arXiv subjects

Chin-Yu Hsiao

Publications and source records attributed to Chin-Yu Hsiao.

At least 19 recordsLinked to original sources

arXiv2026

Localization of Bergman Kernels and the Cheng-Yau Conjecture on Real Analytic Pseudoconvex Domains

In this paper, we establish the localization of Bergman kernels for unbounded pseudoconvex domains near boundary points of finite D'Angelo type. This result was proved by Engliš more than twenty years ago for bounded pseudoconvex domains and had remained as an open question in the unbounded setting. Related foundational work was carried out by Fefferman, Kerzman, Boutet de Monvel--Sjöstrand, Boas, Bell, and others. Combining our localization theorem with an extension theorem of Mir--Zaitsev, we prove that the Bergman metric of a bounded pseudoconvex domain with real-analytic boundary is Einstein if and only if the domain is biholomorphic to the unit ball. This result advances a longstanding conjecture of Cheng and Yau. A key step is to show that the Bergman metric of a smooth, possibly unbounded, pseudoconvex domain cannot be Kähler--Einstein if its boundary contains a non-strongly pseudoconvex $h$-extendible point. We further prove that a bounded weakly pseudoconvex real-analytic domain with a Kähler--Einstein Bergman metric must possess a weakly pseudoconvex $h$-extendible boundary point, thereby reducing the problem to the $h$-extendible setting. This paper draws deeply on, and reveals essential connections among many subfields of microlocal analysis, several complex variables, and complex geometry.

math.CV↗

arXiv2026

Semi-classical heat kernel asymptotics on complex manifolds with boundary

Let $M$ be a relatively compact open subset of a complex manifold $M'$ with smooth boundary $X$ and let $L$ be a holomorphic line bundle over $M'$. Assuming that condition $Z(q)$ holds, we establish the semi-classical asymptotic behavior of $e^{-\frac{t}{k}\Box^{q}_k}$ near the boundary $X$ as $k\to\infty$, where $\Box^{q}_k$ is the $\bar{\partial}$-Neumann Laplacian acting on $(0,q)$-forms on $M$ with values in $L^k$. Our results extend the seminal work of Bismut to complex manifolds with boundary. As applications of our results, we provide a heat kernel-based proof of the holomorphic Morse inequalities for complex manifolds with boundary and derive a semi-classical Weyl law for the $\bar{\partial}$-Neumann Laplacian.

math.CV↗

arXiv2026

Heat kernel asymptotics and analytic torsion on non-degenerate CR manifolds

The existence of small-time asymptotics for the heat kernel of the Kohn Laplacian on a general CR manifold has remained an open problem. In this paper, we resolve the problem in the non-degenerate case. More precisely, let $X$ be a compact oriented CR manifold of dimension $2n+1$, $n \ge 1$, with a nondegenerate Levi form of constant signature $(n_-, n_+)$. Suppose that condition $Y(q)$ holds at each point of $X$, we establish the small-time asymptotics of the heat kernel of Kohn Laplacian. Suppose that condition $Y(q)$ fails, we establish the small-time asymptotics of the kernel of the difference of the heat operator and Szegő projector. As an application we define the analytic torsion on compact oriented nondegenerate CR manifolds and study its dependence on changes of the metrics. Let $L^k$ be the $k$-th power of a CR complex line bundle $L$ over $X$. We establish the asymptotics, as $k \to \infty$, of the analytic torsion with values in $L^k$, under a variant of spectral gap condition. Furthermore, when $X$ admits a transversal CR $S^1$-action, we establish the small-time asymptotics of the $S^1$-equivariant heat kernel of the Kohn Laplacian with values in $L^k$. As an application we define the $S^1$-equivariant Quillen metric with values in $L^k$ and study its dependence on changes of the metrics. Finally, we establish the asymptotics, as $k \to \infty$, of the $S^1$-equivariant analytic torsion with values in $L^k$.

math.DG↗

arXiv2025

On the distribution kernels of Toeplitz operators on CR manifolds

We study the distribution kernel of a Toeplitz operator associated with a classical pseudodifferential operator on a compact, embeddable, strictly pseudoconvex CR manifold. The main result consists of a formula for the values at the diagonal of the second coefficient in the expansion of the symbol of the kernel. We also establish asymptotic expansions for Toeplitz operators on the positive part of a compact not necessary strictly pseudoconvex CR orbifold under certain natural assumptions.

math.CV↗

arXiv2025

Approximation of pseudohermitian structures via embeddings into spheres

Let $(X,T^{1,0}X)$ be a compact strictly pseudoconvex CR manifold which is CR embeddable into the complex Euclidean space. We show that $T^{1,0}X$ can be approximated in $\mathscr{C}^\infty$-topology by a sequence of strictly pseudoconvex CR structures $\{\mathcal{V}^k\}_{k\in \mathbb N}$ such that each $(X,\mathcal{V}^k)$ is CR embeddable into the unit sphere of a complex Euclidean space. Furthermore, as a refinement of this statement, we show that given a one form $α$ on $X$ such that $(X,T^{1,0}X,α)$ is a pseudohermitian manifold we can approximate $(T^{1,0}X,α)$ in $\mathscr{C}^\infty$-topology by a sequence of pseudohermitian structures $\{(\mathcal{V}^k,α^k)\}_{k\in \mathbb N}$ on $X$ such that for each $k\in \mathbb N$ we have that $(X,\mathcal{V}^k,α^k)$ is isomorphic to a real analytic pseudohermitian submanifold of a sphere. A similar result for the Sasakian case was obtained earlier by Loi-Placini. Let $(X,T^{1,0}X,\mathcal{T})$ be a compact Sasakian manifold, i.e. $\mathcal{T}$ is a transversal CR vector field and the one form $α$ defined by $α(\mathcal{T})=1$ and $α(\operatorname{Re}T^{1,0}X)=0$ defines a pseudohermitian structure on $(X,T^{1,0}X)$. Loi-Placini showed that $(T^{1,0}X,\mathcal{T})$ can be smoothly approximated by a sequence of quasi-regular Sasakian structures $\{(\mathcal{V}^k,\mathcal{T}^k)\}_{k\in \mathbb N}$ on $X$ such that each $(X,\mathcal{V}^k,\mathcal{T}^k)$ admits a smooth equivariant CR embedding into a Sasakian sphere. Applying our methods to the Sasakian case we show that it is possible to approximate with a sequence of Sasakian structures having the form $\{(\mathcal{V}^k,\mathcal{T})\}_{k\in \mathbb N}$, i.e. we can keep the vector field $\mathcal{T}$. Further applications concerning Sasakian deformations, the embedding of domains into balls and local approximation results are provided.

math.CV↗

arXiv2025

On the second coefficient in the semi-classical expansion of Toeplitz Operators

Let $X$ be a compact strictly pseudoconvex embeddable CR manifold and let $A$ be the Toeplitz operator on $X$ associated with a Reeb vector field $\mathcal{T}\in\mathscr{C}^\infty(X,TX)$. Consider the operator $χ_k(A)$ defined by functional calculus of $A$, where $χ$ is a smooth function with compact support in the positive real line and $χ_k(λ):=χ(k^{-1}λ)$. It was established recently that $χ_k(A)(x,y)$ admits a full asymptotic expansion in $k$. The second coefficient of the expansion plays an important role in the further study of CR geometry. In this work, we calculate the second coefficient of the expansion.

math.CV↗

arXiv2025

Heat kernel and local index theorem for open complex manifolds with $\mathbb{C}^{\ast }$-action

For a complex manifold $Σ$ with $\mathbb{C}^{\ast }$-action, we define the $m$-th $\mathbb{C}^{\ast }$ Fourier-Dolbeault cohomology group and consider the $m$-index on $Σ$. By applying the method of transversal heat kernel asymptotics, we obtain a local index formula for the $m$-index. We can reinterpret Kawasaki's Hirzebruch-Riemann-Roch formula for a compact complex orbifold with an orbifold holomorphic line bundle by our integral formulas over a (smooth) complex manifold and finitely many complex submanifolds arising from singular strata. We generalize $\mathbb{C}^{\ast }$-action to complex reductive Lie group $G$-action on a compact or noncompact complex manifold. Among others, we study the nonextendability of open group action and the space of all $G$-invariant holomorphic $p$-forms. Finally, in the case of two compatible holomorphic $\mathbb{C}^{\ast }$-actions, a mirror-type isomorphism is found between two linear spaces of holomorphic forms, and the Euler characteristic associated with these spaces can be computed by our $\mathbb{C}^{\ast }$ local index formula on the total space. In the perspective of the equivariant algebraic cobordism theory $Ω_{\ast }^{\mathbb{C}^{\ast }}(Σ),$ a speculative connection is remarked. Possible relevance to the recent development in physics and number theory is briefly mentioned.

math.DG↗

arXiv2025

Quantization and reduction for torsion free CR manifolds

Consider a compact torsion free CR manifold $X$ and assume that $X$ admits a compact CR Lie group action $G$. Let $L$ be a $G$-equivariant rigid CR line bundle over $X$. It seems natural to consider the space of $G$-invariant CR sections in the high tensor powers as quantization space, on which a certain weighted $G$-invariant Fourier-Szegő operator projects. Under certain natural assumptions, we show that the group invariant Fourier-Szegő projector admits a full asymptotic expansion. As an application, if the tensor power of the line bundle is large enough, we prove that quantization commutes with reduction.

math.CV↗

arXiv2024

Induced Fubini-Study metrics on strictly pseudoconvex CR manifolds and zeros of random CR functions

Let $X$ be a compact strictly pseudoconvex embeddable Cauchy-Riemann manifold and let $T_P$ be the Toeplitz operator on $X$ associated with a first-order pseudodifferential operator $P$. In our previous work we established the asymptotic expansion for $k$ large of the kernel of the operators $χ(k^{-1}T_P)$, where $χ$ is a smooth cut-off function supported in the positive real line. By using these asymptotics, we show in this paper that $X$ can be projectively embedded by maps with components of the form $χ(k^{-1}λ)f_λ$, where $λ$ is an eigenvalue of $T_P$ and $f_λ$ 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.

math.CV↗

arXiv2023

Semi-classical asymptotics of partial Bergman kernels on $\mathbb{R}$-symmetric complex manifolds with boundary

Let $M$ be a relatively compact connected open subset with smooth connected boundary of a complex manifold $M'$. Let $(L,h^L)\rightarrow M'$ be a positive line bundle over $M'$. Suppose that $M'$ admits a holomorphic $\mathbb{R}$-action which preserves the boundary of $M$ and lifts to $L$. We establish the asymptotic expansion of a partial Bergman kernel associated to a package of Fourier modes of high frequency with respect to the $\mathbb{R}$-action in the high powers of $L$. As an application, we establish an $\mathbb{R}$-equivariant analogue of Fefferman's and Bell-Ligocka's result about smooth extension up to the boundary of biholomorphic maps between weakly pseudoconvex domains in $\mathbb{C}^n$. Another application concerns the embedding of pseudoconcave manifolds.

math.CV↗

arXiv2023

Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds

Let $X$ be a compact strictly pseudoconvex embeddable CR manifold and let $T_P$ be the Toeplitz operator on $X$ associated with some first order pseudodifferential operator $P$. We consider $χ_k(T_P)$ the functional calculus of $T_P$ by any rescaled cut-off function $χ$ with compact support in the positive real line. In this work, we show that $χ_k(T_P)$ admits a full asymptotic expansion as $k\to+\infty$. As applications, we obtain several CR analogous of results concerning high power of line bundles in complex geometry but without any group action assumptions on the CR manifold. In particular, we establish a Kodaira type embedding theorem, Tian's convergence theorem and a perturbed spherical embedding theorem for strictly pseudoconvex CR manifolds.

math.CV↗

arXiv2023

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 $n$ we consider a Toeplitz operator $T_R$ with symbol a Reeb-like vector field $R$ near the boundary. We show that the kernel of a weighted spectral projection $χ(k^{-1}T_R)$, where $χ$ 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 $k\to+\infty$. More precisely, the restriction to the diagonal $χ(k^{-1}T_R)(x,x)$ decays at the rate $O(k^{-\infty})$ in the interior and has an asymptotic expansion on the boundary with leading term of order $k^{n+1}$ expressed in terms of the Levi form and the pairing of the contact form with the vector field $R$.

math.CV↗

arXiv2023

$G$-invariant Bergman kernel and geometric quantization on complex manifolds with boundary

Let $M$ be a complex manifold with boundary $X$, which admits a holomorphic Lie group $G$-action preserving $X$. We establish a full asymptotic expansion for the $G$-invariant Bergman kernel under certain assumptions. As an application, we get $G$-invariant version of Fefferman's result about regularity of biholomorphic maps on strongly pseudoconvex domains of $\mathbb C^n$. Moreover, we show that the Guillemin-Sternberg map on a complex manifold with boundary is Fredholm by developing reduction to boundary technique, which establish ``quantization commutes with reduction" in this case.

math.CV↗

arXiv2022

Embedding theorems for quantizable pseudo-Kähler manifolds

Given a compact quantizable pseudo-Kähler manifold $(M,ω)$ of constant signature, there exists a Hermitian line bundle $(L,h)$ over $M$ with curvature $-2πi\,ω$. We shall show that the asymptotic expansion of the Bergman kernels for $L^{\otimes k}$-valued $(0,q)$-forms implies more or less immediately a number of analogues of well-known results, such as Kodaira embedding theorem and Tian's almost-isometry theorem.

math.DG↗

arXiv2022

Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds

Given a CR manifold with non-degenerate Levi form, we show that the operators of the functional calculus for Toeplitz operators are complex Fourier integral operators of Szegő type. As an application, we establish semi-classical spectral dimensions for Toeplitz operators. We then consider a CR manifold with a compact Lie group action $G$ and we establish quantization commutes with reduction for Toeplitz operators. Moreover, we also compute semi-classical spectral dimensions for $G$-invariant Toeplitz operators.

math.FA↗

arXiv2022

On the singularities of the Szegő kernels on CR orbifolds

In this paper we study the microlocal properties of the Szegő kernel of a given compact connected orientable CR orbifold whose Kohn Laplacian has closed range. This last assumption is satisfied if certain geometric conditions hold true, as in the smooth case. As applications, we give a pure analytic proof of Kodaira-Bailey theorem and explain how to generalize a CR version of quantization commutes with reduction to orbifolds.

math.CV↗

arXiv2021

Toeplitz operators on CR manifolds and group actions

Let $(X, T^{1,0}X)$ be a connected orientable compact CR manifold of dimension $2n+1$, $n \geq 1$ with non-degenerate Levi curvature. In this paper, we study the algebra of Toeplitz operators on $X$ and we establish star product for some class of symbols on $X$. In the second part of this paper, we consider a compact locally free Lie group $G$ acting on $X$ and we investigate the associated algebra of $G$-invariant Toeplitz operators.

math.CV↗

arXiv2021

Heat kernel asymptotics for Kohn Laplacians on CR manifolds

Let $X$ be an abstract orientable not necessarily compact CR manifold of dimension $2n+1$, $n\geq1$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Suppose that condition $Y(q)$ holds at each point of $X$, we establish asymptotics of the heat kernel of Kohn Laplacian with values in $L^k$. As an application, we give a heat kernel proof of Morse inequalities on compact CR manifolds. When $X$ admits a transversal CR $\mathbb R$-action, we also establish asymptotics of the $\mathbb R$-equivariant heat kernel of Kohn Laplacian with values in $L^k$. As an application, we get $\mathbb R$-equivariant Morse inequalities on compact CR manifolds with transversal CR $\mathbb R$-action.

math.CV↗