arXiv subjects

Rung-Tzung Huang

Publications and source records attributed to Rung-Tzung Huang.

At least 19 recordsLinked to original sources

arXiv2026

Quantization Commutes with Reduction for Embeddable CR Reductions

We prove a canonical quantization-commutes-with-reduction theorem for CR functions. Under the geometric hypotheses of Hsiao--Ma--Marinescu, if the zero moment level is nonempty and the strictly pseudoconvex reduction is globally CR embeddable of real dimension at least three, the CR Guillemin--Sternberg map is a bounded isomorphism at every real Sobolev order. The central new step is an algebraic and complex-analytic elimination of its finite-dimensional Fredholm defect: multiplicativity removes the kernel, while the conductor of the smooth range, integral dependence, and holomorphic removability produce global lifts. Conductor nonvanishing follows from the reduced Szegő singularity; we also prove it using global CR peak functions. The map is an isomorphism of Fréchet algebras, the reduction is connected, and the corresponding maps for holomorphic line bundles positive near the zero level are isomorphisms in every nonnegative tensor degree. Invariant holomorphic sections vanish in every negative degree; when the reduced base has positive complex dimension, the section maps are isomorphisms in every integer degree.

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↗

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↗

arXiv2020

Asymptotics of G-equivariant Szegő kernels

Let $(X, T^{1,0}X)$ be a compact connected orientable CR manifold of dimension $2n+1$ with non-degenerate Levi curvature. Assume that $X$ admits a connected compact Lie group $G$ action. Under certain natural assumptions about the group $G$ action, we define $G$-equivariant Szegő kernels and establish the associated Boutet de Monvel-Sjöstrand type theorems. When $X$ admits also a transversal CR $S^1$ action, we study the asymptotics of Fourier components of $G$-equivariant Szegő kernels with respect to the $S^1$ action.

math.CV↗

arXiv2020

On the coefficients of the equivariant Szegő kernel asymptotic expansions

Let $(X, T^{1,0}X)$ be a compact connected orientable strongly pseudoconvex CR manifold of dimension $2n+1$, $n\geq1$. Assume that $X$ admits a connected compact Lie group $G$ action and a transversal CR $S^1$ action, we compute the coefficients of the first two lower order terms of the equivariant Szegő kernel asymptotic expansions with respect to the $S^1$ action.

math.CV↗

arXiv2020

$S^1$-equivariant Index theorems and Morse inequalities on complex manifolds with boundary

Let $M$ be a complex manifold of dimension $n$ with smooth connected boundary $X$. Assume that $\overline M$ admits a holomorphic $S^1$-action preserving the boundary $X$ and the $S^1$-action is transversal on $X$. We show that the $\overline\partial$-Neumann Laplacian on $M$ is transversally elliptic and as a consequence, the $m$-th Fourier component of the $q$-th Dolbeault cohomology group $H^q_m(\overline M)$ is finite dimensional, for every $m\in\mathbb Z$ and every $q=0,1,\ldots,n$. This enables us to define $\sum^{n}_{j=0}(-1)^j{\rm dim\,}H^q_m(\overline M)$ the $m$-th Fourier component of the Euler characteristic on $M$ and to study large $m$-behavior of $H^q_m(\overline M)$. In this paper, we establish an index formula for $\sum^{n}_{j=0}(-1)^j{\rm dim\,}H^q_m(\overline M)$ and Morse inequalities for $H^q_m(\overline M)$.

math.CV↗

arXiv2020

Morse inequalities for Fourier components of Kohn-Rossi cohomology of CR covering manifolds with $S^1$-action

Let $X$ be a compact connected CR manifold of dimension $2n+1, n \geq 1$. Let $\widetilde{X}$ be a paracompact CR manifold with a transversal CR $S^1$-action, such that there is a discrete group $Γ$ acting freely on $\widetilde{X}$ having $X \, = \, \widetilde{X}/Γ$. Based on an asymptotic formula for the Fourier components of the heat kernel with respect to the $S^1$-action, we establish the Morse inequalities for Fourier components of reduced $L^2$-Kohn-Rossi cohomology with values in a rigid CR vector bundle over $\widetilde{X}$. As a corollary, we obtain the Morse inequalities for Fourier components of Kohn-Rossi cohomology on $X$ which were obtained by Hsiao-Li by using Szegö kernel method.

math.CV↗

arXiv2017

The refined analytic torsion and a well-posed boundary condition for the odd signature operator

In this paper we discuss the refined analytic torsion on an odd dimensional compact oriented Riemannian manifold with boundary under some assumption. For this purpose we introduce two boundary conditions which are complementary to each other and well-posed for the odd signature operator $\mathcal{B}$ in the sense of Seeley. We then show that the zeta-determinants of $\mathcal{B}^2$ and eta-invariants of $\mathcal{B}$ subject to these boundary conditions are well defined by using the method of the asymptotic expansions of the traces of the heat kernels. We use these facts to define the refined analytic torsion on a compact manifold with boundary and show that it is invariant on the change of metrics in the interior of the manifold. We finally describe the refined analytic torsion under these boundary conditions as an element of the determinant line.

math.DG↗

arXiv2017

The anomaly formula of the analytic torsion on CR manifolds with $S^1$ action

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n\ge 1$ with a transversal CR $S^1$-action on $X$. In this paper we introduce the Quillen metric on the determinant line of the Fourier components of the Kohn-Rossi cohomology on $X$ with respect to the $S^1$-action. We study the behavior of the Quillen metric under the change of the metrics on the manifold $X$ and on the vector bundle over $X$. We obtain an anomaly formula for the Quillen metric on $X$ with respect to the $S^1$-action.

math.DG↗

arXiv2017

$G$-invariant Szegö kernel asymptotics and CR reduction

Let $(X, T^{1,0}X)$ be a compact connected orientable CR manifold of dimension $2n+1$ with non-degenerate Levi curvature. Assume that $X$ admits a connected compact Lie group action $G$. Under certain natural assumptions about the group action $G$, we show that the $G$-invariant Szegö kernel for $(0,q)$ forms is a complex Fourier integral operator, smoothing away $μ^{-1}(0)$ and there is a precise description of the singularity near $μ^{-1}(0)$, where $μ$ denotes the CR moment map. We apply our result to the case when $X$ admits a transversal CR $S^1$ action and deduce an asymptotic expansion for the $m$-th Fourier component of the $G$-invariant Szegö kernel for $(0,q)$ forms as $m \to+\infty$. As an application, we show that if $m$ large enough, quantization commutes with reduction.

math.DG↗

arXiv2016

The asymptotics of the analytic torsion on CR manifolds with $S^1$ action

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n \ge 1$ with a transversal CR $S^1$-action on $X$. We introduce the Fourier components of the Ray-Singer analytic torsion on $X$ with respect to the $S^1$-action. We establish an asymptotic formula for the Fourier components of the analytic torsion with respect to the $S^1$-action. This generalizes the asymptotic formula of Bismut and Vasserot on the holomorphic Ray-Singer torsion associated with high powers of a positive line bundle to strongly pseudoconvex CR manifolds with a transversal CR $S^1$-action.

math.DG↗

arXiv2014

Lefschetz fixed point formula on a compact Riemannian manifold with boundary for some boundary conditions

In [8] the authors introduced a pair of new de Rham complexes on a compact oriented Riemannian manifold with boundary by using a pair of new boundary conditions to discuss the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the Lefschetz fixed point formula on these complexes with respect to a smooth map having simple fixed points and satisfying some special condition near the boundary. For this purpose we are going to use the heat kernel method for the Lefschetz fixed point formula.

math.DG↗

arXiv2013

The gluing formula of the zeta-determinants of Dirac Laplacians for certain boundary conditions

The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result of [15] to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$, ${\mathcal P}_{+, {\mathcal L}_{1}}$. We next consider a double of de Rham complexes consisting of differential forms of all degrees with the absolute and relative boundary conditions. Using a similar method, we prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the absolute and relative boundary conditions.

math.DG↗

arXiv2012

The comparison of two constructions of the refined analytic torsion on compact manifolds with boundary

The refined analytic torsion on compact Riemannian manifolds with boundary has been discussed by B. Vertman and the authors, but these two constructions are completely different. Vertman used a double of de Rham complex consisting of the minimal and maximal closed extensions of a flat connection and the authors used well-posed boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$, ${\mathcal P}_{+, {\mathcal L}_{1}}$ for the odd signature operator. In this paper we compare these two constructions by using the BFK-gluing formula for zeta-determinants, the adiabatic method for stretching cylinder part near boundary and the deformation method used in [6], when the odd signature operator comes from a Hermitian flat connection and all de Rham cohomologies vanish.

math.DG↗

arXiv2011

The gluing formula of the refined analytic torsion for an acyclic Hermitian connection

In the previous work ([14]) we introduced the well-posed boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$ and ${\mathcal P}_{+, {\mathcal L}_{1}}$ for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined analytic torsion for an acyclic Hermitian connection with respect to the boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$ and ${\mathcal P}_{+, {\mathcal L}_{1}}$. In this case the refined analytic torsion consists of the Ray-Singer analytic torsion, the eta invariant and the values of the zeta functions at zero. We first compare the Ray-Singer analytic torsion and eta invariant subject to the boundary condition ${\mathcal P}_{-, {\mathcal L}_{0}}$ or ${\mathcal P}_{+, {\mathcal L}_{1}}$ with the Ray-Singer analytic torsion subject to the relative (or absolute) boundary condition and eta invariant subject to the APS boundary condition on a compact manifold with boundary. Using these results together with the well known gluing formula of the Ray-Singer analytic torsion subject to the relative and absolute boundary conditions and eta invariant subject to the APS boundary condition, we obtain the main result.

math.DG↗

arXiv2010

Twisted Cappell-Miller holomorphic and analytic torsions

Recently, Cappell and Miller extended the classical construction of the analytic torsion for de Rham complexes to coupling with an arbitrary flat bundle and the holomorphic torsion for $\bar{\partial}$-complexes to coupling with an arbitrary holomorphic bundle with compatible connection of type $(1,1)$. Cappell and Miller studied the properties of these torsions, including the behavior under metric deformations. On the other hand, Mathai and Wu generalized the classical construction of the analytic torsion to the twisted de Rham complexes with an odd degree closed form as a flux and later, more generally, to the $\mathbb{Z}_2$-graded elliptic complexes. Mathai and Wu also studied the properties of analytic torsions for the $\mathbb{Z}_2$-graded elliptic complexes, including the behavior under metric and flux deformations. In this paper we define the Cappell-Miller holomorphic torsion for the twisted Dolbeault-type complexes and the Cappell-Miller analytic torsion for the twisted de Rham complexes. We obtain variation formulas for the twisted Cappell-Miller holomorphic and analytic torsions under metric and flux deformations.

math.DG↗

arXiv2010

Refined analytic torsion for twisted de Rham complexes

Let $E$ be a flat complex vector bundle over a closed oriented odd dimensional manifold $M$ endowed with a flat connection $\nabla$. The refined analytic torsion for $(M,E)$ was defined and studied by Braverman and Kappeler. Recently Mathai and Wu defined and studied the analytic torsion for the twisted de Rham complex with an odd degree closed differential form $H$, other than one form, as a flux and with coefficients in $E$. In this paper we generalize the construction of the refined analytic torsion to the twisted de Rham complex. We show that the refined analytic torsion of the twisted de Rham complex is independent of the choice of the Riemannian metric on $M$ and the Hermitian metric on $E$. We also show that the twisted refined analytic torsion is invariant (under a natural identification) if $H$ is deformed within its cohomology class. We prove a duality theorem, establishing a relationship between the twisted refined analytic torsion corresponding to a flat connection and its dual. We also define the twisted analogue of the Ray-Singer metric and calculate the twisted Ray-Singer metric of the twisted refined analytic torsion. In particular we show that in case that the Hermtitian connection is flat, the twisted refined analytic torsion is an element with the twisted Ray-Singer norm one.

math.DG↗

arXiv2009

Cappell-Miller analytic torsion for manifolds with boundary

Inspired by the work of Boris Vertman on refined analytic torsion for manifolds with boundary, in this paper we extend the construction of the Cappell-Miller analytic torsion to manifolds with boundary. We also compare it with the refined analytic torsion on manifolds with boundary. As a byproduct of the gluing formula for refined analytic torsion and the comparison theorem for the Cappell-Miller analytic torsion and the refined analytic torsion, we establish the gluing formula for the Cappell-Miller analytic torsion in the case that the Hermitian metric is flat.

math.DG↗