Geometric effects of hyperbolic cohomology classes on K\"ahler manifolds (with an appendix by Beno\^it Claudon)

Francesco Bei; Simone Diverio; Stefano Trapani

REVIEW 3 major objections 5 minor 2 cited by

Geometric effects of hyperbolic cohomology classes on K\"ahler manifolds (with an appendix by Beno\^it Claudon)

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A purely topological condition on Kähler manifolds forces nonzero holomorphic sections of adjoint bundles.

desk verdict A new 'topologically hyperbolic' class for Kähler manifolds, with strong spectral and non-vanishing consequences; the core proof uses the standard Vafa–Witten trick and the softest spot is reliance on cited index-theoretic results. read the letter →

arxiv 2506.09907 v2 pith:LJYHAP57 submitted 2025-06-11 math.CV math.AGmath.DGmath.SP

Francesco Bei , Simone Diverio , Stefano Trapani This is my paper

classification math.CVmath.AGmath.DGmath.SP MSC 32Q1558J5032L2053C21

keywords KählertopologicallyhyperboliccohomologyclassspectralgapNakanopositivityGriffithseffectivenon-vanishingL2indextheoremKodairadimension

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Compact Kähler manifolds are usually called hyperbolic when a Kähler class lifts to a bounded exact form on the universal cover. This paper strips the positivity away: a manifold is Kähler topologically hyperbolic when it carries any real degree-2 cohomology class whose pullback to the universal cover is d-exact with a bounded primitive, and whose top self-intersection is nonzero. The main theorem says that on such a manifold, any Hermitian holomorphic vector bundle that is Nakano positive on a full-measure open set has nonzero holomorphic sections twisted by the canonical bundle, with the dimension of those sections equal to a positive Euler characteristic. These conclusions are new even in the classical Kähler hyperbolic setting, and they lead to effective non-vanishing results for adjoint line bundles and to quantitative curvature obstructions. The appendix identifies hyperbolic degree-2 classes with those whose cocycles are bounded in the first variable for each fixed second variable, and shows that Kähler topologically hyperbolic surfaces are of general type.

What carries the argument

The central object is a hyperbolic cohomology class in degree 2: a class [η]∈H²(M,R) whose pullback π*η to the universal cover is exact with a bounded primitive β (dβ=π*η), together with homological non-singularity ∫_M η^m≠0. The argument is carried by the Gromov–Vafa–Witten trick: twist a spin-c Dirac operator by the family of connections ∇_s=∇_0+isη, use heat-kernel traces to compute the L²-Γ_s index, and apply the local index theorem to write that index as a polynomial in s whose leading coefficient is ∫_M η^m. Because that coefficient is nonzero, the polynomial is not identically zero, so there is an interval of s where the twisted kernels are nonzero; a comparison result then pushes zero into the spectrum of the untwisted Dirac operator. A separate spectral-gap theorem for elliptic operators on Galois coverings (Theorem 5.1) rules zero out of the spectrum on the positive-Nakano pieces, forcing the zero eigenvalue to sit in the (m,0)-part, which is exactly $H^{{m,0}}$_{∂E}(M,E).

What would settle it

Compute χ(M,K_M⊗E) for a compact Kähler topologically hyperbolic manifold with a Hermitian holomorphic vector bundle Nakano positive on a full-measure open set; an example with χ(M,K_M⊗E) ≤ 0 and $H^{{m,0}}$_{∂E}(M,E)={0} would directly refute Theorem 7.1. Alternatively, exhibit a Kähler topologically hyperbolic surface of Kodaira dimension less than 2, contradicting the appendix's Theorem A.10.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the Gromov–Vafa–Witten mechanism works with only the top self-intersection of a bounded-primitive 2-class, replacing the Kähler, nef, and big positivity assumptions used in earlier notions of hyperbolicity. Theorem 7.1 states that if (M,h) is Kähler topologically hyperbolic and (E,τ)→M is a Hermitian holomorphic vector bundle that is Nakano positive over an open subset A⊂M of full measure, then $H^{{m,0}}$_{∂E}(M,E)≠{0} and $h^{{m,0}}$_{∂E}(M,E)=χ(M,K_M⊗E)>0. For line bundles this is read as a special case of the effective non-vanishing conjecture, since the Nakano-positivity hypothesis forces E to be big and nef. The same spectral control gives curvature inequalities such as min_M scal_h ≤ −4λ̃_{0,h} for weakly Kähler hyperbolic manifolds, and the appendix proves that Kähler topologically hyperbolic surfaces have Kodaira dimension 2.

Load-bearing premise

The proof stands on the L² index-theoretic step in Proposition 7.3(3): the index of the twisted spin-c Dirac operator must coincide with the polynomial in the twisting parameter given by the local index theorem (with nonzero leading coefficient), and the cited comparison result [Eys97, Prop. 7.1.2] must force 0 into the spectrum of the untwisted operator on the infinite-volume universal cover.

Editorial extensions

If this is right

A Kähler topologically hyperbolic manifold cannot be uniruled: no dominant meromorphic map from P¹×N can cover it, and in the projective case the canonical bundle is pseudoeffective.
It cannot be bimeromorphic to a compact Kähler manifold with trivial first real Chern class, ruling out complex tori, Calabi–Yau, and hyperkähler factors up to finite étale covers.
Every Hermitian holomorphic vector bundle that is Nakano positive on a full-measure open set yields nonzero sections of K_M⊗E, and the dimension equals χ(M,K_M⊗E)>0; for line bundles, K_M⊗E^p is big and h^{m,0}(M,E^p)>0 for every positive p.
Every Kähler metric on a weakly Kähler hyperbolic manifold satisfies min_M scal_h ≤ −4λ̃_{0,h}, with equality iff scal_h is constantly −4λ̃_{0,h}; in the topologically hyperbolic case a Ricci lower bound a forces a ≤ −λ̃_{0,h}/m.
Kähler topologically hyperbolic surfaces are of general type.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, if the paper's Conjecture 4.13 holds, then Kähler topological hyperbolicity would be a cohomological obstruction to being special in the sense used in the classification of Kähler manifolds.
Beyond the paper, the appendix's cochain description makes it plausible that topological hyperbolicity can be checked on the fundamental group alone for finitely presented groups, using only group-cohomology computations.
Beyond the paper, the polynomial index formula suggests a quantitative refinement: the dimension of H^{m,0}(M,K_M⊗E) may be governed by the size of ∫_M η^m and by curvature bounds, giving explicit constants in effective non-vanishing statements.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper introduces Kähler topologically hyperbolic manifolds (a degree-2 real class whose pullback to the universal cover has a bounded primitive and whose top power is nonzero) and proves that this purely topological condition controls birational geometry, spectral gaps, non-vanishing of H^{m,0} for adjoint bundles, and curvature bounds. The main theorem, 7.1, is a genuine advance: the absence of positivity on the cohomology class is compensated by Nakano positivity of the bundle on a full-measure open set, and the Gromov–Vafa–Witten trick still works. The proof is a coherent chain from the spectral criterion (Thm 5.1) through closed-range arguments to the index-theoretic step.

What's new: the definition itself, Theorem 5.1 as a general spectral gap statement for elliptic operators with zeroth-order potential, and the applications to non-uniruledness, non-Calabi–Yau bimeromorphic models, and scalar/Ricci curvature estimates. The appendix by Claudon gives a clean cochain description of degree-2 hyperbolic classes and shows surfaces in the class are of general type. The paper is well-written, organized, and unusually honest: the authors explicitly state that they have no examples separating topological hyperbolicity from weak Kähler hyperbolicity.

Soft spots, in order. (1) The keystone Prop 7.3(3) is where a careful referee should focus. The Γ_s-index formula and the passage from nonzero twisted kernels to 0 ∈ spectrum are cited to [Eys97] rather than proved. This is a standard tool for the Vafa–Witten trick, and the concern that it might fail on infinite-volume universal covers is not substantiated—η is an L∞ 1-form, so the twisted operator is a bounded perturbation of the untwisted one. But because the result is cited, the paper inherits the risk. (2) Several key lemmas, including the pullback of hyperbolic classes over classifying maps, come from the same authors' BCDT24 preprint. That is not circular—[BCDT24] is independent—but a referee might want those statements stated precisely. (3) Minor typos, e.g., 'topologically Kähler hyperbolic' in Prop 4.5. Nothing load-bearing.

Verdict: this deserves a serious referee. The central claim is likely correct and the ideas are transferable. I would send it out, and I expect a competent referee to come back with requested clarifications on the index step rather than a rejection.

Referee Report

3 major / 5 minor

Summary. The paper introduces the notion of Kähler topologically hyperbolic manifold, defined by the existence of a degree-2 hyperbolic cohomology class with non-zero top self-intersection. It proves that this property is bimeromorphically and homotopically invariant, that such manifolds are not uniruled and are not bimeromorphic to compact Kähler manifolds with trivial first real Chern class, and that the product behaves functorially. The main analytic result is Theorem 7.1: on a Kähler topologically hyperbolic manifold, any Hermitian holomorphic vector bundle that is Nakano positive on a full-measure open subset has H^{m,0}(M, K_M ⊗ E) ≠ 0 and h^{m,0}(M, K_M ⊗ E) = χ(M, K_M ⊗ E) > 0. The proof combines a spectral gap theorem for elliptic operators on Galois coverings (Theorem 5.1), the Gromov–Vafa–Witten twist, and an L² Γ_s-index computation. The paper also derives pointwise curvature bounds for Kähler metrics on such manifolds. Claudon's appendix gives an explicit description of degree-2 hyperbolic classes for finitely presented groups and shows that Kähler topologically hyperbolic surfaces are of general type.

Significance. If the main theorem is correct, it is a substantial result: a purely topological condition forces non-vanishing of holomorphic sections of adjoint bundles, without any algebraic positivity assumption on the hyperbolic class itself. The definition is natural and unifies Gromov's Kähler hyperbolicity and weak Kähler hyperbolicity, and the homotopy invariance proved in Proposition 4.5 is a strong feature. The paper is largely self-contained for the spectral tool Theorem 5.1, and the appendix by Claudon provides an explicit, checkable description of hyperbolic classes in terms of ℓ∞-cocycles, which is independently useful. There are no fitted parameters in the main derivation. The weakest point is the index-theoretic step in Proposition 7.3(3), where the L² Γ_s-index formula is imported from references whose hypotheses are not stated; this is load-bearing for Theorem 7.1 and needs to be made explicit before the main claim can be considered fully established.

major comments (3)

[§7, Prop. 7.3(3), Eq. (9)] The L² Γ_s-index formula (9) is the keystone of Theorem 7.1, but the proof is a citation to [Eys97, §7.2.2] together with the local index theorem. The group Γ_s sits in the extension 1 → U(1) → Γ_s → π1(M) → 1, so it is not a discrete group, whereas the L²-index theory of [Ati76] invoked at [Ati76, Prop. 2.4] is normally formulated for countable discrete groups. Moreover the universal cover M̃ has infinite volume, so the heat semigroup is not trace class, and the definition of the L² Γ_s-index as an integral of heat-kernel traces over a fundamental domain requires justification. Please state the exact hypotheses of [Eys97, §7.2.2] and [Ati76, Prop. 2.4] and verify them for this non-discrete central extension acting on the line-bundle fiber.
[§7, Prop. 7.3(3), spectral passage] The step 'we can find ε > 0 such that ker D^s_{E,m} ≠ {0} for each s ∈ (0, ε) and thus, thanks to [Eys97, Prop. 7.1.2], 0 ∈ σ(ð_{E,m})' is not self-contained. The cited proposition is not stated, so the reader cannot check whether its hypotheses hold when the twist is by a flat bundle with connection ∇s = ∇0 + isη on an infinite-volume cover and when the kernels are taken in the U(1)-equivariant sense. Since this is the mechanism that produces the nonzero L²-harmonic (m,0)-form, the argument should be written out or the cited result quoted with its hypotheses.
[§7, proof of Th. 7.1] The assertion that Nakano positivity on a full-measure open subset A implies H^{m,q}_{∂E}(M,E) = 0 for q ≥ 1 is used to identify h^{m,0}(M,E) with the positive Euler characteristic. Standard Nakano vanishing is usually stated under positivity everywhere; by continuity the curvature is only semipositive on the zero-measure complement. Please provide a precise reference for a Demailly-type semipositive vanishing theorem or a short argument showing the vanishing under this weaker hypothesis.

minor comments (5)

[§4, Prop. 4.5] The phrase 'M is topologically Kähler hyperbolic' should read 'M is Kähler topologically hyperbolic'.
[§5, proof of Th. 5.1] The final passage from the lower bound on im(E(1)) to all w ∈ D(P) is only justified by a page reference to [BDET24, p. 28]; since L does not commute with the spectral projection E(1), please expand the decomposition argument.
[Appendix A, Prop. A.8] In the statement of Proposition A.8, the second summand should be H²_hyp(G₂,R), not H²_hyp(G,R).
[Introduction, p. 4] The authors honestly note that no explicit examples of Kähler topologically hyperbolic manifolds beyond weakly Kähler hyperbolic ones are currently known; this limitation is worth preserving in the final version, as it contextualizes the new results.
[Remark 4.8] The chain of inclusions contains a repeated equality; a short sentence indicating which inclusions are known strict and which are conjectural would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the non-vanishing theorem follows from the external Gromov–Vafa–Witten/Eyssidieux index argument and the definition's own ∫ μ^m ≠ 0 input; only secondary birational/curvature lemmas are self-cited.

full rationale

The central derivation is not circular. Theorem 7.1 is proved from Proposition 7.3, whose keystone step (3) is the Gromov–Vafa–Witten twisting argument: the L2 Γ_s-index formula (9) is obtained by citing Gromov [Gro91], Eyssidieux [Eys97, §7.2.2 and Prop. 7.1.2], and Duistermaat [Dui11], none of which is authored by the present authors; the nonzero leading coefficient uses the hypothesis ∫_M μ^m ≠ 0 exactly as an input, not as a disguised conclusion. No parameter is fitted and no cohomology group is assumed nonzero. The paper's own previous papers (BDET24, BCDT24) are cited for peripheral but load-bearing lemmas: pullback of d-bounded forms, birational invariance of weak Kähler hyperbolicity, bigness of K_M, and existence of L2 holomorphic (m,0)-forms under weak Kähler hyperbolicity. These are prior results with independent proofs and are not used to define the target conclusion. The appended limitation note in the Introduction ('it is honest to say that we don’t dispose for the moment of explicit remarkable examples of Kähler topologically hyperbolic manifolds which are not already weakly Kähler hyperbolic') is an explicit scope caveat and not a circular step. Overall, there is no significant circularity; the score of 2 reflects the presence of several self-citations to the authors' earlier program, not a reduction of the main theorem to its own assumptions.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no new physical or geometric entities. Its central claim rests on standard theorems in spectral theory, Hodge theory, birational geometry, and index theory, plus several results from the authors' prior papers that are cited but not reproved.

assumptions (10)

domain assumption Any degree-2 hyperbolic cohomology class on a modification is the pullback of a hyperbolic class from the classifying space of the fundamental group ([BCDT24, Cor. 2.8]).
Invoked in Theorem 4.6 to prove birational invariance. This is a same-author preprint result not reproved in the paper.
domain assumption Hyperbolic classes are preserved by pullback via smooth maps and homotopy equivalences ([BDET24, Lemma 2.28]).
Used in Propositions 4.3 and 4.5 to transfer hyperbolicity along products and homotopy equivalences.
standard math A compact Kähler manifold with trivial first real Chern class has a finite étale cover that splits as T × CY × HK.
Used in Theorem 4.11 to rule out bimeromorphic targets with c1 = 0.
standard math A projective manifold that is not uniruled has pseudoeffective canonical class ([BDPP13]).
Used in Corollary 4.10 and Corollary 7.7 to conclude that K_M is pseudoeffective.
standard math Atiyah's L2-index theorem identifies χ(M,K_M⊗E) with the alternating sum of L2 Hodge numbers over the universal cover.
Used in Theorem 7.1 and Proposition 7.3 to convert spectral information into a topological Euler characteristic.
standard math A holomorphic line bundle with positive curvature on an open set of full measure is big (Siu-Demailly).
Used in Remark 7.2 and Corollary 7.7 to conclude that E is big and M is projective.
standard math For twisted Dirac operators on universal covers, non-vanishing kernels of twisted operators for small twists imply that zero lies in the spectrum of the untwisted operator ([Eys97, Prop. 7.1.2]).
Load-bearing in Proposition 7.3(3), Corollary 8.3, and Proposition 8.4.
standard math The local index theorem for twisted spin-c Dirac operators gives the index formula (9) as an integral of characteristic classes.
Used in Proposition 7.3 to show that the twisted index is a nonconstant polynomial in the twisting parameter s.
standard math Cheeger's inequality and (refined) Kato inequalities relate spectral gaps to isoperimetric and curvature data.
Used in Proposition 6.1, Corollary 6.3, Theorem 8.5, and Corollary 8.1 to bound the bottom of the spectrum and curvature terms.
standard math Hopf's theorem identifies H^2(G,R) with the kernel of the map H^2(X,R) to H^2 of the universal cover for a manifold X with fundamental group G.
Basis for the explicit description of degree-2 hyperbolic classes for finitely presented groups in Appendix A.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Geometric effects of hyperbolic cohomology classes on K\"ahler manifolds (with an appendix by Beno\^it Claudon)." pith.science (2026). https://pith.science/paper/LJYHAP57

@misc{pith2026250609907,
  author       = {Pith},
  title        = {Pith review of: Geometric effects of hyperbolic cohomology classes on K\"ahler manifolds (with an appendix by Beno\^it Claudon)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LJYHAP57}},
  note         = {Machine review of arXiv:2506.09907}
}

abstract

We introduce the notion of K\"ahler topologically hyperbolic manifold, as a"topological" generalization of K\"ahler [Gro91] and weakly K\"ahler [BDET24] hyperbolic manifolds. Analogously to [BCDT24], we show the birational invariance of this property and then that K\"ahler topologically hyperbolic manifolds are not uniruled nor bimeromorphic to compact K\"ahler manifolds with trivial first real Chern class. Then, we prove spectral gap theorems for positive holomorphic Hermitian vector bundles on K\"ahler topologically hyperbolic manifolds, obtaining in particular effective non vanishing results \`a la Kawamata for adjoint line bundles. We finally explore the effects of K\"ahler topologically hyperbolicity on Ricci and scalar curvature of K\"ahler metrics. In the appendix, it is given an explicit description of degree~$2$ hyperbolic classes for finitely presented groups, and an algebro-geometric consequence for K\"ahler topologically hyperbolic surfaces: they are necessarily of general type.

Discussion (0). Sign in to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Geometric Rigidity via Almost-Harmonic Twisted Spinors
math.DG 2026-06 unverdicted novelty 7.0 of 10

For spin manifolds with a suitable closed two-form, scalar curvature is bounded above by the bottom of the spectrum on the universal cover, with equality implying the metric is Einstein and the cover hyperbolic.
Holomorphic Deformations of Compact K\"ahler Hyperbolic Manifolds
math.AG 2025-08 unverdicted novelty 4.0 of 10

The paper proposes modified versions of Kähler hyperbolicity as a first step toward deciding whether Gromov's property is open under holomorphic deformation; the body text is unreadable as posted.

Reference graph

Works this paper leans on

34 extracted references · 32 canonical work pages · cited by 2 Pith papers

[1]

M. F. Atiyah. Elliptic operators, discrete groups and von N eumann algebras. In Colloque `` A nalyse et T opologie'' en l' H onneur de H enri C artan ( O rsay, 1974) , volume No. 32-33 of Ast\'erisque , pages 43--72. Soc. Math. France, Paris, 1976

work page 1974
[2]

ahler manifolds . ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Z\

Werner Ballmann. Lectures on K \"ahler manifolds . ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Z\"urich, 2006

work page 2006
[3]

Weak K \"ahler hyperbolicity is birational

Francesco Bei, Beno \^ t Claudon, Simone Diverio, and Stefano Trapani. Weak K \"ahler hyperbolicity is birational. arXiv e-print , 2024

work page 2024
[4]

Weakly K \"ahler hyperbolic manifolds and the G reen- G riffiths- L ang conjecture

Francesco Bei, Simone Diverio, Philippe Eyssidieux, and Stefano Trapani. Weakly K \"ahler hyperbolic manifolds and the G reen- G riffiths- L ang conjecture. J. Reine Angew. Math. , 807:257--297, 2024

work page 2024
[5]

Introduction to H odge theory , volume 8 of SMF/AMS Texts and Monographs

Jos\'e Bertin, Jean-Pierre Demailly, Luc Illusie, and Chris Peters. Introduction to H odge theory , volume 8 of SMF/AMS Texts and Monographs . American Mathematical Society, Providence, RI; Soci\'et\'e Math\'ematique de France, Paris, 2002. Translated from the 1996 French original by James Lewis and Peters

work page 2002
[6]

The pseudo-effective cone of a compact K \"ahler manifold and varieties of negative K odaira dimension

S\'ebastien Boucksom, Jean-Pierre Demailly, Mihai P a un, and Thomas Peternell. The pseudo-effective cone of a compact K \"ahler manifold and varieties of negative K odaira dimension. J. Algebraic Geom. , 22(2):201--248, 2013

work page 2013
[7]

Sobolev spaces and B ochner L aplacian on complex projective varieties and stratified pseudomanifolds

Francesco Bei. Sobolev spaces and B ochner L aplacian on complex projective varieties and stratified pseudomanifolds. J. Geom. Anal. , 27(1):746--796, 2017

work page 2017
[8]

el Broustet and Andreas H\

Ama\"el Broustet and Andreas H\"oring. Effective non-vanishing conjectures for projective threefolds. Adv. Geom. , 10(4):737--746, 2010

work page 2010

Show all 34 references

[9]

Brunnbauer, D

M. Brunnbauer, D. Kotschick, and L. Sch\"onlinner. On atoroidal and hyperbolic cohomology classes. Topology Appl. , 344:Paper No. 108830, 9, 2024

2024
[10]

David M. J. Calderbank, Paul Gauduchon, and Marc Herzlich. Refined K ato inequalities and conformal weights in R iemannian geometry. J. Funct. Anal. , 173(1):214--255, 2000

2000
[11]

Isoperimetric inequalities , volume 145 of Cambridge Tracts in Mathematics

Isaac Chavel. Isoperimetric inequalities , volume 145 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2001. Differential geometric and analytic perspectives

2001
[12]

Examples of a non-vanishing conjecture of K awamata

Yu-Lin Chang. Examples of a non-vanishing conjecture of K awamata. Internat. J. Math. , 18(5):527--533, 2007

2007
[13]

Refined K ato inequalities for harmonic fields on K \"ahler manifolds

Daniel Cibotaru and Peng Zhu. Refined K ato inequalities for harmonic fields on K \"ahler manifolds. Pacific J. Math. , 256(1):51--66, 2012

2012
[14]

Higher-dimensional algebraic geometry

Olivier Debarre. Higher-dimensional algebraic geometry . Universitext. Springer-Verlag, New York, 2001

2001
[15]

Champs magn\'etiques et in\'egalit\'es de M orse pour la d'' -cohomologie

Jean-Pierre Demailly. Champs magn\'etiques et in\'egalit\'es de M orse pour la d'' -cohomologie. Ann. Inst. Fourier (Grenoble) , 35(4):189--229, 1985

1985
[16]

Vanishing theorems for tensor powers of a positive vector bundle

Jean-Pierre Demailly. Vanishing theorems for tensor powers of a positive vector bundle. In Geometry and analysis on manifolds ( K atata/ K yoto, 1987) , volume 1339 of Lecture Notes in Math. , pages 86--105. Springer, Berlin, 1988

1987
[17]

Vanishing theorems for tensor powers of an ample vector bundle

Jean-Pierre Demailly. Vanishing theorems for tensor powers of an ample vector bundle. Invent. Math. , 91(1):203--220, 1988

1988
[18]

Vanishing theorems for square-integrable harmonic forms

Jozef Dodziuk. Vanishing theorems for square-integrable harmonic forms. Proc. Indian Acad. Sci. Math. Sci. , 90(1):21--27, 1981

1981
[19]

Demailly and H

J.-P. Demailly and H. Skoda. Relations entre les notions de positivit\'es de P . A . G riffiths et de S . N akano pour les fibr\'es vectoriels. In S\'eminaire P ierre L elong- H enri S koda ( A nalyse). A nn\'ees 1978/79 ( F rench) , volume 822 of Lecture Notes in Math. , page...

1978
[20]

auser Classics. Birkh\

J. J. Duistermaat. The heat kernel L efschetz fixed point formula for the spin- c D irac operator . Modern Birkh\"auser Classics. Birkh\"auser/Springer, New York, 2011. Reprint of the 1996 edition

2011
[21]

La caract\'eristique d' E uler du complexe de G auss- M anin

Philippe Eyssidieux. La caract\'eristique d' E uler du complexe de G auss- M anin. J. Reine Angew. Math. , 490:155--212, 1997

1997
[22]

Syst\`emes lin\'eaires adjoints L^2

Philippe Eyssidieux. Syst\`emes lin\'eaires adjoints L^2 . Ann. Inst. Fourier (Grenoble) , 49(1):vi, ix--x, 141--176, 1999

1999
[23]

M. Gromov. K\"ahler hyperbolicity and L_2 - H odge theory. J. Differential Geom. , 33(1):263--292, 1991

1991
[24]

On a conjecture of B eltrametti and S ommese

Andreas H \" o ring. On a conjecture of B eltrametti and S ommese. J. Algebraic Geom. , 21(4):721--751, 2012

2012
[25]

Scalar curvature and uniruledness on projective manifolds

Gordon Heier and Bun Wong. Scalar curvature and uniruledness on projective manifolds. Comm. Anal. Geom. , 20(4):751--764, 2012

2012
[26]

Shafarevich maps and automorphic forms

J\'anos Koll\'ar. Shafarevich maps and automorphic forms . M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1995

1995
[27]

Operators of F uchs type, conical singularities, and asymptotic methods , volume 136 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]

Matthias Lesch. Operators of F uchs type, conical singularities, and asymptotic methods , volume 136 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics] . B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1997

1997
[28]

L^2 -invariants: theory and applications to geometry and K -theory , volume 44 of Ergebnisse der Mathematik und ihrer Grenzgebiete

Wolfgang L \"u ck. L^2 -invariants: theory and applications to geometry and K -theory , volume 44 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern ...

2002
[29]

Vanishing theorems for ample vector bundles

Laurent Manivel. Vanishing theorems for ample vector bundles. Invent. Math. , 127(2):401--416, 1997

1997
[30]

Elliptic operators, topology and asymptotic methods , volume 395 of Pitman Research Notes in Mathematics Series

John Roe. Elliptic operators, topology and asymptotic methods , volume 395 of Pitman Research Notes in Mathematics Series . Longman, Harlow, second edition, 1998

1998
[31]

A vanishing theorem for semipositive line bundles over non- K \"ahler manifolds

Yum Tong Siu. A vanishing theorem for semipositive line bundles over non- K \"ahler manifolds. J. Differential Geom. , 19(2):431--452, 1984

1984
[32]

Some recent results in complex manifold theory related to vanishing theorems for the semipositive case

Yum Tong Siu. Some recent results in complex manifold theory related to vanishing theorems for the semipositive case. In Workshop B onn 1984 ( B onn, 1984) , volume 1111 of Lecture Notes in Math. , pages 169--192. Springer, Berlin, 1985

1984
[33]

Nonvanishing theorems on an algebraic variety with large fundamental group

Shigeharu Takayama. Nonvanishing theorems on an algebraic variety with large fundamental group. J. Algebraic Geom. , 8(1):181--195, 1999

1999
[34]

Hodge theory and complex algebraic geometry

Claire Voisin. Hodge theory and complex algebraic geometry. I , volume 76 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, english edition, 2007. Translated from the French by Leila Schneps

2007

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

M. F. Atiyah. Elliptic operators, discrete groups and von N eumann algebras. In Colloque `` A nalyse et T opologie'' en l' H onneur de H enri C artan ( O rsay, 1974) , volume No. 32-33 of Ast\'erisque , pages 43--72. Soc. Math. France, Paris, 1976

work page 1974

[2] [2]

ahler manifolds . ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Z\

Werner Ballmann. Lectures on K \"ahler manifolds . ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Z\"urich, 2006

work page 2006

[3] [3]

Weak K \"ahler hyperbolicity is birational

Francesco Bei, Beno \^ t Claudon, Simone Diverio, and Stefano Trapani. Weak K \"ahler hyperbolicity is birational. arXiv e-print , 2024

work page 2024

[4] [4]

Weakly K \"ahler hyperbolic manifolds and the G reen- G riffiths- L ang conjecture

Francesco Bei, Simone Diverio, Philippe Eyssidieux, and Stefano Trapani. Weakly K \"ahler hyperbolic manifolds and the G reen- G riffiths- L ang conjecture. J. Reine Angew. Math. , 807:257--297, 2024

work page 2024

[5] [5]

Introduction to H odge theory , volume 8 of SMF/AMS Texts and Monographs

Jos\'e Bertin, Jean-Pierre Demailly, Luc Illusie, and Chris Peters. Introduction to H odge theory , volume 8 of SMF/AMS Texts and Monographs . American Mathematical Society, Providence, RI; Soci\'et\'e Math\'ematique de France, Paris, 2002. Translated from the 1996 French original by James Lewis and Peters

work page 2002

[6] [6]

The pseudo-effective cone of a compact K \"ahler manifold and varieties of negative K odaira dimension

S\'ebastien Boucksom, Jean-Pierre Demailly, Mihai P a un, and Thomas Peternell. The pseudo-effective cone of a compact K \"ahler manifold and varieties of negative K odaira dimension. J. Algebraic Geom. , 22(2):201--248, 2013

work page 2013

[7] [7]

Sobolev spaces and B ochner L aplacian on complex projective varieties and stratified pseudomanifolds

Francesco Bei. Sobolev spaces and B ochner L aplacian on complex projective varieties and stratified pseudomanifolds. J. Geom. Anal. , 27(1):746--796, 2017

work page 2017

[8] [8]

el Broustet and Andreas H\

Ama\"el Broustet and Andreas H\"oring. Effective non-vanishing conjectures for projective threefolds. Adv. Geom. , 10(4):737--746, 2010

work page 2010