arXiv subjects

Bingxiao Liu

Publications and source records attributed to Bingxiao Liu.

10 recordsLinked to original sources

arXiv2026

Nakano-Griffiths inequality, holomorphic Morse inequalities, and extension theorems for $q$-concave domains

We establish a general Nakano-Griffiths inequality with boundary conditions and apply it to derive holomorphic Morse inequalities for domains satisfying analytic convexity assumptions. The proof of these inequalities relies on analyzing the spectral spaces of the Laplace operator with $\overline{\partial }$-Neumann boundary conditions. As an application, we obtain a criterion for Moishezon $1$-concave manifolds. We further apply the holomorphic Morse inequalities on Levi $q$-concave domains in conjunction with the Kohn-Rossi extension theorem to obtain the following result. Let $X$ be a compact complex manifold of dimension $n$ equipped with a holomorphic line bundle that is semi-positive everywhere and positive at least at one point, and let $D\subset X$ be a smooth $q$-concave domain ($1 \le q \le n-1$). We prove that every $\bar{\partial }_{b}$-closed $(0,\ell )$-form on $bD$ with values in a holomorphic vector bundle, admits a meromorphic extension to $D$ for all $q \le \ell \le n-1$.

math.CV↗

arXiv2026

Asymptotic expansion of induced Grassmannian Chern forms and distribution of random degeneracy sets

For the Grassmannian embeddings defined by the spaces $H^0(X,L^p\otimes E)$, where $L$ is a positive line bundle and $E$ is a holomorphic vector bundle over a compact complex manifold, we prove a complete asymptotic expansion of the induced Grassmannian Chern forms and compute the first coefficients explicitly. As an application of the first-order asymptotics and of the theory of meromorphic transforms by Dinh and Sibony, we prove that on a compact Kähler manifold, the normalized currents of integration over the loci where several random sections become linearly dependent converge almost surely to the corresponding power of the curvature form of the positive line bundle, with a quantitative estimate for the speed of convergence. Moreover, in the determinant case, we additionally present an alternative method based on the Wishart distribution, together with variance estimates.

math.CV↗

arXiv2025

Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections

Let $(X,ω)$ be a compact Kähler manifold, $(L,h^L)$ be a positive line bundle, and $(E,h^E)$ be a Hermitian holomorphic vector bundle of rank $r$ on $X$. We prove that the pullback by the Kodaira embedding associated to $L^p\otimes E$ of the $k$-th Chern class of the dual of the universal bundle over the Grassmannian converges as $p\to\infty$ to the $k$-th power of the Chern form $c_1(L,h^L)$, for $0\leq k\leq r$. If $c_1(L,h^L)=ω$ we also determine the second term in the semiclassical expansion, which involves $c_1(E,h^E)$. As a consequence we show that the limit distribution of zeros of random sequences of holomorphic sections of high powers $L^p\otimes E$ is $c_1(L,h^L)^r$. Furthermore, we compute the expectation of the currents of integration along degeneracy sets of random holomorphic sections.

math.CV↗

arXiv2024

Semipositive line bundles on punctured Riemann surfaces: Bergman kernels and random zeros

We give an extensive study on the Bergman kernel expansions and the random zeros associated with the high tensor powers of a semipositive line bundle on a complete punctured Riemann surface. We prove several results for the zeros of Gaussian holomorphic sections in the semi-classical limit, including the equidistribution, large deviation estimates, central limit theorem, and number variances.

math.CV↗

arXiv2024

Integral invariants for framed 3-manifolds associated to trivalent graphs possibly with self-loops

Bott--Cattaneo's theory defines the integral invariants for a framed rational homology 3-sphere equipped with an acyclic orthogonal local system, in terms of graph cocycles without self-loops. The 2-loop term of their invariants is associated with the theta graph. Their definition requires a cohomological condition. Cattaneo--Shimizu removed this cohomological condition and gave a 2-loop invariant associated with a linear combination of the theta graph and the dumbbell graph, the 2-loop trivalent graph with self-loops. In this paper, we are concerned with an acyclic local system given by the adjoint representation of a semi-simple Lie group composed with a representation of the fundamental group of a closed 3-manifold, and we show that through a cohomological construction eventually the integral associated with the dumbbell graph vanishes. Based on this idea, we construct a theory of graph complexes and cocycles, so that higher-loop invariants can be defined by two different but equivalent methods: the graph cocycles without self-loops as in Bott--Cattaneo's theory, and the ones with self-loops that extend Cattaneo--Shimizu's 2-loop invariants. As a consequence, we prove that the generating series of Chern--Simons perturbation theory gives rise to topological invariants for framed 3-manifolds in our setting, which admits a formula in terms of only trivalent graphs without self-loops.

math.GT↗

arXiv2024

On full asymptotics of analytic torsions for compact locally symmetric orbifolds

We consider a certain sequence of flat vector bundles on a compact locally symmetric orbifold, and we evaluate explicitly the associated asymptotic Ray-Singer real analytic torsion. The basic idea is to computing the heat trace via Selberg's trace formula, so that a key point in this paper is to evaluate the orbital integrals associated with nontrivial elliptic elements. For that purpose, we deduce a geometric localization formula, so that we can rewrite an elliptic orbital integral as a sum of certain identity orbital integrals associated with the centralizer of that elliptic element. The explicit geometric formula of Bismut for semisimple orbital integrals plays an essential role in these computations.

math.DG↗

arXiv2024

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.

math.CV↗

arXiv2023

Gaussian holomorphic sections on noncompact complex manifolds

We give two constructions of Gaussian-like random holomorphic sections of a Hermitian holomorphic line bundle $(L,h_{L})$ on a Hermitian complex manifold $(X,Θ)$. In particular, we are interested in the case where the space of $\mathcal{L}^2$-holomorphic sections $H^{0}_{(2)}(X,L)$ is infinite dimensional. We first provide a general construction of Gaussian random holomorphic sections of $L$, which, if $\dim H^{0}_{(2)}(X,L)=\infty$, are almost never $\mathcal{L}^2$-integrable on $X$. The second construction combines the abstract Wiener space theory with the Berezin-Toeplitz quantization and yields a random $\mathcal{L}^2$-holomorphic section. Furthermore, we study their random zeros in the context of semiclassical limits, including their equidistribution, large deviation estimates and hole probabilities.

math.CV↗

arXiv2022

Hypoelliptic Laplacian and twisted trace formula

We give an explicit geometric formula for the twisted orbital integrals using the method of the hypoelliptic Laplacian developed by Bismut. Combining with the twisted trace formula, we can evaluate the equivariant trace of the heat operators of the Laplacians on a compact locally symmetric space. In particular, we revisit the equivariant local index theorems and twisted $L_{2}$-torsions for locally symmetric spaces.

math.DG↗

arXiv2021

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.

math.CV↗