Hermitian Holomorphic Line Bundle Research Articles

Geometry of analytic continuation on complex manifolds – history, survey, and report

Abstract Beginning with the state of art around 1953, solutions of the Levi problem on complex manifolds will be recalled at first up to Takayama’s result in 1998. Then, the activity of extending the results by the L 2 {L}^{2} method in these decades will be reported. The method is by exploiting the finite dimensionality of certain L 2 {L}^{2} ∂ ¯ \bar{\partial } -cohomology groups to prove that a Hermitian holomorphic line bundle L L over a complex manifold M M is bimeromorphically equivalent to an ample bundle when it is restricted to a bounded locally pseudoconvex domain Ω ⋐ M \Omega \hspace{0.15em}\Subset \hspace{0.15em}M under the positivity of L ∣ ∂ Ω {L| }_{\partial \Omega } and the regularity of ∂ Ω \partial \Omega .

Asymptotic normality of divisors of random holomorphic sections on non-compact complex manifolds

We prove a central limit theorem for divisors of Gaussian i.i.d. centered holomorphic sections of tensor powers of a Hermitian holomorphic line bundle over a non-compact Hermitian manifold.

Bergman kernels and equidistribution for sequences of line bundles on Kähler manifolds

Given a sequence of positive Hermitian holomorphic line bundles (Lp,hp) on a Kähler manifold X, we establish the asymptotic expansion of the Bergman kernel of the space of global holomorphic sections of Lp, under a natural convergence assumption on the sequence of curvatures c1(Lp,hp). We then apply this to study the asymptotic distribution of common zeros of random sequences of m-tuples of sections of Lp as p→+∞.

Exponential rarefaction of maximal real algebraic hypersurfaces

Given an ample real Hermitian holomorphic line bundle $L$ over a real algebraic variety $X$, the space of real holomorphic sections of $L^{\otimes d}$ inherits a natural Gaussian probability measure. We prove that the probability that the zero locus of a real holomorphic section $s$ of $L^{\otimes d}$ defines a maximal hypersurface tends to $0$ exponentially fast as $d$ goes to infinity. This extends to any dimension a result of Gayet and Welschinger (2011) valid for maximal real algebraic curves inside a real algebraic surface. The starting point is a low degree approximation property which relates the topology of the real vanishing locus of a real holomorphic section of $L^{\otimes d}$ with the topology of the real vanishing locus a real holomorphic section of $L^{\otimes d'}$ for a sufficiently smaller $d'\<d$. Such a statement is inspired by the recent work of Diatta and Lerario (2022).

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.

A Natural Hermitian Line Bundle on the Moduli Space of Semistable Representations of a Quiver

This paper describes the construction of a natural Hermitian holomorphic line bundle on the stratified moduli space of complex representations of a finite quiver, which are semistable with respect to a fixed rational weight and have a fixed type. It is shown that the curvature of this Hermitian line bundle on each stratum of the moduli space is essentially the Kahler form of that stratum.

Universality results for zeros of random holomorphic sections

In this work we prove a 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 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 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 X .

Convexity with respect to weakly $q$-concave line bundles and reduction

We reconsider intermediate holomorphic convexity introduced by Barlet and Silva in the more general case of convexity with respect to hermitian holomorphic line bundles whose weight functions are (weakly) $q$-convex. Also we give completeness and Kahler properties of the canonical reduction.

Holomorphic aspects of moduli of representations of quivers

This article describes some complex-analytic aspects of the moduli space of the finite-dimensional complex representations of a finite quiver, which are stable with respect to a fixed rational weight. We construct a natural structure of a complex manifold on this moduli space, and a K\"ahler metric on the complex manifold. We then define a Hermitian holomorphic line bundle on the moduli space, and show that its curvature is a rational multiple of the K\"ahler form.

Flat line bundles and the Cappell-Miller torsion in Arakelov geometry

In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology side, the idea goes back to Fay's holomorphic extension of determinants of Dolbeault laplacians, and it is shown here to be equivalent to the holomorphic Cappell-Miller torsion. On the Deligne pairing side, the logarithm is a refinement of the intersection connections considered in previous work. The construction naturally leads to an Arakelov theory for flat line bundles on arithmetic surfaces and produces arithmetic intersection numbers valued in ${\mathbb C}/\pi i {\mathbb Z}$. In this context we prove an arithmetic Riemann-Roch theorem. This realizes a program proposed by Cappell-Miller to show that the holomorphic torsion exhibits properties similar to those of the Quillen metric proved by Bismut, Gillet and Soule. Finally, we give examples that clarify the kind of invariants that the formalism captures; namely, periods of differential forms.

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.

Berezin–Toeplitz quantization for lower energy forms

ABSTRACTLet 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 nondegenerate. In particular, we quantize any manifold admitting a positive line bundle. The quantum spaces are the spectral spaces corresponding to [0,k−N], where N>1 is fixed, of the Kodaira Laplace operator acting on forms with values in tensor powers Lk. We establish the asymptotic expansion of associated Toeplitz operators and their composition in the semiclassical limit k→∞ 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.

Szegő kernels and Poincaré series

Let \(M = {{\widetilde M} \mathord{\left/ {\vphantom {{\widetilde M} \Gamma }} \right. \kern-\nulldelimiterspace} \Gamma }\) be a Kahler manifold, where Γ ~ π1(M) and \(\widetilde M\) is the universal Kahler cover. Let (L, h) → M be a positive hermitian holomorphic line bundle. We first prove that the L2 Szegő projector \({\widetilde \Pi _N}\) for L2-holomorphic sections on the lifted bundle \({\widetilde L^N}\) is related to the Szegő projector for H0(M, LN) by \({\widehat \Pi _N}\left( {x,y} \right) = \sum\nolimits_{\gamma \in \Gamma } {{{\widetilde {\widehat \Pi }}_N}} \left( {\gamma \cdot x,y} \right)\). We then apply this result to give a simple proof of Napier’s theorem on the holomorphic convexity of \(\widetilde M\) with respect to \({\widetilde L^N}\) and to surjectivity of Poincare series.

Chern–Simons line bundle on Teichmüller space

Let $X$ be a non-compact geometrically finite hyperbolic 3-manifold without cusps of rank 1. The deformation space $\mc{H}$ of $X$ can be identified with the Teichmuller space $\mc{T}$ of the conformal boundary of $X$ as the graph of a section in $T^*\mc{T}$. We construct a Hermitian holomorphic line bundle $\mc{L}$ on $\mc{T}$, with curvature equal to a multiple of the Weil-Petersson symplectic form. This bundle has a canonical holomorphic section defined by $e^{\frac{1}{\pi}{\rm Vol}_R(X)+2\pi i\CS(X)}$ where ${\rm Vol}_R(X)$ is the renormalized volume of $X$ and $\CS(X)$ is the Chern-Simons invariant of $X$. This section is parallel on $\mc{H}$ for the Hermitian connection modified by the $(1,0)$ component of the Liouville form on $T^*\mc{T}$. As applications, we deduce that $\mc{H}$ is Lagrangian in $T^*\mc{T}$, and that ${\rm Vol}_R(X)$ is a Kahler potential for the Weil-Petersson metric on $\mc{T}$ and on its quotient by a certain subgroup of the mapping class group. For the Schottky uniformisation, we use a formula of Zograf to construct an explicit isomorphism of holomorphic Hermitian line bundles between $\mc{L}^{-1}$ and the sixth power of the determinant line bundle.

Critical values of random analytic functions on complex manifolds

We study the asymptotic distribution of critical values of random holomorphic `polynomials' s_n on a Kaehler manifold M as the degree n tends to infinity. By `polynomial' of degree n we mean a holomorphic section of the nth power of a positive Hermitian holomorphic line bundle $(L, h). In the special case M = CP^m and L = O(1), and h is the Fubini-Study metric, the random polynomials are the SU(m + 1) polynomials. By a critical value we mean the norm ||s_n||_h of s_n at a non-zero critical point of the norm. The metric and Kahler form endow the polynomials with a Hilbert space structure and we consider the associated Gaussian random polynomials and the spherical ensemble where ||s_n|| = 1 is chosen from Haar measure. Our main result is that the expected limit distribution of critical values as n tends to infinity in the spherical ensemble is universal (i.e. is independent of the choice of h), and we give an explicit formula for it. The limit distribution is the same as for suitably normalized Gaussian measures.

Resolution of singularities for a class of Hilbert modules

Let M be the completion of the polynomial ring C(z) under bar] with respect to some inner product, and for any ideal I subset of C (z) under bar], let I] be the closure of I in M. For a homogeneous ideal I, the joint kernel of the submodule I] subset of M is shown, after imposing some mild conditions on M, to be the linear span of the set of vectors {p(i)(partial derivative/partial derivative(w) over bar (1),...,partial derivative/partial derivative(w) over bar (m)) K-I] (., w)vertical bar(w=0), 1 <= i <= t}, where K-I] is the reproducing kernel for the submodule 2] and p(1),..., p(t) is some minimal ``canonical set of generators'' for the ideal I. The proof includes an algorithm for constructing this canonical set of generators, which is determined uniquely modulo linear relations, for homogeneous ideals. A short proof of the ``Rigidity Theorem'' using the sheaf model for Hilbert modules over polynomial rings is given. We describe, via the monoidal transformation, the construction of a Hermitian holomorphic line bundle for a large class of Hilbert modules of the form I]. We show that the curvature, or even its restriction to the exceptional set, of this line bundle is an invariant for the unitary equivalence class of I]. Several examples are given to illustrate the explicit computation of these invariants.

Exponential rarefaction of real curves with many components

Given a positive real Hermitian holomorphic line bundle L over a smooth real projective manifold X, the space of real holomorphic sections of the bundle L d inherits for every d∈ℕ∗ a L 2-scalar product which induces a Gaussian measure. When X is a curve or a surface, we estimate the volume of the cone of real sections whose vanishing locus contains many real components. In particular, the volume of the cone of maximal real sections decreases exponentially as d grows to infinity.

The distribution of normalized zero-sets of random meromorphic functions

This paper is concerned with the distribution of normalized zero-sets of random meromorphic functions. The normalization of the zero-set plays the same role as the counting function for a meromorphic function in Nevanlinna theory. The results generalize the theory of Shiffman and Zelditch on the distribution of the zeroes of random holomorphic sections of powers of positive Hermitian holomorphic line bundles. As in a very special case, our paper resembles a form of First Main Theorem in classical Nevanlinna Theory.

EFFECTIVE ŁOJASIEWICZ INEQUALITY FOR ARITHMETICALLY DEFINED VARIETIES AND A GEOMETRIC APPLICATION TO BIHOMOGENEOUS POLYNOMIALS

We establish two versions of effective Łojasiewicz inequality for arithmetically defined affine varieties. As an application, we consider bihomogeneous polynomials on the complex Euclidean space which are positive along the affine cone of an arithmetically defined projective variety, and we obtain effective estimates on certain modifications needed to turn them into sums of squares of pointwise norms of homogeneous polynomials. The latter can be interpreted as an effective result on isometric embeddings for the associated indefinite Hermitian holomorphic line bundles.

Legendrian tori and the semi-classical limit

Let X be C P n or a compact smooth quotient of the n-dimensional complex hyperbolic space, n > 1 . Let L be a hermitian holomorphic line bundle (with hermitian connection) on X chosen as follows: if X = C P n then L is the hyperplane bundle, and in the second case L is chosen so that L ⊗ ( n + 1 ) = K X ⊗ E , where K X is the canonical line bundle and E is a flat line bundle. The unit circle bundle P in L ∗ is a contact manifold. Let k ′ be a fixed positive integer. We construct certain Legendrian tori in P (the construction depends, in particular, on the choice of k ′ ) and sequences { u k } , k = k ′ m , m = 1 , 2 , … , of holomorphic sections of L ⊗ k associated to these tori. We study asymptotics of the norms ‖ u k ‖ k as m → + ∞ and, in particular, apply this result to construct explicitly certain non-trivial holomorphic automorphic forms on the n-dimensional complex hyperbolic space. We obtain an n > 1 analogue of the classical period formula (this is a well-known statement for automorphic forms on the upper half plane, n = 1 ).