arXiv research

This thesis predicts the distribution of smoothed zeros of random sections on line bundles.

The paper connects bundle curvature to random zero currents.

Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.

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 r…

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.

Study Bergman kernels and zero distributions of random sections on Kähler manifolds.

The study proves a central limit theorem for Gaussian holomorphic sections on Kähler manifolds.

In this paper we develop an integration theory for zero sets of polyfold Fredholm sections. The results are needed in the application of the polyfold theory. We use it for example in the construction of symplectic field theory.

Constructs metrics with zero eigenvalues for Hodge-Laplacian.

Study improves variance calculation for random zero sets on complex manifolds.

Formula for sections on complex manifolds with non-isolated components.

We introduce the natural and fairly general notion of a subanalytic bundle (with a finite dimensional vector space $P$ of sections) on a subanalytic subset $X$ of a real analytic manifold $M$, and prove that when $M$ is compact, there is a Baire subset $U$ of sections in $P$ whose zero-loci in $X$ have tubular neighbou…

Starting with an orientable compact real-analytic Riemannian manifold $(L,g)$ with $χ(L)=0$, we show that a small neighbourhood $\textrm{Op}(L)$ of the zero section in the cotangent bundle $T^{*}L$ carries a Calabi-Yau structure such that the zero section is an isometrically embedded special Lagrangian submanifold.

New curvature obstruction for Killing vector fields on Lorentzian manifolds.

Compactness theorem for Fueter sections yields non-zero harmonic 1-forms.

Researchers confirm conjecture for complex nilmanifolds in higher dimensions.

We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp's theorem, every torus of dimension at least three admits Riemannian metrics with negative Ricci curvature. We show that the sectional cu…

Study shows normal distribution in divisor counts of random sections on complex manifolds.

Formula for sectional curvatures on matrix groups.

We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension $-\infty$. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although th…

Establishes metrics with positive curvature on projective line bundles.

We exhibit sufficient conditions for a finite collection of periodic orbits of a Reeb flow on a closed $3$-manifold to bound a positive global surface of section with genus zero. These conditions turn out to be $C^\infty$-generically necessary. Moreover, they involve linking assumptions on periodic orbits with Conley-Z…

The paper studies random systems of holomorphic sections on compact Kähler manifolds and proves equidistribution results.

We obtain a locally symmetric Kaehler Einstein structure on the cotangent bundle of a Riemannian manifold of negative constant sectional curvature. Similar results are obtained on a tube around zero section in the cotangent bundle, in the case of a Riemannian manifold of positive constant sectional curvature. The obtai…

Paper computes Atiyah class for DG manifolds of amplitude +1.

We study harmonic sections of a Riemannian vector bundle whose total space is equipped with a 2-parameter family of metrics which includes both the Sasaki and Cheeger-Gromoll metrics. This enables the theory of harmonic unit sections to be extended to bundles with non-zero Euler class.

The paper extends metrics and solitons on toric Fano manifolds with irregular Sasaki-Einstein metrics.

Odd-dimensional manifolds have contact maps of non-zero degree.

The article confirms a complex geometry conjecture for a specific type of manifold.

Surveying random sections on Kähler manifolds, leading to metrics.

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative c…

Study of random sections on complex spaces converging to equilibrium metrics.

In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we prove a Schwarz Lemma from a complete Riemannian manifold to a complex Finsler manifold. We also show that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identical…

The paper equidistributes zeros of random polynomials and sections on manifolds.

The paper confirms a conjecture for Bismut torsion parallel metrics.

Tian's theorem connects Chern classes of bundles to random section zeros and degeneracy sets.

We mainly study 3-dimensional complete gradient Ricci solitons with positive sectional curvature, whose scalar curvature attains its maximum at some point. In section 2, we estimate the area growth of level sets and the volume growth of sublevel sets of a Ricci potential. In section 3, we show that the scalar curvature…

The Weil-Petersson metric for the moduli space of Riemann surfaces has negative sectional curvature. Surfaces represented in the complement of a compact set in the moduli space have short geodesics. At such surfaces the Weil-Petersson metric is approximately a product metric. An almost product metric has sections with …

Paper proves Chern flat for 3D Hermitian manifolds with zero real bisectional curvature.

We consider cocycles of isometries on spaces of nonpositive curvature $H$. We show that the supremum of the drift over all invariant ergodic probability measures equals the infimum of the displacements of continuous sections under the cocycle dynamics. In particular, if a cocycle has uniform sublinear drift, then there…

Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold $(X, π)$, we prove that there exists a holomorphic symplectic struct…

Rough and Hodge Laplacians eigenvalues approach zero with fixed volume.

The paper studies sections of time-like twistor spaces with specific covariant derivatives.

Suppose $(X_n)$ is a sequence of positive-dimensional smooth projective complete intersections over $\mathbb{F}_q$ with dimensions bounded from above and with characteristic zero lifts $(\tilde{X}_n)$ to smooth projective geometrically connected varieties. Suppose each complex variety $\tilde{X}^{an}_n$ has (underlying…

Let $L$ be a holomorphic line bundle over a compact Kähler manifold $X$ endowed with a singular Hermitian metric $h$ with curvature current $c_1(L,h)\geq0$. In certain cases when the wedge product $c_1(L,h)^k$ is a well defined current for some positive integer $k\leq\dim X$, we prove that $c_1(L,h)^k$ can be approxima…

Reduces Poisson manifolds with Hamiltonian Lie algebroids.

Over the moduli space of rank $n$ semi-stable lattices is a universal family of tori. Along the fibers, there are natural differential operators and differential equations, particularly, the heat equations and the Fokker-Planck equations in statistical mechanics. In this paper, we explain why, by taking averages over t…

Lorentzian Ptolemy inequality linked to curvature bounds.