dt dx = I 1 + I 2 ,
where

I 1 = R d 0 -ν F ν (t, x) dt dx , I 2 = R d -ν -∞
F ν (t, x) dt dx .

Clearly, |I 1 | ≤ C 2 ν. To estimate I 2 consider the internal integral

J(t) = e tΓ R d
f (x)e -iν -1 |t|g(x) dx , and apply to it the stationary phase method with λ = ν -1 |t| ≥ 1. By (6.2) and (6. The constant K(f, g) depends on C, f C 2 , g C 3 and C # (g).

  s_3

( 4 . 1 )

 41 3.9

  (

1 2 1 ΣF

 11 |k|) -1 µ(η, θ)m(dη)dθ, µ(η, 0) ≡ 0. The proof in Sections 2-4 simplifies and leads to the asymptoticI ν -πν -(z)/Γ(z) dz ≤ Const,valid for C 2 -functions F and Γ, satisfying some mild restriction.

6

  AppendixLet ϕ(x) and g(x) be smooth functions on R n and ϕ has a compact support. Consider the integralI(λ) = R n ϕ(x)e iλg(x) dx, λ ≥ 1.

  stph_est with C depending on ϕ C 2 , g C 3 and C # (g). Now let f (x) ∈ C 2 0 (R d ) and g(x) ∈ C 3 (R d ) be such that |f | ≤ C, meas( suppf ) ≤ C.Let an x 0 be the unique critical point of g(x) andC -1 ≤ | det Hess g(x 0 )| ≤ C. (6.3) hessConsider the integral

  3), |J(t)| is bounded by e tΓ K 1 (f, g)(ν -1 |t|) -d/2 . So |I 2 | = ν -∞ J(t) dt ≤ K 1 (f, g)ν d/2 ν -∞ |t| -d/2 e tΓ dt ≤ K 2 (f, g)ν d/2 ν -d/2+1 χ d (ν) since Γ > 0, where χ d (ν) is defined in (1.6). Thus |I(ν)| ≤ |I 1 | + |I 2 | ≤ K(f, g) ν χ d (ν) . (6.4) Aa3

corresponding to the Riemann structure on Σ * , obtained by the restricting to Σ * the standard Riemann stricture on R 2d .

So the integrand F k depends on the parameter k ∈ R d . This dependence should be controlled, which can be done with some extra efforts.

as ∂/∂t ⊥ S t and S t ⊃ Σ t .

Since the vector ∂/∂t is perpendicular to Σ t and lies in T (η,t,0) Σ * , and ∂/∂θ is proportional to the vector N (η,t,0) , normal to Σ * at (η, t, 0).