Holomorphic Morse Inequalities Research Articles

Spectral kernels and holomorphic Morse inequalities for sequence of line bundles

Spectral kernels and holomorphic Morse inequalities for sequence of line bundles

On Bergman kernel functions and weak holomorphic Morse inequalities

On Bergman kernel functions and weak holomorphic Morse inequalities

SINGULAR HOLOMORPHIC MORSE INEQUALITIES ON NON-COMPACT MANIFOLDS

We obtain asymptotic estimates of the dimension of cohomology on possibly non-compact complex manifolds for line bundles endowed with Hermitian metrics with algebraic singularities. We give a unified approach to establishing singular holomorphic Morse inequalities for hyperconcave manifolds, pseudoconvex domains, q-convex manifolds and q-concave manifolds, and we generalize related estimates of Berndtsson. We also consider the case of metrics with more general than algebraic singularities.

The closures of test configurations and algebraic singularity types

Given a Kähler manifold X with an ample line bundle L, we consider the metric space of finite energy geodesic rays associated to the Chern class c1(L). We characterize rays that can be approximated by ample test configurations. At the same time, we also characterize the closure of algebraic singularity types among all singularity types of quasi-plurisubharmonic functions, pointing out the very close relationship between these two seemingly unrelated problems.By Bonavero's holomorphic Morse inequalities, the arithmetic and non-pluripolar volumes of algebraic singularity types coincide. We show that in general the arithmetic volume dominates the non-pluripolar one, and equality holds exactly on the closure of algebraic singularity types. Analogously, we give an estimate for the Monge–Ampère energy of a general finite energy ray in terms of the arithmetic volumes along its Legendre transform. Equality holds exactly for rays approximable by test configurations.Various other cohomological and potential theoretic characterizations are given in both settings. As applications, we give a concrete formula for the non-Archimedean Monge–Ampère energy in terms of asymptotic expansion, and show the continuity of the projection map from L1 rays to non-Archimedean rays.

Orbifold hyperbolicity

AbstractWe define and study jet bundles in the geometric orbifold category. We show that the usual arguments from the compact and the logarithmic settings do not all extend to this more general framework. This is illustrated by simple examples of orbifold pairs of general type that do not admit any global jet differential, even if some of these examples satisfy the Green–Griffiths–Lang conjecture. This contrasts with an important result of Demailly (Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Pure Appl. Math. Q.7(2011), 1165–1207) proving that compact varieties of general type always admit jet differentials. We illustrate the usefulness of the study of orbifold jets by establishing the hyperbolicity of some orbifold surfaces, that cannot be derived from the current techniques in Nevanlinna theory. We also conjecture that Demailly's theorem should hold for orbifold pairs with smooth boundary divisors under a certain natural multiplicity condition, and provide some evidence towards it.

Holomorphic Morse inequalities for orbifolds

We prove that Demailly’s holomorphic Morse inequalities hold true for complex orbifolds by using a heat kernel method. Then we introduce the class of Moishezon orbifolds and as an application of our inequalties, we give a geometric criterion for a compact connected orbifold to be a Moishezon orbifolds, thus generalizing Siu’s and Demailly’s answers to the Grauert–Riemenschneider conjecture to the orbifold case.

Local Demailly-Bouche's holomorphic morse inequalities

Let $(X,\omega)$ be a Hermitian manifold and let $(E,h^E)$, $(F,h^F)$ be two Hermitian holomorphic line bundle over $X$. Suppose that the maximal rank of the Chern curvature $c(E)$ of $E$ is $r$, and the kernel of $c(E)$ is foliated, i.e. there is a foliation $Y$ of $X$, of complex codimension $r$, such that the tangent space of the leaf at each point $x\in X$ is contained in the kernel of $c(E)$. In this paper, local versions of Demailly-Bouche's holomorphic Morse inequalities (which give asymptotic bounds for cohomology groups $H^{q}(X,E^k\otimes F^l)$ as $k,l,k/l\rightarrow \infty$) are presented. The local version holds on any Hermitian manifold regardless of compactness and completeness. The proof is a variation of Berman's method to derive holomorphic Morse inequalities on compact complex manifolds with boundary.

$G$-invariant holomorphic Morse inequalities

Consider an action of a connected compact Lie group on a compact complex manifold $M$, and two equivariant vector bundles $L$ and $E$ on $M$, with $L$ of rank $1$. The purpose of this paper is to establish holomorphic Morse inequalities a la Demailly for the invariant part of the Dolbeault cohomology of tensor powers of $L$ twisted by $E$. To do so, we define a moment map $\mu$ by the Kostant formula and we define the reduction of $M$ under a natural hypothesis on $\mu^{-1} (0)$. Our inequalities are given in term of the curvature of the bundle induced by $L$ on this reduction.

G-invariant holomorphic Morse inequalities

Consider an action of a connected compact Lie group on a compact complex manifold M, and two equivariant vector bundles L and E on M, with L of rank 1. In this note, we give holomorphic Morse inequalities in the spirit of Demailly for the invariant part of the Dolbeault cohomology of high tensor powers of L twisted by E, via the induced geometric data on the reduced space.

Remark on the Off-Diagonal Expansion of the Bergman Kernel on Compact Kähler Manifolds

In this short note, we compare our previous works on the off-diagonal expansion of the Bergman kernel and the recent preprint of Lu-Shiffman (arxiv.1301.2166). In particular, we note that the vanishing of the coefficient of p^{-1/2} is implicitly contained in Dai-Liu-Ma's work (J. Differential Geom. 72 (2006), no. 1, 1-41) and was explicitly stated in our book (Holomorphic Morse inequalities and Bergman kernels, Progress in Math., vol. 254, Birkh\"auser, 2007).

Szegö kernel asymptotics and Morse inequalities on CR manifolds

Let X be an abstract compact orientable CR manifold of dimension \({2n-1, n\,\geqslant\,2}\), and let L k be the k-th tensor power of a CR complex line bundle L over X. We assume that condition Y(q) holds at each point of X. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in L k, for large k. After integration, this gives weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities. We apply the strong Morse inequalities to the embedding of some convex–concave manifolds.

Holomorphic Morse Inequalities and Asymptotic Cohomology Groups: A Tribute to Bernhard Riemann

The goal of this note is to present the potential relationships between certain Monge-Ampere integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of line bundles, as recently introduced by algebraic geometers. The expected most general statements, which are still conjectural, certainly owe a debt to Riemann’s pioneering work, which led to the concept of Hilbert polynomials and to the Hirzebruch-Riemann-Roch formula during the XX-th century.

The ∂-cohomology Groups, Holomorphic Morse Inequalities, and Finite Type Conditions

The ∂-cohomology Groups, Holomorphic Morse Inequalities, and Finite Type Conditions

Book Review: Holomorphic Morse inequalities and Bergman kernels

Book Review: Holomorphic Morse inequalities and Bergman kernels

Corrigendum to: Holomorphic Morse inequalities on manifolds with boundary

Nous corrigeons l’énoncé qui affirme que les inégalités de Morse holomorphes pour un fibré hermitien en droites L sur X sont optimales tant que L s’étend comme un fibré semi-positif sur un remplissage Stein, paru dans l’article “Inégalités de Morse holomorphes sur des variétés à bord”. Nous ajoutons certaines conditions et considérons une situation plus générale.

Holomorphic Morse Inequalities on Manifolds with Boundary

Let X be a compact complex manifold with boundary and let L-k be a high power of a hermitian holomorphic line bundle over X. When X has no boundary, Demailly's holomorphic Morse inequalities give asymptotic bounds on the dimensions of the Dolbeault cohomology groups with values in Lk, in terms of the curvature of L. We extend Demailly's inequalities to the case when X has a boundary by adding a boundary term expressed as a certain average of the curvature of the line bundle and the Levi curvature of the boundary. Examples are given that show that the inequalities are sharp.

Bergman kernels and local holomorphic Morse inequalities

Let (X,ω) be a hermitian manifold and let Lk be a high power of a hermitian holomorphic line bundle over X. Local versions of Demailly’s holomorphic Morse inequalities (that give bounds on the dimension of the Dolbeault cohomology groups associated to Lk), are presented - after integration they give the usual holomorphic Morse inequalities. The local weak inequalities hold on any hermitian manifold (X,ω), regardless of compactness and completeness. The proofs, which are elementary, are based on a new approach to pointwise Bergman kernel estimates, where the kernels are estimated by a model kernel in Open image in new window

On the Instanton Complex of Holomorphic Morse Theory

Consider a holomorphic torus action on vector bundles over a complex manifold which lifts to a holomorphic vector bundle. When the connected components of the fixed-point set are partially ordered, we construct, using sheaf-theoretical techniques, two spectral sequences that converges to the twisted Dolbeault cohomology groups and those with compact support, respectively. These spectral sequences are the holomorphic counterparts of the instanton complex in standard Morse theory. The results proved imply holomorphic Morse inequalities and fixed-point formulas on a possibly non-compact manifold. Finally, a number of examples and applications are given.

ON THE VOLUME OF A LINE BUNDLE

Using the Calabi–Yau technique to solve Monge-Ampère equations, we translate a result of T. Fujita on approximate Zariski decompositions into an analytic setting and combine this to the holomorphic Morse inequalities in order to express the volume of a line bundle as the maximum of the mean curvatures of all the singular Hermitian metrics on it, with a way to pick an element at which the maximum is reached and satisfying a singular Monge–Ampère equation. This enables us to introduce the volume of any (1,1)-class on a compact Kähler manifold, and Fujita's theorem is then extended to this context.

HOLOMORPHIC QUANTIZATION FORMULA IN SINGULAR REDUCTION

We show that the holomorphic Morse inequalities proved by Tian and the author [8, 9] are in effect equalities by refining the analytic arguments in [8, 9].