We study the existence of
holomorphic sections of
invariant line bundles over Galois coverings. We show that the von
Neumann dimension of the space of
holomorphic sections is bounded
below under weak curvature conditions. We also give criteria for a compact
complex space with isolated singularities and some related strongly pseudoconcave
manifolds to be Moishezon. As applications we prove the stability of the previous
Moishezon pseudoconcave manifolds under perturbation of complex structures
as well as weak Lefschetz theorems.
Morse Inequalities on covering manifolds On holomorphic sections of bundles over weakly pseudoconvex manifolds The Dirac operator on high tensor powers of line bundles. We study the asymptotic of the spectrum of the spin-c Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin-Uribe [Asymptotic Anal. 1 (1988), no. 2, 105-113] which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin.On Moishezon spaces with isolated singularities. On compactification of hyperconcave ends and the theorems of Siu-Yau and Nadel We show that the `pseudoconcave holes' of some naturally arising class of manifolds,
called hyperconcave ends, can be filled in, including the case of complex dimension two.
As a consequence we obtain a stronger version of the compactification theorem of Siu-Yau
and extend Nadel's theorems to dimension two.If you don't have acces to the published versions from the journal web page, you could
download my local copies (Invent. Math. or CRAS)Existence of holomorphic sections and perturbation of positive line bundles over q-concave manifolds By using holomorphic Morse inequalities we prove that sufficiently small deformations of a pseudoconcave domain in a projective manifold is Moishezon.Embeddability of some strongly pseudoconvex manifolds We obtain an embedding theorem for compact strongly pseudoconvex CR manifolds which are bounadries of some complete Hermitian manifolds. We use this to compactify some negatively curved Kähler manifolds with compact strongly pseudoconvex boundary. An embedding theorem for Sasakian manifolds is also derived.Generalized Bergman kernels on symplectic manifolds We establish the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian (introduced by Guillemin-Uribe) on high tensor powers of a positive line bundle over acompact symplectic manifold. The Bergman kernel is the smooth kernel of the projection on the spectral space corresponding to small eigenvalues of the renormalized Bochner-Laplacian. We also discuss some applications (calculation of the density of states of the renormalized Bochner-Laplacian, a symplectic version of the convergence of the induced Fubini-Study metric and Kodaira embedding theorem, as well generalizations for non-compact or singular manifolds).The Laplace Operator on High tensor Powers of a Line Bundle The first coefficients of the asymptotic expansion of the Bergman kernel of the Dirac operator We establish the existence of the asymptotic expansion of the Bergman kernel associated to the spin-c Dirac operators acting on high tensor powers of line bundles with non-degenarate mixed curvature (negative and positive eigenvalues) by extending the work of Dai, Liu, Ma [ J. Differential Geom. 72 (2006), no. 1, 1-41]. We compute the second coefficient in the asymptotic expansion using the method of our paper Generalized Bergman kernels on symplectic manifolds.Holomorphic Morse Inequalities and Bergman Kernels In the present book we give a self-contained and unified approach of the holomorphic Morse inequalities and the asymptotic expansion of the Bergman kernel by using the heat kernel, and we present also various applications. Our point of view is from the local index theory,especially from the analytic localization techniques developed by Bismut-Lebeau. Basically, the holomorphic Morse inequalities are a consequence of the small time asymptotic expansion of the heat kernel. The Bergman kernel corresponds to the limit of the heat kernel when time goes to infinity, and is more sophisticate. A simple principle in this book is that the existence of the spectral gap of the operators implies the existence of the asymptotic expansion of the corresponding Bergman kernel whether the manifold is compact or not, or singular, or with boundary. Moreover, we present a general and algorithmic way to compute the coefficients in the expansion.
The book was awarded the 2006 Ferran Sunyer i Balaguer Prize. It appeared in July 2007 in the series Progress in Mathematics (Birkhäuser Verlag).Before rushing to buy the book it is wise to have a look at the Contents and Introduction.
Toeplitz operators on symplectic manifolds We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for non-compact manifolds and orbifolds are also established.Szegö kernel asymptotics and Morse inequalities on CR manifolds We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for theSzegő kernel on (0, q)-forms with values in the high tensor powers of the line bundle. This gives after integration weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities which we apply to the embedding of some convex-concave manifolds.Berezin-Toeplitz quantization on Kähler manifolds We study Berezin-Toeplitz quantization on Kähler manifolds. We explain first how to compute various associated asymptotic expansions, then we compute explicitly the first terms of the expansion of the kernel of the Berezin-Toeplitz operators, and of the composition of two Berezin-Toeplitz operators. As an application, we estimate the norm of Donaldson's Q-operator.Berezin-Toeplitz Quantization and its kernel expansion We survey recent results about the asymptotic expansion of Toeplitz operators and their kernels, as well as Berezin-Toeplitz quantization. We deal in particular with calculation of the first coefficients of these expansions.Equidistribution of zeros of holomorphic sections in the non compact setting We consider N-tensor powers of a positive Hermitian line bundle L over a non-compact complex manifold X. In the compact case, B. Shiffman and S. Zelditch proved that the zeros of random sections become asymptotically uniformly distributed with respect to the natural measure coming from the curvature of L, as N tends to infinity. Under certain boundedness assumptions on the curvature of the canonical line bundle of X and on the Chern form of L we prove a non-compact version of this result. We give various applications, including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of SL2(Z) and to the hyperbolic measure, the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity. We also give estimates for the speed of convergence of the currents of integration on the zero-divisors.Equidistribution results for singular metrics on line bundles Let L be a holomorphic line bundle with a positively curved singular Hermitian metric over a complex manifold X. One can define naturally the sequence of Fubini-Study currents associated to the space of square integrable holomorphic sections of the p-th tensor powers of L. Assuming that the singular set of the metric is contained in a compact analytic subset of X and that the logarithm of the Bergman kernel function associated to the p-th tensor power of L (defined outside the singular set) grows like o(p) as p tends to infinity, we prove the following: 1) the k-th power of the Fubini-Study currents converge weakly on the whole X to the k-th power of the curvature current of L. 2) the expectations of the common zeros of a random k-tuple of square integrable holomorphic sections converge weakly in the sense of currents to to the k-th power of the curvature current of L. Here k is so that the codimension of the singular set of the metric is greater or equal as k. Our weak asymptotic condition on the Bergman kernel function is known to hold in many cases, as it is a consequence of its asymptotic expansion. We also prove it here in a quite general setting. We then show that many important geometric situations (singular metrics on big line bundles, Kaehler-Einstein metrics on Zariski-open sets, artihmetic quotients) fit into our framework.Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degenerate. As application we obtain the Bergman kernel asymptotics for adjoint semi-positive line bundles over complete Kähler manifolds, on the set where the curvature is positive. We also prove the asymptotics for big line bundles endowed with singular Hermitian metrics with strictly positive curvature current. In this case the full asymptotics holds outside the singular locus of the metric.Convergence of Fubini-Study currents for orbifold line bundles We discuss positive closed currents and Fubini-Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini-Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curvature is strictly positive, generalizing a well-known theorem of Tian. We include applications to the asymptotic distribution of zeros of random holomorphic sections.On the approximation of positive closed currents on compact Kähler manifolds Let L be a holomorphic line bundle with a positively curved singular Hermitian metric over compact Kähler manifold X of dimension n. In certain cases when the k-th power of the curvature current of L is a well defined current for some positive integer k (less or equal than n), we prove that it can be approximated by averages of currents of integration over the common zero sets of k-tuples of holomorphic sections over X of the high powers of L. In the second part of the paper we study the convergence of the Fubini-Study currents and the equidistribution of zeros of square integrable holomorphic n-forms with values in the high powers of L. As an application, we obtain an approximation theorem for the k-th power of the curvature current of L using currents of integration over the common zero sets of k-tuples of such n-forms.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, Birkhäuser, 2007).Semi-classical properties of Berezin-Toeplitz operators with C^k symbol We obtain the semi-classical expansion of the kernels and traces of Toeplitz operators with C^k symbol on a symplectic manifold. We also give a semi-classical estimate of the distance of a Toeplitz operator to the space of self-adjoint and multiplication operators.Exponential estimate for the asymptotics of Bergman kernels We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. Further, we give Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line bundles.Scaling asymptotics of heat kernels of line bundles We consider a general Hermitian holomorphic line bundle L on a compact complex manifold M and the Kodaira Laplacian on (0,q) forms with values in L^p. The main result is a complete asymptotic expansion for the semi-classically scaled heat kernel along the diagonal. It is a generalization of the Bergman/Szegö kernel asymptotics in the case of a positive line bundle, but no positivity is assumed. We give two proofs, one based on the Hadamard parametrix for the heat kernel on a principal bundle and the second based on the analytic localization of the Dirac-Dolbeault operator.On the singularities of the Szegő projections on lower energy forms Let X be an abstract not necessarily compact orientable CR manifold of dimension 2n-1, n>1. We show that the spectral function of the Kohn Laplacian on (0,q)-forms admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if X is compact and the Levi form is non-degenerate of constant signature on X, then the spectrum of the Kohn Laplacian consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation of the associated Szegő kernel admits an asymptotic expansion under a local closed range condition. As applications, we establish the Szegő kernel asymptotic expansions on some weakly pseudoconvex CR manifolds and on CR manifolds with transversal CR circle actions. By using these asymptotics, we establish some local embedding theorems on CR manifolds and we give an analytic proof of a theorem of Lempert asserting that a compact strictly pseudoconvex CR manifold of dimension three with a transversal CR circle action can be CR embedded into a complex Euclidean space.Equdistribution and convergence speed for zeros of holomorphic sections of singular Hermitian line bundles We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.Berezin-Toeplitz quantization for lower energy forms Let M be an arbitrary complex manifold and let L be a Hermitian holomorphic line bundle over M. We introduce the Berezin-Toeplitz quantization of the open set of M where the curvature on L is non-degenerate. The quantum spaces are the spectral spaces corresponding to [0,k^{-N}] (N>1 fixed), of the Kodaira Laplace operator acting on forms with values in tensor powers L^k. We establish the asymptotic expansion of associated Toeplitz operators and their composition as k goes to infinity and we define the corresponding star-product. If the Kodaira Laplace operator has a certain spectral gap this method yields quantization by means of harmonic forms. As applications, we obtain the Berezin-Toeplitz quantization for semi-positive and big line bundles.Equidistribution for sequences of line bundles on normal Kähler spaces We study the asymptotics of Fubini-Study currents and zeros of random holomorphic sections associated to a sequence of singular Hermitian line bundles on a compact normal Kähler complex space.Szegö kernel asymptotics and Kodaira embedding theorems of Levi-flat CR manifolds Let X be an orientable compact Levi-flat CR manifold and let L be a positive CR complex line bundle over X. We prove that certain microlocal conjugations of the associated Szegő kernel admit an asymptotic expansion with respect to high powers of L. As an application, we give a Szegő kernel proof of the Kodaira type embedding theorem on Levi-flat CR manifolds due to Ohsawa and Sibony.Hölder singular metrics on big line bundles and equidistribution We show that normalized currents of integration along the common zeros of random m-tuples of sections of powers of m singular Hermitian big line bundles on a compact Kähler manifold distribute asymptotically to the wedge product of the curvature currents of the metrics. If the Hermitian metrics are Hölder with singularities we also estimate the speed of convergence.Quantum Hall effect and Quillen metric We study the generating functional, the adiabatic curvature and the adiabatic phase for the integer quantum Hall effect (QHE) on a compact Riemann surface. For the generating functional we derive its asymptotic expansion for the large flux of the magnetic field, i.e., for the large degree k of the positive Hermitian line bundle L^k. The expansion consists of the anomalous and exact terms. The anomalous terms are the leading terms of the expansion. This part is responsible for the quantization of the adiabatic transport coefficients in QHE. We then identify the anomalous part of the expansion with the Quillen metric on the determinant line bundle, and the subleading exact part with the asymptotics of the regularized spectral determinant of the Laplacian for the line bundle L^k, at large k. Finally, we show how the generating functional of the integer QHE is related to the gauge and gravitational (2+1)d Chern-Simons functionals. We observe the relation between the Bismut-Gillet-Soulé curvature formula for the Quillen metric and the adiabatic curvature for the electromagnetic and geometric adiabatic transport of the integer Quantum Hall state. Then we relate the adiabatic phase in QHE to the eta invariant and show that the geometric part of the adiabatic phase is given by the Chern-Simons functional.On the first order asymptotics of partial Bergman kernels We show that under very general assumptions the partial Bergman kernel function of sections vanishing along an analytic hypersurface has exponential decay in a neighborhood of the vanishing locus. Considering an ample line bundle, we obtain a uniform estimate of the Bergman kernel function associated to a singular metric along the hypersurface. Finally, we study the asymptotics of the partial Bergman kernel function on a given compact set and near the vanishing locus.Equivariant Kodaira embedding of CR manifolds with circle action We consider a compact CR manifold with a transversal CR locally free circle action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion. As a consequence, we establish an equivariant Kodaira embedding theorem.Bergman kernels on punctured Riemann surfaces In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincaré metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the punctured unit disc endowed with the standard Poincaré metric. One of the technical tools is a new weighted elliptic estimate near the punctures, which is uniform with respect to the tensor power. As a consequence, we obtain an optimal uniform estimate of the supremum norm of the Bergman kernel, involving a fractional growth order of the tensor power. This holds in particular for the Bergman kernel of cusp forms of high weight of non-cocompact geometrically finite Fuchsian groups of first kind without elliptic elements.On the stability of equivariant embedding of compact CR manifolds with circle action We prove the stability of the equivariant embedding of compact strictly pseudoconvex CR manifolds with transversal CR circle action under circle invariant perturbations of the CR structures.Bergman kernel asymptotics for singular metrics on punctured Riemann surfaces We consider singular metrics on a punctured Riemann surface and on a line bundle and study the behavior of the Bergman kernel in the neighbourhood of the punctures. The results have an interpretation in terms of the asymptotic profile of the density of states function of the lowest Landau level in quantum Hall effect.Approximation and equidistribution results for pseudo-effective line bundles We study the distribution of the common zero sets of m-tuples of holomorphic sections of powers of m singular Hermitian pseudo-effective line bundles on a compact Kähler manifold. As an application, we obtain sufficient conditions which ensure that the wedge product of the curvature currents of these line bundles can be approximated by analytic cycles.Optimal convergence speed of Bergman metrics on symplectic manifolds It is known that a compact symplectic manifold endowed with a prequantum line bundle can be embedded in the projective space generated by the eigensections of low energy of the Bochner Laplacian acting on high p-tensor powers of the prequantum line bundle. We show that the Fubini-Study metrics induced by these embeddings converge at speed rate 1/p^2 to the symplectic form.Donaldson's Q-operators for symplectic manifolds We prove an estimate for Donaldson's Q-operator on a prequantized compact symplectic manifold. This estimate is an ingredient in the recent result of Keller and Lejmi about a symplectic generalization of Donaldson's lower bound for the L^2-norm of the Hermitian scalar curvature.Berezin-Toeplitz quantization for eigenstates of the Bochner-Laplacian on symplectic manifolds We study the Berezin-Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic manifold. We show that this quantization has the correct semiclassical behavior and construct the corresponding star-product.Universality results for zeros of random holomorphic sections In this work we prove an universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular Hermitian holomorphic line bundles on a compact Kähler complex space X. Namely, under mild moment assumptions, we show that the asymptotic distribution of zeros of random holomorphic sections is independent of the choice of the probability measure on the space of holomorphic sections. In the case when X is a compact Kähler manifold, we also prove an off-diagonal exponential decay estimate for the Bergman kernels of a sequence of positive line bundles on X.Generalized Bergman kernels on symplectic manifolds of bounded geometry We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded geometry. First, we establish the off-diagonal exponential estimate for the generalized Bergman kernel. As an application, we obtain the relation between the generalized Bergman kernel on a Galois covering of a compact symplectic manifold and the generalized Bergman kernel on the base. Then we state the full off-diagonal asymptotic expansion of the generalized Bergman kernel, improving the remainder estimate known in the compact case to an exponential decay. Finally, we establish the theory of Berezin-Toeplitz quantization on symplectic orbifolds associated with the renormalized Bochner-Laplacian.A survey on zeros of random holomorphic sections We survey results on the distribution of zeros of random polynomials and of random holomorphic sections of line bundles, especially for large classes of probability measures on the spaces of holomorphic sections. We provide furthermore some new examples of measures supported in totally real subsets of the complex probability space.Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the relation of the Bochner Laplacian on tensor powers with the sub-Riemannian (sR) Laplacian.Geometric quantization on CR manifolds Let X be a compact connected orientable CR manifold with the action of a connected compact Lie group G. Under natural pseudoconvexity assumptions we show that the CR Guillemin-Strernberg map is Fredholm at the level of Sobolev spaces of CR functions. As application we study this map for holomorphic line bundles which are positive near the inverse image of 0 by the momentum map. We also show that ``quantization commutes with reduction'' for Sasakian manifolds.Holomorphic sections of line bundles vanishing along subvarieties We consider a compact normal complex space endowed with a holomorphic line bundle L and study the space of global holomorphic sections of tensor powers L^p of the line bundle that vanish p along an l-tuple of analytic subsets. We prove a bigness criterion for this space, analogous to Ji-Shiffman's criterion for big line bundles. We give estimates of the partial Bergman kernel, investigate the convergence of the Fubini-Study currents and their potentials, and the equilibrium distribution of normalized currents of integration along zero divisors of random holomorphic sections in this space for large p. Regularity results for the equilibrium envelope are also included.On the singularities of the Bergman projections for lower energy forms on complex manifolds with boundary Let M be a complex manifold of dimension n with smooth boundary X. We consider the dbar-Neumann Laplacian for (0,q) forms and show that its spectral kernel of admits a full asymptotic expansion near the non-degenerate part of the boundary X and the Bergman projection admits an asymptotic expansion under a certain local closed range condition. As applications, we establish the Bergman kernel asymptotic expansions for some domains with weakly pseudoconvex boundary and the S^1-equivariant Bergman kernel asymptotic expansions and embedding theorems for domains with holomorphic S^1-action.Quotient of Bergman kernels on punctured Riemann surfaces In this paper we consider a punctured Riemann surface endowed with a Hermitian metric that equals the Poincaré metric near the punctures, and a holomorphic line bundle that polarizes the metric. We show that the quotient of the Bergman kernel of high tensor powers of the line bundle and of the Bergman kernel of the Poincaré model near the singularity tends to one up to arbitrary negative powers of the tensor power.Moser-Trudinger inequalities and complex Monge-Ampère equation Our aim is to give a version of the Moser-Trudinger inequality in the setting of complex geometry. As a very particular case, our result already gives a new Moser-Trudinger inequality for functions in the Sobolev space W^{1,2} of a domain in R^2. We also deduce a new necessary condition for the complex Monge-Ampère equation for a given measure on a compact Kahler manifold to admit a Holder continuous solution.Szegő kernel asymptotics on some non-compact complete CR manifolds We establish Szegő kernel asymptotic expansions on non-compact strictly pseudoconvex complete CR manifolds with transversal CR R-action under certain natural geometric conditions.Bergman kernels and equidistribution for sequences of line bundles on Kähler manifolds Given a sequence of positive Hermitian holomorphic line bundles (L_p,h_p) on a Kähler manifold X, we establish the asymptotic expansion of the Bergman kernel of the space of global holomorphic sections of L_p, under a natural convergence assumption on the sequence of curvatures c_1(L_p,h_p). We then apply this to study the asymptotic distribution of common zeros of random sequences of m-tuples of sections of L_p as p goes to infinity.Equidistribution for weakly holomorphic sections of line bundles on algebraic curves We prove the convergence of the normalized Fubini-Study measures and the logarithms of the Bergman kernels of various Bergman spaces of holomorphic and weakly holomorphic sections associated to a singular Hermitian holomorphic line bundle on an algebraic curve. Using this, we study the asymptotic distribution of the zeros of random sequences of sections in these spaces.Large deviations for zeros of holomorphic sections on punctured Riemann surfaces In this article we obtain large deviation estimates for zeros of random holomorphic sections on punctured Riemann surfaces. These estimates are then employed to yield estimates for the respective hole probabilities. A particular case of relevance that is covered by our setting is that of cusp forms on arithmetic surfaces. Most of the results we obtain also allow for reasonably general probability distributions on holomorphic sections, which shows the universal character of these estimates. Finally, we also extend our results to the case of certain higher dimensional complete Hermitian manifolds, which are not necessarily assumed to be compact.Semi-classical Bergman kernel asymptotics on complex manifolds with boundary Let M be a relatively compact connected open subset with smooth connected boundary of a complex manifold M'. Let L be a positive line bundle over M'. Suppose that M' admits a holomorphic R-action which preserves the boundary of M and the R-action can be lifted to L. In this work, we establish an asymptotic expansion for the Bergman kernel on M with respect to high powers of L.Bergman kernel functions associated to measures supported on totally real submanifolds We prove that the Bergman kernel function associated to a smooth measure supported on a piecewise-smooth maximally totally real submanifold K in C^n is of polynomial growth (e.g, in dimension one, K is a finite union of transverse Jordan arcs in C). Our bounds are sharp when K is smooth. We give an application to equidistribution of zeros of random polynomials extending a result of Shiffman-Zelditch to the higher dimensional setting.Gaussian holomorphic sections on noncompact complex manifolds We give two constructions of Gaussian-like random holomorphic sections of a Hermitian holomorphic line bundle on a Hermitian complex manifold . In particular, we are interested in the case where the space of -holomorphic sections is infinite dimensional. We first provide a general construction of Gaussian random holomorphic sections of , which, if the space of -holomorphic sections is infinite dimensional, are almost never -integrable on . The second construction combines the abstract Wiener space theory with the Berezin-Toeplitz quantization and yields a random -holomorphic section. Furthermore, we study their random zeros in the context of semiclassical limits, including their equidistribution, large deviation estimates and hole probabilities.Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds We introduce a semi-classical approach to the spectral theory of Toeplitz operators of Boutet de Monvel and Guillemin. Let be a compact strictly pseudoconvex embeddable CR manifold and let be the Toeplitz operator on associated with a certain first order pseudodifferential operator . Using the theory of Boutet de Monvel-Sjöstrand we treat as a Fourier integral operator of order one of Melin-Sjöstrand type. We consider the operator defined by functional calculus, where is a smooth function with compact support in and . We show that admits a full asymptotic expansion as a semi-classical Fourier integral operator as . As applications, we obtain several analogues of results concerning the high powers of line bundles in complex geometry for strictly pseudoconvex CR manifolds: Kodaira embedding theorem, Tian's convergence theorem and embedding into perturbed spheres.Singular holomorphic Morse inequalities on non-compact manifolds We study 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, -convex manifolds and -concave manifolds, and we generalize related estimates of Berndtsson. We also consider the case of metrics with more general than algebraic singularities.Semi-classical spectral asymptotics of Toeplitz operators on strictly pseudodonvex domains On a relatively compact strictly pseudoconvex domain with smooth boundary in a complex manifold of dimension we consider a Toeplitz operator with symbol a Reeb-like vector field near the boundary. We show that the kernel of a weighted spectral projection where is a cut-off function with compact support in the positive real line, is a semi-classical Fourier integral operator with complex phase, hence admits a full asymptotic expansion as . More precisely, the restriction to the diagonal decays at the rate in the interior and has an asymptotic expansion on the boundary with leading term of order expressed in terms of the Levi form and the pairing of the contact form with the vector field .Restricted spaces of holomorphic sections vanishing along subvarieties Geometric quantization results for semi-positive line bundles on a Riemann surface In earlier work the authors proved the Bergman kernel expansion for semipositive line bundles over a Riemann surface whose curvature vanishes to atmost finite order at each point. Here we explore the related results and consequences of the expansion in the semipositive case including: Tian's approximation theorem for induced Fubini-Study metrics, leading order asymptotics and composition for Toeplitz operators, asymptotics of zeroes for random sections and the asymptotics of holomorphic torsion.Induced Fubini-Study metrics on strictly pseudoconvex CR manifolds and zeros of random CR functions Let be a compact strictly pseudoconvex embeddable Cauchy-Riemann manifold and let be the Toeplitz operator on associated with a first-order pseudodifferential operator . In our previous work we established the asymptotic expansion for large of the kernel of the operators , where is a smooth cut-off function supported in the positive real line and . By using these asymptotics, we show in this paper that can be projectively embedded by maps with components of the form , where is an eigenvalue of and is a corresponding eigenfunction. We establish the asymptotics of the pull-back of the Fubini-Study metric by these maps and we obtain the distribution of the zero divisors of random Cauchy-Riemann functions. We then establish a version of the Lelong-Poincaré formula for domains with boundary and obtain the distribution of the zero divisors of random holomorphic functions on strictly pseudoconvex domains.Zeros of random holomorphic sections of big line bundles with continuous metrics Let be a compact normal complex space, be a big holomorphic line bundle on and be a continuous Hermitian metric on . We consider the spaces of holomorphic sections endowed with the inner product induced by and a volume form on , and prove that the corresponding sequence of normalized Fubini-Study currents converge weakly to the curvature current of the equilibrium metric associated to . We also show that the normalized currents of integration along the zero divisors of random sequences of holomorphic sections converge almost surely to , for very general classes of probability measures on .Toeplitz operators and zeros of square-integrable random holomorphic sections We use the theory of abstract Wiener spaces to construct a probabilistic model for Berezin-Toeplitz quantization on a complete Hermitian complex manifold endowed with a positive line bundle. We associate to a function with compact support (a classical observable) a family of square-integrable Gaussian holomorphic sections. Our focus then is on the asymptotic distributions of their zeros in the semiclassical limit, in particular, we prove equidistribution results, large deviation estimates, and central limit theorems of the random zeros on the support of the given function. One of the key ingredients of our approach is the local asymptotic expansions of Berezin-Toeplitz kernels with non-smooth symbols.Tian's theorem for Moishezon spaces We prove that the Fubini-Study currents associated to a sequence of singular Hermitian holomorphic line bundles on a compact normal Moishezon space distribute asymptotically as the curvature currents of their metrics.A survey on asymptotic equilibrium distribution of zeros of random holomorphic sections This article surveys the asymptotic equilibrium distribution of the zeros of random holomorphic polynomials and of sections of large tensor powers of a positive line bundle. It concentrates on the contribution of pluripotential theory to this area, particularly in the authors' recent work.Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections Let be a compact Kähler manifold, be a positive line bundle, and be a Hermitian holomorphic vector bundle of rank on . We prove that the pullback by the Kodaira embedding associated to of the -th Chern form of the dual of the universal bundle over the Grassmannian converges as to the -th power of the Chern form , for . If we also determine the second term in the semiclassical expansion, which involves . As a consequence, we show that the limit distribution of zeros of random sequences of holomorphic sections of high powers is . Furthermore, we compute the expectation of the currents of integration along degeneracy sets of random holomorphic sections.Nakano-Griffiths inequality, holomorphic Morse inequalities, and extension theorems for q-concave domains We consider a compact complex manifold of dimension with a holomorphic line bundle that is semipositive everywhere and positive at some point, together with a smooth domain whose boundary Levi form has at least negative eigenvalues, where . Every -closed -form on the boundary, with values in a holomorphic vector bundle, is shown to have a meromorphic extension whenever . The proof combines holomorphic Morse inequalities on Levi -concave domains with the Kohn–Rossi extension theorem. The Morse inequalities are obtained from spectral spaces of the Laplacian with -Neumann boundary conditions, using a general boundary version of the Nakano–Griffiths inequality. This framework also gives a common approach to holomorphic Morse inequalities and geometric vanishing theorems for -concave and -convex manifolds or domains.Bergman kernels and Poincaré series Let be a bounded-geometry Hermitian manifold on which a discrete isometry group acts, with a quotient of finite volume. We identify the quotient's Bergman kernel with the sum, over , of the Bergman kernel upstairs. When is a Hermitian symmetric space, this identity implies that a large family of relative Poincaré series is nonzero. The result extends earlier theorems of Borthwick–Paul–Uribe and Barron (formerly Foth) to general finite-volume locally symmetric spaces.Berezin-Toeplitz Quantization of non-compact manifolds We construct Berezin–Toeplitz quantization for a noncompact Hermitian manifold , a positive holomorphic line bundle , and a Hermitian holomorphic vector bundle . If the Kodaira Laplacian on -forms with values in has a spectral gap growing linearly with , the Bergman projection onto the square-integrable holomorphic sections has the standard off-diagonal decay and a complete local expansion on compact sets as . Smooth endomorphism-valued symbols that are constant off a compact set then produce a closed Toeplitz algebra with a full composition expansion, a star product, and the expected semiclassical commutator formula. We also characterize Toeplitz families whose kernels have compact support. Fundamental estimates for give geometric spectral-gap criteria on broad classes of complete and noncompact manifolds. Finally, a Szegő-type theorem describes the eigenvalue distribution of the compact Toeplitz operators associated with bounded, compactly supported symbols.Semi-classical heat kernel asymptotics on complex manifolds with boundary Let be relatively compact in a complex manifold , with smooth boundary , and let be a holomorphic line bundle over . Assuming , we derive the semiclassical behavior near of as , where denotes the -Neumann Laplacian on -forms with coefficients in . This extends Bismut's work to complex manifolds with boundary. It also yields a heat-kernel proof of the holomorphic Morse inequalities in the boundary setting and a semiclassical Weyl law for the -Neumann Laplacian.Asymptotic expansion of induced Grassmannian Chern forms and distribution of random degeneracy sets For the Grassmannian embeddings determined by , where is positive and is a holomorphic vector bundle on a compact complex manifold, we establish a complete asymptotic expansion for the induced Grassmannian Chern forms and explicitly determine the leading coefficients. Combining the first-order expansion with Dinh and Sibony's theory of meromorphic transforms, we show on a compact Kähler manifold that the normalized integration currents over the loci where several random sections become linearly dependent converge almost surely to the appropriate power of the curvature form of , with a quantitative convergence rate. In the determinant case, we also provide a second proof based on the Wishart distribution, together with variance bounds.