Henri Guenancia

QUASI-PROJECTIVE MANIFOLDS WITH NEGATIVE HOLOMORPHIC SECTIONAL CURVATURE by

Let (M, ω) be a compact Kähler manifold with negative holomorphic sectional curvature. It was proved by Wu-Yau and Tosatti-Yang that M is necessarily projective and has ample canonical bundle. In this paper, we show that any irreducible subvariety of M is of general type, thus confirming in this particular case a celebrated conjecture of Lang. Moreover, we can extend the theorem to the quasi-negative curvature case building on earlier results of Diverio-Trapani. Finally, we investigate the more general setting of a quasi-projective manifold X • endowed with a Kähler metric with negative holomorphic sectional curvature and we prove that such a manifold X • is necessarily of log general type.

1.2. The general quasi-projective case. -Another way to think of the situation of Theorem A is to view Y reg as a quasi-projective manifold endowed with a Kähler metric ω such that 1. ω has negative holomorphic sectional curvature; 2. ω extends smoothly to a (singular) compactification.

1. Introduction 1.1. Singular subvarieties. -Let M be a compact Kähler manifold of dimension n and let ω be a Kähler metric on M such that its holomorphic sectional curvature is negative; that is, for every x ∈ M and any [v] ∈ P(T M,x ), one has HSC ω (x, [v]) < 0.

Recall that if (R i jk ¯ ) is the curvature tensor of ω in some holomorphic coordinates (z i ) and if v = v i ∂ ∂zi is a non-zero tangent vector at x, then the holomorphic sectional curvature of (M, ω) at (x, [v]) is defined by

HSC ω (x, [v]) := 1 |v| 4 ω • i,j,k, R i jk ¯ v i vj v k v .
Under the assumptions on (M, ω) above, the Ahlfors-Schwarz lemma shows that M is Brody hyperbolic; that is, every holomorphic map f : C → M is constant. Hyperbolicity for projective (or merely compact Kähler) manifolds is conjectured to be related to algebraic properties. More precisely, S. Lang formulated the following Conjecture [START_REF] Lang | Hyperbolic and Diophantine analysis[END_REF]Conj. 5.6]. -A projective manifold X is hyperbolic if and only if each of its subvarieties (including X itself ) is of general type.

2000 Mathematics Subject Classification. -32Q05, 32Q20, 32Q45.

The author is partially supported by the NSF Grant DMS-1510214 and the project PEPS "Jeune chercheuse, jeune chercheur" funded by the CNRS.

Recall that an irreducible projective variety Y is said to be of general type if the canonical bundle K Y of any smooth birational model Y of Y is big; that is, Y has maximal Kodaira dimension. Thirty years after its formulation, Lang's conjecture remains mostly open. Besides the trivial case of curves, the known cases of the conjecture are:

• Surfaces with some specific geometry [START_REF] Deschamps | Courbes de genre géométrique borné sur une surface de type général [d'après F. A. Bogomolov[END_REF][START_REF] Green | Two applications of algebraic geometry to entire holomorphic mappings[END_REF][START_REF] Mori | The uniruledness of the moduli space of curves of genus 11[END_REF][START_REF] Mcquillan | Diophantine approximations and foliations[END_REF].

• Generic hypersurfaces of high degree in P n . By the work of Clemens [START_REF] Clemens | Curves on generic hypersurfaces[END_REF], Ein [START_REF] Ein | Subvarieties of generic complete intersections[END_REF][START_REF]Subvarieties of generic complete intersections. II[END_REF] and Voisin [START_REF] Voisin | On a conjecture of Clemens on rational curves on hypersurfaces[END_REF] later improved by Pacienza [START_REF] Pacienza | Subvarieties of general type on a general projective hypersurface[END_REF], their subvarieties are of general type. Moreover, they are hyperbolic thanks to the recent breakthroughs by Siu [START_REF] Siu | Hyperbolicity of generic high-degree hypersurfaces in complex projective space[END_REF] and Brotbek [START_REF] Brotbek | On the hyperbolicity of general hypersurfaces[END_REF] independently; cf also Demailly [START_REF] Demailly | Recent results on the Kobayashi and Green-Griffiths-Lang conjectures[END_REF].

• Quotients of bounded domains (Boucksom and Diverio [START_REF] Boucksom | A note on lang's conjecture for quotients of bounded domains[END_REF]).

Let us go back to the case of a compact Kähler manifold (M, ω) with negative holomorphic sectional curvature. It was proved by Wu and Yau [START_REF] Wu | Negative holomorphic curvature and positive canonical bundle[END_REF] that K M is ample provided that M is a projective manifold. Shortly after, Tosatti and Yang [START_REF] Tosatti | An extension of a theorem of Wu-Yau[END_REF] extended the result to the general Kähler case. In particular, under those general assumptions, M is automatically projective. Now, if Y ⊂ M is a smooth subvariety of M , then the decreasing property of the holomorphic (bi)sectional curvature shows that K Y is ample again. However, in view of Lang's conjecture, it is crucial to control the geometry of singular subvarieties of M as well. That is the precisely the object of the first main result of this paper given below.

Theorem A. -Let (M, ω) be a compact Kähler manifold with negative holomorphic sectional curvature and let Y ⊆ M be a possibly singular, irreducible subvariety of M . Then, Y is of general type.

It follows from Theorem A that Lang's conjecture holds for compact manifolds M admitting a Kähler metric with negative holomorphic sectional curvature.

About the proof. The main original idea is to construct on a desingularization Y of Y a family of singular Kähler-Einstein metrics (ω b ) b>0 having generically cone singularities along a given ample divisor B and whose cone angle 2π(1 -b) is meant to tend to 2π. These metrics are relatively well understood only on the log canonical model of ( Y , bB) and the heart of the proof consists in working on these varying birational models and to show that the volume of ω b does not go to 0 when b approaches 0. The general idea of using a continuity method and Royden's Laplacian estimate originates from [START_REF] Wu | Negative holomorphic curvature and positive canonical bundle[END_REF], but the degree of technicality in the singular setting is significantly higher. For instance, the Ricci curvature blows down to -∞, thus prohibiting the use of a maximum principle. Also, as the computations are performed on spaces which depend on the parameter b, establishing the volume estimate requires a delicate analysis.

The quasi-negative curvature case. Theorem A generalizes to the case of quasi-negative holomorphic sectional curvature, where one needs to use as an important first step a result of Diverio-Trapani [START_REF] Diverio | Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle[END_REF]. We refer to § 3 and Theorem 3.1 for a statement and a proof.

Log terminal subvarieties. In the setting of the Theorem A, one can additionally show that if Y has log terminal singularities, then K Y is an ample Q-line bundle, cf Remark 2.3.

Given this point of view, it is natural to ask to which extent Theorem A generalizes to arbitrary quasi-projective manifolds. More precisely, given a projective manifold X, a reduced divisor D with simple normal crossings and a Kähler metric ω on X • := X\D with negative holomorphic curvature, is it true that (X, D) is of log general type; that is, K X + D is big?

This question is in part motivated by recent results of Cadorel [START_REF] Cadorel | Symmetric differentials on complex hyperbolic manifolds with cusps[END_REF] who proved that given a projective log smooth pair (X, D) such that X • admits a Kähler metric ω with negative holomorphic sectional curvature and non-positive holomorphic bisectional curvature, then Ω X (log D) is big, and, moreover, Ω X is big provided that ω is bounded near D.

His proof involves working on P(Ω X (log D)) and considering the tautological line bundle O(1) on it. By the assumption on the bisectional curvature, ω induces a smooth, non-negatively curved hermitian metric h on O(1) away from (the inverse image of) D. Moreover, the Alhfors-Schwarz lemma guarantees that h extends across D as a singular metric with non-negative curvature. Using a result of Boucksom [START_REF] Boucksom | On the volume of a line bundle[END_REF] on a metric characterization of bigness then completes the proof.

One cannot expect such a strong result on the logarithmic cotangent bundle if one drops the assumption on the bisectional curvature. However, it seems reasonable to expect it for the logarithmic canonical bundle. The main difficulty is that one does not get from ω a positively curved metric on K X + D even on a Zariski open set. So one has to produce such a metric out of other methods, like the continuity method, cf [START_REF] Wu | Negative holomorphic curvature and positive canonical bundle[END_REF]. However, one faces several new difficulties compared to the setting of Theorem A: 1. To start the continuity method, one needs K X + D to be pseudo-effective. In the case D = 0, this is a consequence of the absence of rational curves (Ahlfors-Schwarz lemma) combined with Mori's bend and break and [START_REF] Boucksom | The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension[END_REF]. If D is not empty then one only knows that X • has no entire curves hence X has no rational curve meeting D at at most two points.

To conclude, one would then need to have a logarithmic version of Mori's bend and break, but unfortunately it is not known as of now, cf Remark 4.3. To circumvent the difficulty and inspired by the proof of [START_REF] Campana | Orbifold generic semi-positivity: an application to families of canonically polarized manifolds[END_REF]Thm. 4.1], we modify the boundary D into D + sB for some ample B and some s > 0 to make K X + D + sB psef. Only at the very end of the argument, one will see that K X + D is pseudoeffective. 2. The finiteness of the log canonical ring, known for klt pairs and crucial to understanding the deforming Kähler-Einstein metrics, is not known for lc pairs like (X, D). The idea is then to deform (X, D) into a klt pair (X, ∆ b,s := (1 -b)D + (b + s)B) that makes it klt and of log general type. The price to pay is that we have to carry on an additional error term in the volume estimate (compare Proposition 2.1 and Theorem 4.4).

Give or take these adjustements, one can still run the strategy of Theorem A mutatis mutandis; it will tell us that the volume of K X + (1 -b)D + (b + s)B is bounded away from zero uniformly in b, s > 0. A very important point is that the behavior of ω near D is not arbitrary, as ω must be dominated by a metric with Poincaré singularities along D thanks to Ahlfors-Schwarz lemma. However, one needs to look early on at ω on birational models of (X, D) where the Kähler-Einstein metrics are better understood, and ω will pick up singularities along exceptional divisors which will complicate the argument. In the end, the result is the following Theorem B. -Let (X, D) be a pair consisting of a projective manifold X and a reduced divisor D = i∈I D i with simple normal crossings. Let ω be a Kähler metric on X • := X\D such that there exists κ 0 > 0 satisfying

∀(x, v) ∈ X • × T X,x \{0}, HSC ω (x, [v]) < -κ 0 .
Then, the pair (X, D) is of log general type; that is, K X + D is big. If additionnally ω is assumed to be bounded near D, then K X is big.

In particular, Theorem A is a corollary of Theorem B. However, we chose to state and prove Theorem A separately in order to better highlight the new ideas that are necessary for Theorem A and then only later add a layer of technicality to go from Theorem A to the more general Theorem B. Another reason for this choice is that the proof of Theorem B does not seem to extend to the quasi-negative case.

Acknowledgements. I would like to thank Simone Diverio for introducing me to this problem and for the many related insightful discussions. I am very much indebted to Sébastien Boucksom for his comments on a preliminary draft of this paper and for suggesting me to consider the quasiprojective case. Finally, I am grateful to Benoît Cadorel for interesting exchanges about this topic.

Proof of Theorem A

Let (M, ω) as in the Theorem, and let Y ⊂ M be an irreducible subvariety of dimension m. One considers p : X → Y a desingularization of Y , and the goal is to show that X is of general type using the special Kähler metric ω| Y . The important observation is that p * (ω| Y ) is a smooth closed (1, 1)-form on X which is positive on a Zariski open set Ω of X. Moreover, there exists κ 0 > 0 such that the Kähler metric (p * (ω| Y ))| Ω has holomorphic sectional curvature bounded above by -κ 0 . This is because the holomorphic sectional curvature of the Kähler metric ω| Yreg admits such a bound by the compactness of M and the decreasing property of the bisectional curvature. These observations lead us to consider the following setting.

2.1. Setting. -Let X be a smooth, complex projective variety of dimension m. Let ω be a smooth, closed, semipositive (1, 1)-form on X such that there exists a Zariski open subset Ω ⊂ X satisfying:

1. The restriction ω| Ω is a Kähler metric on Ω. 2. There exists κ 0 > 0 such that for any (x, [v]) ∈ Ω × P(T X,x ), one has

HSC ω (x, [v]) -κ 0 .
Moreover, let B be a smooth divisor such that K X +bB is a big Q-divisor for some rational number b ∈ [0, 1). Let ω KE,b be the Kähler-Einstein metric associated to the pair (X, bB). That is, ω KE,b is a closed, positive current with minimal singularities in c 1 (K X +bB) satisfying the Einstein equation

Ric ω KE,b = -ω KE,b + b[B]
cf [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF]. That current defines a smooth Kähler metric on the Zariski open set Amp(K X + bB)\B thanks to the techniques of loc. cit. (cf. also [START_REF] Guenancia | Kähler-Einstein metrics with cone singularities on klt pairs[END_REF]) and the existence of a log canonical model for (X, bB), cf [START_REF] Birkar | Existence of minimal models for varieties of log general type[END_REF].

The following proposition is the crucial estimate needed for the proof of the main Theorem.

Proposition 2.1. -In the setting 2.1 above, there exists a constant

C = C(m, κ 0 ) independent of b such that Amp(K X +bB)\B tr ω KE,b ω • ω m KE,b C vol(K X + bB).
Using the proposition above, Theorem A follows relatively quickly.

Corollary 2.2. -In the setting 2.1 above, X is of general type, ie K X is big.

As hinted in the introduction, the idea of the proof of the Corollary is to consider an ample divisor B on X and analyze the family of singular Kähler-Einstein metrics ω KE,b of the pairs of log general type (X, bB) when b > 0 approaches zero. More precisely, the main point is to show that the volume of these singular metrics does not go to zero when b → 0. The metrics ω KE,b are not so well understood directly on X, but become much more manageable when seen on the log canonical model X can,b of the pair (X, bB) whose existence is guaranteed by the fundamental results of [START_REF] Birkar | Existence of minimal models for varieties of log general type[END_REF]. However, these models vary with b, hence it is crucial that the estimates be obtained on the fixed manifold X, which is the essence of Proposition 2.1.

Proof of the volume estimate.

-This section is devoted to the proof of Proposition 2.1.

Proof of Proposition 2.1. -By [START_REF] Birkar | Existence of minimal models for varieties of log general type[END_REF], there exists a canonical model (X can , bB can ) of (X, B) with klt singularities such that K Xcan + bB can is ample. Let us consider a resolution Z of the graph of the birational map φ : X X can as summarized in the diagram below

Z µ ν ! ! X φ / / X can Then, there exists a Q-divisor B Z = r i=0 b i B i with snc support, coefficients b i ∈ (0, 1), with b 0 = b, µ * B 0 = B and B i being ν-exceptional for i = 1, . . . , r such that K Z + B Z = ν * (K Xcan + bB can ) + E Z for some effective, ν-exceptional Q-divisor E Z = d j=0 a j E j .
Let us stress here that µ is an isomorphism over the Zariski open set Amp(K X + bB) given that φ is defined there and induces an isomorphism onto its image when restricted to that set.

Let A := K Xcan + bB can and let ω Z be a background Kähler metric on Z. For any t ∈ [0, 1], the cohomology class c 1 (ν * A + t{ω Z }) is semi-positive and big (it is even Kähler if t > 0). Thus, it follows from [START_REF] Eyssidieux | Singular Kähler-Einstein metrics[END_REF] that there exists a unique singular Kähler-Einstein metric

ω t ∈ c 1 (ν * A + t{ω Z }) solving Ric ω t = -ω t + tω Z + [B Z ] -[E Z ]
Moreover, the current ω t has bounded potentials for any t ∈ [0, 1] and there exists an effective,

µ-exceptional Q-divisor F on Z such that (2.1) µ * ω KE,b = ω 0 + [F ].
Step 1. Approximate KE metrics on a birational model In the following, we will introduce a family of smooth approximations (ω t,ε ) ε>0 of ω t defined as follows. Let us choose on O Z (B i ) (resp. O Z (E j )) a holomorphic section s i (resp. t j ) cutting out B i (resp. E j ) and a smooth hermitian metric h Bi (resp. h Ej ) with Chern curvature Θ h B i (resp. Θ h E j ). In order to lighten notation, one sets

|s i | 2 := h Bi (s i , s i ) (resp. |t j | 2 := h Ej (t j , t j )). For any ε ∈ (0, 1), one defines θ B ε := r i=0 b i (Θ h B i +dd c log(|s i | 2 +ε 2 )) and similarly θ E ε := d j=0 a j (Θ h E j + dd c log(|t j | 2 + ε 2 )). The smooth (1, 1)-form θ B ε represents c 1 (B Z
) and converges weakly to the current of integration [B Z ] when ε → 0, and similarly for θ E ε . Thanks to [START_REF] Aubin | Équations du type Monge-Ampère sur les variétés kählériennes compactes[END_REF][START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I[END_REF], there exists for any t, ε > 0 a unique smooth, Kähler metric

ω t,ε ∈ c 1 (ν * A + t{ω Z }) such that (2.2) Ric ω t,ε = -ω t,ε + tω Z + θ B ε -θ E ε
In terms of Monge-Ampère equations, this is equivalent to saying that ω t,ε = ν * ω A + tω Z + dd c ϕ t,ε solves

(ν * ω A + tω Z + dd c ϕ t,ε ) m = d j=0 (|t j | 2 + ε 2 ) aj d i=0 (|s i | 2 + ε 2 ) bi e ϕt,ε dV
where ω A ∈ c 1 (A) is a Kähler form on X can and dV is a smooth volume form chosen such that

Ric dV = -ν * ω A +tω Z + b i Θ h B i -a j Θ h E j
. By the proof of [GP16, Prop. 1] and the estimates of [GP16, Sect. 4], there exists a constant C t independent of ε > 0 such that

(2.3) ω t,ε C t ω B Z ,ε
where ω B Z ,ε is an approximate conical metric along B Z , cf. e.g. [GP16, Sect. 3].

Step 2. Bounding the Ricci curvature from below The heart of the proof relies on the following formula due to Royden, cf [WY16, Prop. 9], valid on the Zariski open set U ⊂ Z defined by U := µ -1 (Ω ∩ Amp(K X + bB)), and where ω := µ * ω.

(2.4) ∆ ωt,ε log tr ωt,ε ω κ• tr ωt,ε ω -λ where κ := n+1 2n • κ 0 and λ : Z → R + is any function such that Ric ω t,ε -λω t,ε . The first step is to get an explicit expression for λ, and then to write a global regularized version of (2.4) that we could integrate over the whole Z.

Keeping in mind that we want to get a lower bound of Ric ω t,ε , it is clear from (2.2) that θ B ε and θ E ε will not play the same role. We first deal with the easier term

θ B ε = d i=0 b i ε 2 (|s i | 2 + ε 2 ) 2 • Ds i , Ds i + ε 2 |s i | 2 + ε 2 • Θ h B i -f B ε ω Z where f B ε := C d i=0 ε 2 |si| 2 +ε 2
for some C > 0 large enough. In particular, one gets

(2.5) θ B ε -(f B ε tr ωt,ε ω Z )• ω t,ε Similarly, one can decompose θ E ε = α ε + β ε where α ε 0 and ±β ε C j ε 2 |tj | 2 +ε 2 • ω Z for some uniform constant C > 0. More precisely, α ε = j a j ε 2 (|tj | 2 +ε 2 ) 2 • Dt j , Dt j and β ε = j a j ε 2 |tj | 2 +ε 2 • Θ h E j . If we define f E ε := C j ε 2 |tj | 2 +ε 2
for some large C > 0, then we have

θ E ε α ε + f E ε ω Z tr ωt,ε (α ε + f E ε ω Z ) • ω t,ε = tr ωt,ε (θ E ε + (f E ε ω Z -β ε ))• ω t,ε tr ωt,ε (θ E ε + 2f E ε ω Z )• ω t,ε Let us now set χ ε := f B ε + 2f E ε ;
this is a smooth, positive function bounded uniformly when ε → 0 and such that χ ε → 0 almost everywhere. From (2.2), (2.5) and the inequality above, one deduces that Ric ω t,ε -1 + tr ωt,ε (θ E ε + χ ε ω Z ) • ω t,ε which, along with (2.4), yields the following formula valid on U

(2.6) ∆ ωt,ε log tr ωt,ε ω κ• tr ωt,ε ω -tr ωt,ε (θ E ε + χ ε ω Z ) -1
Step 3. Integration by parts.

Because ω might vanish outside of U , the left-hand side of (2.6) might become singular across Z U . So let us choose δ > 0; it is easy to deduce from (2.6) the following inequality

(2.7) ∆ ωt,ε log(u + δ) κ• u 2 u + δ -v• u u + δ
where u := tr ωt,ε ω and v = tr ωt,ε (θ E ε + χ ε ω Z ) + 1 are smooth, nonnegative functions on the whole Z which depend on t, ε > 0. Indeed, the inequality (2.6) can be rewritten as

∆u κu 2 + 1 u |∇u| 2 -vu hence ∆ log(u + δ) 1 u + δ • (κu 2 + 1 u |∇u| 2 -vu) - 1 (u + δ) 2 • |∇u| 2 = κ• u 2 u + δ -v• u u + δ + 1 u(u + δ) - 1 (u + δ) 2 • |∇u| 2 κ• u 2 u + δ -v• u u + δ and (2.7) follows.
As both sides of (2.7) are continuous on Z (remember that t, ε, δ > 0 are fixed for the time being), the inequality extends across Z \ U . Then, one can multiply each side by ω m t,ε and integrate over Z. We get

Z κ• u 2 u + δ ω m t,ε Z v• u u + δ ω m t,ε
By dominated convergence, one can pass to the limit in the integrals when δ → 0 to get (2.8)

Z κ• tr ωt,ε ω • ω m t,ε Z (tr ωt,ε (θ E ε + χ ε ω Z ) + 1) ω m t,ε
Step 4. Computing the error terms Let us now analyze the right-hand side of (2.8), which coincides with (2.9)

m Z θ E ε ∧ ω m-1 t,ε + m Z χ ε ω Z ∧ ω m-1 t,ε + {ν * ω A + tω Z } m
The first and last terms of (4.10) are cohomological. The first term is equal to

m E Z • (ν * A + t{ω Z }) m-1 = mt m-1 E Z • {ω Z } m-1
as E Z is ν-exceptional, hence it converges to zero when t → 0. The last one converges to (A m ) = vol(K X + bB) when t → 0. As for the second term, it can be estimated at t > 0 fixed thanks to (2.3) by the integral

C t Z χ ε ω m B Z
where ω B Z is a metric with conical singularities along B Z . In particular, ω m B Z = gω m Z for some density g ∈ L 1 (ω m Z ). As χ ε is uniformly bounded and tends to 0 almost everywhere when ε approaches 0, the dominated convergence theorem asserts that

lim ε→0 Z χ ε ω Z ∧ ω m-1 t,ε = lim ε→0 Z χ ε ω m B Z = 0.
In conclusion, one gets (2.10) lim sup

t→0 lim sup ε→0 Z κ• tr ωt,ε ω • ω m t,ε vol(K X + bB)
Step 5. Conclusion Let us fix a relatively compact open set K Amp(K X + bB)\B. Given (4.1), we know that on µ -1 (K), µ * ω KE is the smooth limit of ω t,ε when t, ε approach zero. Therefore

K κ• tr ω KE,b ω • ω m KE,b = µ -1 (K) κ• tr µ * ωKE ω • (µ * ω KE ) m = lim sup t→0 lim sup ε→0 µ -1 (K) κ• tr ωt,ε ω • ω m t,ε lim sup t→0 lim sup ε→0 Z κ• tr ωt,ε ω • ω m t,ε vol(K X + bB)
and as this holds for any K, we get the desired inequality. Proposition 2.1 is proved.

2.3. End of the proof. -This section is devoted to the proof of Corollary 2.2.

Proof of Corollary 2.2. -We first claim that K X is pseudoeffective. Indeed, observe that if f : P 1 → X is a rational curve whose image hits Ω, then there exists a finite set Σ ⊂ P 1 such that f (P 1 \Σ) ⊆ Ω. Then one can apply the inequality [Roy80, Prop. 4] to f :

(P 1 \Σ, ω FS ) → (Ω, ω) to get ∆ ωFS log tr ωFS (f * ω) κ tr ωFS (f * ω) + 2
where κ := m+1 m • κ 0 . In particular, the function log tr ωFS (f * ω) on P 1 \Σ is subharmonic and bounded above. Therefore it extends to a subharmonic function on P 1 , hence it has to be constant which is a contradiction. This shows that every rational curve on X is contained in the Zariski closed proper subset X\Ω, hence K X is pseudoeffective by [START_REF] Boucksom | The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension[END_REF]. Note that we only used the boundedness from above of tr ωFS (f * ω) near the complement of Ω and not its smoothness across X\Ω. This will be useful later, cf Step 6 on page 16.

Let B be an ample divisor on X. For any rational number b > 0, the Q-line bundle K X + bB is big, hence there exists a unique Kähler-Einstein metric )

ω b ∈ c 1 (K X + bB) on X solving Ric ω b = -ω b + b[B] cf [BEGZ10] or [Gue13, Thm. 2.2]. In terms of Monge-Ampère equation, if θ (resp. θ B ) is a smooth representative of c 1 (K X ) (resp. c 1 (B)), then ω b = θ + bθ B + dd c ϕ b solves (θ + bθ B + dd c ϕ b ) m = e ϕ b |s|
Ω b tr ω b ω • ω m b C vol(K X + bB)
Les us define M b := e sup X ϕ b and u b := ϕ b -sup X ϕ b so that (u b ) b>0 is a family of sup-normalized Cω X -psh functions for some C > 0 large enough, independent of b. In particular, (u b ) b>0 is relatively compact in L 1 (dV ), hence by the dominated convergence theorem, there exists C > 0 independent of b ∈ (0, 1/2) such that

C -1 X e u b |s| 2b dV C hence (2.12) vol(K X + bB) = M b X e u b |s| 2b dV ∈ [C -1 M b , CM b ]
In particular, (2.11) allows us to conclude that (2.13)

Ω b tr ω b ω • ω m b CM b
On Ω b , one has the following standard inequality

tr ω b ω ω m ω m b 1/m = (ω m /dV ) 1/m (M b e u b /|s| 2b ) -1/m
Now let K Ω be a relatively compact open subset which is located away from the degeneracy locus of ω so that (ω m /dV ) 1/m C -1 > 0 on K, up to taking C larger. Then, one has

K∩Ω b tr ω b ω ω m b C -1 M 1-1/m b K∩Ω b e (1-1 m )u b |s| 2b( 1 m -1) dV = C -1 M 1-1/m b K e (1-1 m )u b |s| 2b( 1 m -1) dV C -1 M 1-1/m b
for some C > 0 independent of b as K\(K ∩ Ω b ) has zero Lebesgue measure. Combined with (2.13) one gets that

M b C -1 M 1-1/m b
for some uniform C > 0. In particular, M b is uniformly bounded from below away from zero, hence one deduces from (2.12) the existence of η > 0 independent of b > 0 such that vol(K X + bB) > η.

By the continuity of the volume function, cf [Laz04, Thm. 2.2.37], one deduces that vol(K X ) > 0, hence K X is big and X is of general type.

Let us finish this section with the following

The quasi-negative case

The argument in the proof of Theorem A is relatively robust and allows us to work with a weaker assumption on the holomorphic section curvature of (M, ω). More precisely, let us consider a compact Kähler manifold (M, ω) with quasi-negative holomorphic sectional curvature; that is (i) For any pair (x, [v]) ∈ M × P(T M,x ), one has HSC ω (x, [v]) 0.

(ii) There exists x 0 ∈ M such that for any [v] ∈ P(T M,x0 ), one has HSC ω (x 0 , [v]) < 0. In this setting, Diverio-Trapani [START_REF] Diverio | Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle[END_REF] proved that the conclusions of [WY16, TY17] hold as well, namely M is projective and K M is ample. Introducing the (open) negative curvature locus

W := {x ∈ M ; ∀v ∈ T M,x \{0}, HSC ω (x, [v]) < 0}
one can use again the decreasing property of the holomorphic bisectional curvature to conclude that any smooth subvariety Y ⊂ M such that Y ∩ W = ∅ satisfies that K Y is ample. The goal of this section is to extend this result to singular subvarieties: Theorem 3.1. -Let (M, ω) be a compact Kähler manifold with quasi-negative holomorphic sectional curvature. Let Y be a possibly singular irreducible subvariety

Y ⊂ M such that Y ∩ W = ∅. Then Y is of general type.
The proof of Theorem 3.1 is very much similar to the proof of Theorem A. Considering a desingularization of Y , one gets a smooth projective manifold X which is not uniruled as M contains no rational curve. Again, using [START_REF] Boucksom | The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension[END_REF], K X is pseudo-effective. Then one considers an ample line bundle B on X and a rational number b > 0 so that K X + bB is big, hence there is a unique KE metric ω b ∈ c 1 (K X + bB). The pull-back of the Kähler metric on X will still be denoted ω, as in the case of Theorem A. Let us point out the main adjustments that need to be performed in the quasi-negative case.

Step 1. There is no change to be made here, as we consider the same metrics ω t,ε on Z.

Step 2.

On the Zariski open set U := µ -1 (Ω ∩ Amp(K X + bB)) ⊂ Z, the Laplacian inequality now becomes (3.1) ∆ ωt,ε log tr ωt,ε ω κ• tr ωt,ε ω -λ where κ : U → R + is a function such that -n+1 2n • κ(z) is a nonpositive upper bound for the holomorphic sectional curvature of ω z and λ : Z → R + is a function such that Ric ω t,ε -λω t,ε , as before. The continuous function κ : U → R + does not necessarily extend to a continuous function on Z. However it easy to construct a continuous function κ : Z → R + along with two small neighborhoods W ⊂ W of Z U with the following properties

• κ| W ≡ 0 • κ = κ on U W • κ κ on U • (U W ) ∩ (p • µ) -1 (Y ∩ W) = ∅ Because of
the third point, the formula (2.7) remains true if one replaces κ by κ.

Steps 3-5. No change is needed here. The conclusion we get is

(3.2) Amp(K X +bB)\B κ• tr ω b ω • ω m b vol(K X + bB).
Moving on to the last part of the proof, one can pick a relatively compact subset K Ω ∩ p -1 (W) ⊂ X such that on K, one has κ C -1 and (ω m /dV ) 1/m C -1 . Therefore

K∩Ω b tr ω b ω • ω m b C K∩Ω b κ• tr ω b ω • ω m b C vol(K X + bB)
At this point, the same arguments as before show that vol(K

X + bB) ∈ [C -1 M b , CM b ] as well as K∩Ω b tr ω b ω ω m b C -1 M 1-1/m b
from which the uniform positive lower bound on vol(K X + bB) follows.

4. The quasi-projective case 4.1. Setting. -Let X be a smooth, complex projective variety of dimension m and let D = p k=0 D i be a reduced divisor with simple normal crossings. Let X • := X\D and let ω be a Kähler form on X • such that there exists κ 0 > 0 such that

∀ (x, [v]) ∈ X • × P(T X,x ), HSC ω (x, [v]) -κ 0 .
In this setting, one can deduce from the Ahlfors-Schwarz lemma the following Lemma 4.1. -In the setting 4.1 above, the following statements hold 1. Every holomorphic map f : C → X • is constant. 2. The Kähler metric ω is dominated by a Kähler metric ω P on X • with Poincaré singularities along D.

Recall that a Kähler metric ω P on X • is said to have Poincaré singularities along D if for any x ∈ D and any coordinate chart U ∆ m around x where D is given by (z 1 • • • z r = 0), ω| |U is quasi-isometric to the model Poincaré metric

ω mod := r k=1 i dz k ∧ dz k |z k | 2 log 2 |z k | 2 + m k=r+1 i dz k ∧ dz k Proof of Lemma 4.1. -The first item is a consequence of [Roy80, Cor. 1]. The second one is a consequence of [Roy80, Thm. 1] applied to f = id : ((∆ * ) r × ∆ m-r , ω mod ) → ((∆ * ) r × ∆ m-r , ω) where one identifies U ∩ X • with (∆ * ) r × ∆ m-r .
Remark 4.2. -At this point, one would like to conclude that K X +D is pseudoeffective. Indeed, if K X + D were to fail to be pseudo-effective, then by [START_REF] Boucksom | The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension[END_REF], one would obtain a covering family of curves (C t ) such that (K X + D)• C t < 0. Following Mori's bend and break, one could deform each curve C t into a new reducible curve containing a rational curve C t passing through a given point. From the second item of Lemma 4.1, one would obtain a contradiction if one knew that C t intersects D in a most two points. Therefore, the pseudoeffectiveness of K X + D would be a consequence of the following general conjecture of Keel-McKernan Conjecture 4.3. -(Logarithmic bend and break, cf [KM99, 1.11]) Let (X, D) be a pair consisting of a smooth projective complex variety X and a reduced divisor D with simple normal crossings. If C ⊂ X is a curve such that (K X + D)• C < 0 and C D, then through a general point of C there is a rational curve meeting D at most once.

The logarithmic bend and break is known in dimension two by [KM99, 1.12]. Let us also mention that Lu-Zhang [LZ17, Thm. 1.4] and McQuillan-Pacienza [MP12, Rem. 1.1] proved the above conjecture assuming that for any non-empty subset J ⊂ I, any holomorphic map f :

C → j∈J D i \ k / ∈J D k is constant.
4.2. The main statement. -Let B be a smooth divisor on X such that 1. B + D has simple normal crossings 2. The line bundles associated to B and B -D are ample 3. There exists s 0 ∈ (0, 1 2 ) such that K X + D + s 0 B is pseudo-effective. Now, let 0 s < 1/2 be any rational number such that the Q-line bundle K X + D + sB is pseudoeffective. Up until the very end, the number s will be fixed. By the assumptions on B above, one knows that for any rational number b > 0, the Q-line bundle

K X + D + sB + b(B -D) = K X + (1 -b)D + (b + s)B is big. Let ∆ b,s := (1 -b)D + (b + s)B
The pair (X, ∆ b,s ) is klt and is of log general type whenever b ∈ (0, 1/2). By [START_REF] Eyssidieux | Singular Kähler-Einstein metrics[END_REF], (X, ∆ b,s ) admits a unique Kähler-Einstein metric ω KE,b,s . That is, ω KE,b,s is a closed, positive current in c 1 (K X + ∆ b,s ) with bounded potentials satisfying the Einstein equation

Ric ω KE,b,s = -ω KE,b,s + [∆ b,s ]
in the weak sense. Thanks to [START_REF] Birkar | Existence of minimal models for varieties of log general type[END_REF], ω KE,b,s defines a smooth Kähler metric on the Zariski open set Amp(K X + ∆ b,s )\(D ∪ B).

Indeed, thanks to loc. cit., there exists a canonical model (X can,b,s , ∆ can,b,s ) of (X, ∆ b,s ) with klt singularities such that K X can,b,s + ∆ can,b,s is ample. Let us consider a resolution Z of the graph of the birational map φ : X X can,b,s as summarized in the diagram below

K Z + ∆ b,s = ν * (K X can,b,s + ∆ can,b,s ) + E
Moreover, one can assume that Exc(µ) is divisorial and that the support of ∆ b,s + E has simple normal crossings. Up to setting some a j 's to zero, one can also assume that Exc(µ) ⊆ d j=0 E j . Let us stress here that µ is an isomorphism over the Zariski open set Amp(K X + ∆ b,s ) given that φ is defined there and induces an isomorphism onto its image when restricted to that set.

Let A := K X can,b,s + ∆ can,b,s and let ω Z be a background Kähler metric on Z. For any t ∈ [0, 1], the cohomology class c 1 (ν * A + t{ω Z }) is semi-positive and big (it is even Kähler if t > 0). Thus, it follows from [START_REF] Eyssidieux | Singular Kähler-Einstein metrics[END_REF] that there exists a unique singular Kähler-Einstein metric

ω t ∈ c 1 (ν * A + t{ω Z }) solving Ric ω t = -ω t + tω Z + [∆ b,s ] -[E]
The current ω t is smooth outside Supp(∆ b,s + E) and, moreover, there exists an effective, µexceptional Q-divisor F on Z such that (4.1)

µ * ω KE,b,s = ω 0 + [F ].
In particular, ω KE,b,s is smooth on Amp(K X + ∆ b,s )\(D ∪ B).

As in the earlier setting, the key point is the following volume estimate Theorem 4.4. -In the setting 2.1 above, given an ample line bundle H on X, there exists a constant C depending only on X, D, H, ω -but not b or s-such that

Amp(K X +∆ b,s )\(D∪B) tr ω KE,b,s ω • ω m KE,b,s C (K X + ∆ b,s ) m + b (K X + ∆ b,s ) m-1 • H
where • is the movable intersection product, cf [BDPP13, §3] and the references therein. Furthermore, the line bundle K X + D is big.

4.3. Proof of Theorem 4.4. -The strategy of the proof is similar to that of Proposition 2.1, but it gets more technical. We will only indicate what are the main changes to perform.

Step 1.

For t, ε > 0 we now instead consider the current

ω t,ε = ν * ω A + tω Z + dd c ϕ t,ε ∈ c 1 (ν * A + t{ω Z }) solving (4.2) (ν * ω A + tω Z + dd c ϕ t,ε ) m = d j=0 (|t j | 2 + ε 2 ) aj |s D | 2(1-b) • |s B | 2(b+s) • e ϕt,ε dV
where t j , s D , s B are respectively sections of O Z (E j ), O Z (D ), O Z (B ) cutting out E j , D , B and the smooth hermitian metrics chosen on the various bundles are such that the following equation holds.

(4.3) Ric ω t,ε = -ω t,ε + tω Z + [∆ b,s ] -θ E ε
where θ E ε := d j=0 a j (Θ h E j + dd c log(|t j | 2 + ε 2 )). In the following, one sets Z • := Z\(D ∪ B ). By the proof of [GP16, Prop. 2.1], ω t,ε is a Kähler metric on Z • with conical singularities along ∆ b,s , and it is uniformly (in ε) dominated by a Kähler metric on Z • \ ∪ aj <0 E j with conic singularities along ∆ b,s + aj <0 (-a j )E j . In particular, there exists f ∈ L 1 (ω m Z ) independent of ε such that

ω Z ∧ ω m-1 t,ε f ω m Z
By Lebesgue dominated convergence theorem, one gets

(4.4) ∀t > 0, ∀j = 0 . . . d, lim ε→0 Z • ε 2 |s Ej | 2 + ε 2 ω Z ∧ ω m-1 t,ε = 0.
Finally, remember from Ahlfors-Schwarz lemma, cf Lemma 4.1, that the Kähler metric ω := (µ| Z • ) * ω on Z • has at most Poincaré singularities along D + E j . In particular, one has (4.5) sup

Z •   |s D | 2b • d j=0 |s Ej | 2 • tr ωt,ε ω   < +∞.
Step 2.

The following Laplacian inequality holds on Z

• (4.6) ∆ ωt,ε log tr ω t,ε,δ ω κ• tr ωt,ε ω -tr ωt,ε (θ E ε + χ ε ω Z ) -1 where χ ε = C d j=0 ε 2 |s E j | 2 +ε 2 for some large C independent of ε. Moreover, one has ∆ ωt,ε   log |s D | 2b + d j=0 log |s Ej | 2   = tr ωt,ε (dd c (log |s D | 2b + d j=0 log |s Ej | 2 )) -b • tr ωt,ε Θ D - d j=0 tr ωt,ε Θ Ej
where Θ D , Θ Ej are the Chern curvature form of the smooth hermitian metrics chosen on the respective associated line bundles. In the end, one gets the following identity, holding on Z

• (4.7) ∆ ωt,ε log |s D | 2b • d j=0 |s Ej | 2 • tr ω t,ε,δ ω κ• tr ωt,ε ω -tr ωt,ε θ E ε +χ ε ω Z +bΘ D + d j=0 Θ Ej -1
Step 3.

As before, one starts by choosing δ > 0 and deduce from (4.6) the following

∆ ωt,ε log(u + δ) κ• u 2 u + δ -v• u u + δ where u := |s D | 2b • d j=0 |s Ej | 2 • tr ω t,ε,δ ω and v = tr ωt,ε θ E ε + χ ε ω Z + bΘ D + d j=0 Θ Ej + 1
. By the observation (4.5) above, all the terms involved are smooth on Z • and globally bounded. In particular, the dominated convergence theorem shows that (4.8)

Z • (κu -v) ω m t,ε = lim δ→0 Z • κ• u 2 u + δ -v• u u + δ ω m t,ε
Combining (4.8) with Lemma 4.5 below, one eventually gets (4.9)

Z • κ tr ωt,ε ω ω m t,ε Z • tr ωt,ε θ E ε + χ ε ω Z + bΘ D + d j=0 Θ Ej + 1 ω m t,ε Lemma 4.5. -Let f, g ∈ L ∞ (Z • ) ∩ C ∞ (Z • ) such that ∆ ωc f g on Z • ,
where ω c is some Kähler metric with conic singularities along ∆ b,s . Then By assumption, the function g is integrable with respect to ω m c and by dominated convergence, one has

Z • g ω n c = lim α→0 Z ξ α g ω m c
But that last integral is dominated by

Z ξ α • ∆ ωc f ω m c = m• Z f dd c ξ α ∧ ω m-1 c C• sup Z |f |• Vol ωP (Supp(ξ α ))
where C is such m dd c ξ α ∧ ω m-1 c Cω m P . Finally, the right-hand side tends to zero when α approaches zero. The Lemma is proved.

Step 4. The right-hand side of (4.9) can be rewritten as

(4.10) m Z (θ E ε + bΘ D + d j=0 Θ Ej ) ∧ ω m-1 t,ε + m Z χ ε ω Z ∧ ω m-1 t,ε + {ν * ω A + tω Z } m
The first term is cohomological and coincides with m

(E + E j + bD )• ({ν * ω A + tω Z }) m-1 , which is independent of ε.
For the second, one has the limit computation (4.4). As E j is ν-exceptional, one gets

(4.11) lim sup t→0 lim sup ε→0 Z • κ• tr ωt,ε ω • ω m t,ε mb(D • (ν * A) m-1 ) + (K X + ∆ b,s ) m
Finally, let p > 0 such that pH -D is effective. Then, one has

D • (ν * A) m-1 (µ * D• (ν * A) m-1 ) p (µ * H• (ν * A) m-1 ) = p H• (K X + ∆ b,s ) m-1
which ends the proof of the first part of Theorem 4.4.

Step 5. Bigness of K X + D.

As ω is dominated by a metric with Poincaré singularities along D, Skoda-El Mir extension theorem implies that the current ω on X • can be extended to a closed, positive (1, 1)-current on X putting no mass on D. We still denote it by ω, and set α := {ω}; this is a pseudoeffective class. As ω has no zero Lelong numbers, Demailly's regularization theorem shows that α is even nef, but we will not use this fact. We claim that for any t > 0, one has (4.12)

Z • tr ωt,ε ω • ω m t,ε = m({ω m-1 t,ε }• µ * α)
Indeed, the integral on the left-hand side can be rewritten as m Z ω ∧ ω m-1 t,ε

given that ω has at most Poincaré singularities. Moreover, for any t, ε > 0, the metric ω t,ε has conic singularities along ∆ b,s and can be regularized into a family of smooth Kähler metrics (ω t,ε,δ ) δ>0 in the same cohomology class {ω t,ε } such that ω t,ε,δ C t,ε ω t,ε for some C t,ε > 0 independent of δ. By Lebesgue dominated convergence theorem, one deduces that

Z ω ∧ ω m-1 t,ε = lim δ→0 Z ω ∧ ω m-1 t,ε,δ .
Now, the total mass on Z of a closed, positive (1, 1)-current with respect to a given Kähler metric only depends on the cohomology class of that current. From the identity above, one deduces

Z ω ∧ ω m-1 t,ε = {ω m-1 t,ε }• { ω}
which prove (4.12).

When t, ε approach zero, the right-hand side of (4.12) converges to m((ν * A) m-1 • µ * α) which coincides with the movable intersection product m (K X + ∆ b,s ) m-1 • α . As a result, one obtains

(K X + ∆ b,s ) m-1 • α bp κ (K X + ∆ b,s ) m-1 • H + 1 κm (K X + ∆ b,s ) m
Let us now try to analyze the class α. Because ω is smooth and Kähler on a Zariski open set, α is big thanks to [START_REF] Boucksom | On the volume of a line bundle[END_REF]. In particular, for b small enough, one has an inequality of (1, 1) cohomology classes

α - bp κ H 1 2 α.
By the increasing and superadditive properties of the movable intersection [BDPP13, Thm. 3.5 (ii)], one has The inequality above holds for any rational number s 0 such that K X +D+sB is pseudoeffective.

(K X + ∆ b,s ) m-1 • α - bp κ (K X + ∆ b,s ) m-1 • H 1 2 (K X +
If we can show that K X + D is pseudo-effective, then we are done as (4.13) would show that K X + D is big. But if K X + D is not pseudoeffective, there exists a real number s ∞ > 0 such that K X + D + s ∞ B is pseudoeffective but not big. Taking a sequence of rational numbers (s n ) decreasing to s ∞ , one has that K X + D + s n B is big with vol(K X + D + s n B)

κm 2 m • α m .
By continuity of the volume function, one gets vol(K X + D + s ∞ B) > 0 which is a contradiction.

Step 6. The case where ω is bounded.

Here the pseudoeffectivity of K X comes almost for free by the exact same argument as the one in the first step of the proof of Corollary 2.2 (p. 8) by setting Ω := X\D.

From there, one can reproduce almost verbatim the arguments of the proof of Theorem A. The only difference is in Step 3. as the quantity tr ωt,ε ω is no longer smooth across D but merely bounded. However, the integration by parts technique of Lemma 4.5 still applies as the family (ξ α of cut-off functions satisfies ±dd c ξ α ∧ ω m-1 sm ω m P where ω sm is a smooth Kähler form on X and ω P is some Kähler form on X\D with Poincaré singularities along D. In particular, X |∆ ωsm ξ α | ω m sm converges to 0 as α approaches zero.

  2b dV where dV is a fixed smooth volume form such that Ric dV = -θ, s is a section of O X (B) cutting out B and |• | is a smooth hermitian metric on O X (B) whose curvature is equal to θ B . Thanks to loc. cit., ω b has full mass; that is X ω m b = vol(K X + bB) and, moreover, ω b is a genuine smooth Kähler-Einstein metric on the Zariski open set Ω b := Amp(K X + bB)\B. Combining this with the content of the Proposition, one gets a uniform constant C > 0 such that the following inequality holds (2.11

/

  / X can,b,s Let ∆ b,s := (1 -b)D + (b + s)B where D (resp. B ) is the strict transform of D (resp. B) by µ. There exist a ν-exceptional Q-divisor E := d j=0 a j E j with snc support and coefficients a j ∈ (-1, +∞) such that

  Proof of Lemma 4.5. -It is well-known that the complex codimension one set D ∪B ⊆ Z admits a family of cut-off functions (ξ α ) α>0 such that lim supα→0 sup Z |dd c ξ α | ωP < +∞where ω P is a metric with Poincaré singularities along D + B , cf e.g. [CGP13, Sect. 9].

  ∆ b,s ) m-1 • α and therefore, using the Teissier-Hovanskii inequalities [BDPP13, Thm. 3.5 (iii)], one gets(K X + ∆ b,s ) m κm 2 (K X + ∆ b,s ) m-1 • α κm 2 (K X + ∆ b,s ) m 1-1/m • α m 1/mor equivalently(K X + ∆ b,s ) m κm 2 m • α mand the right-hand side is positive, independent of both b and s. In conclusion, one gets(4.13) vol(K X + D + sB) = lim b→0 vol(K X + ∆ b,s ) κm 2 m • α m .

  Remark 2.3. -In the setting of the Theorem A, one can additionally see that if Y has log terminal singularities (see e.g. [KM98, Def. 2.34] for a definition), then K Y is an ample Q-line bundle. To see this, first observe that K Y is a big Q-line bundle because a desingularization Y of Y is of general type and there is a natural inclusion H 0 ( Y , mK Y ) ⊆ H 0 (Y, mK Y ) for any integer m divisible enough. By [BBP13, Thm A.(ii)], the augmented base locus of K

Y is uniruled, hence empty, as M does not contain any rational curve.