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Dexie Lin

Publications and source records attributed to Dexie Lin.

10 recordsLinked to original sources
arXiv2025

Operator Models via Compact Embedding

In this paper, we extend several function-theoretic and geometric constructions to the realm of multi-variable operator theory. A commuting tuple of bounded linear operators on a Hilbert space, equipped with a cyclic vector, is abbreviated as a cyclic commuting tuple. We encode the complete information of such a tuple into a single positive compact operator on the Fock space. Drawing parallels with spectral geometry, we investigate how the spectral data of this positive compact operator -- its eigenvalues and eigenfunction -- reflect fundamental properties of the operator tuple. Our main contributions are threefold. First, we establish a Weyl-type approximation formula for certain operator tuples, demonstrating that the asymptotic behavior of eigenvalues carries geometric information about the vanishing variety. Second, we construct two kernel functions derived from the eigenvalues and eigenfunctions: the first extends the Bergman kernel, while the second extends the Fourier-Laplace transformation. We prove that the Fourier-Laplace kernel defines a reproducing kernel Hilbert space (RKHS) on which the coordinate differential operators are unitarily equivalent to the adjoint tuple. Consequently, we show that the convergence points of the Bergman-type kernel characterize the joint eigenvalues of the adjoint tuple. Finally, we obtain a Paley-Wiener-Schwartz type theorem for cyclic commuting tuples, characterizing cyclic commuting tuples whose associated Agler's linear functional are distributions. For tuples consisting of matrices, we obtains a more explicit characterization.

math.FA↗
arXiv2025

A note on $\tmd$-operator

In almost K\"ahler manifolds, one of the challenges is to construct an elliptic operator on functions that plays a role analogous to the $\partial\bar{\partial}$ operator in complex or K\"ahler manifolds. One of the aims of this paper is to revisit the $\tmd$-operator introduced in \cite{TWZZ}. We will provide some local analysis estimates and highlight several difficulties that remain to be addressed. Additionally, we use the Atiyah-Hitchin-Singer operator to demonstrate that every $d$-exact $(1,1)$-form is globally $\tmd$-exact for any compact taming symplectic $4$-manifold.

math.DG↗
arXiv2023

Compatible almost complex structures on the Hard Lefschetz condition

For a compact K\"ahler manifold, it is well-established that its de Rham cohomology satisfies the Hard Lefschetz condition, which is reflected in the equality between the Betti numbers and the Hodge numbers. A special subclass of symplectic manifolds also adheres to this condition. Cirici and Wilson \cite{CW20} employ the variant Hodge number to propose a sufficient criterion for compact almost K\"ahler manifolds to satisfy this condition. In this paper, we show that this condition is only sufficient by presenting examples of compact almost K\"ahler manifolds that fulfill the Hard Lefschetz condition while violating the equality between the variant Hodge numbers and Betti numbers, that is, \[b^1>2h^{1,0}.\] This phenomenon contrasts with the behavior observed in compact K\"ahler manifolds.

math.DG↗
arXiv2023

A Kodaira type conjecture on almost complex 4 manifolds

Not long ago, Cirici and Wilson defined a Dolbeault cohomology on almost complex manifolds to answer Hirzebruch's problem. In this paper, we define a refined Dolbeault cohomology on almost complex manifolds. We show that the condition $\tilde h^{1,0}=\tilde h^{0,1}$ implies a symplectic structure on a compact almost complex $4$ manifold, where $\tilde h^{1,0}$ and $\tilde h^{0,1}$ are the dimensions of the refined Dolbeault cohomology groups with bi-degrees $(1,0)$ and $(0,1)$ respectively. Combining the partial answer to Donaldson's tameness conjecture, we offer a sufficient condition for a compact almost complex $4$ manifold to become an almost K\"ahler one. Moreover, we prove that the condition $\tilde{h}^{1,0}=\tilde h^{0,1}$ is equivalent to the generalized $\partial\bar\partial$-lemma. This can be regarded as an analogue of the Kodaira's conjecture on almost complex $4$ manifolds. As an application, we show that the Kodaira-Thurston manifold satisfies the $\partial\bar\partial$-lemma. Meanwhile, we show that the Fr\"olicher-type equality does not hold on a general almost complex $4$ manifold, which is different to the case of compact complex surfaces.

math.DG↗
arXiv2022

Notes on of Seiberg-Witten map on manifold with flat scalar curvature

In this paper, we focus on the moduli space of Seiberg-Witten equation on non-compact manifold with periodic end. Suppose that the scalar curvature on the periodic end is identically zero and the topological conditions: the first de-Rham cohomology and the self-dual cohomology restricting on the periodic end vanish. Then, we will show that the moduli space of the perturbed Seiberg-Witten equation is compact.

math.DG↗
arXiv2021

Transverse $\mathcal F^T$-entropy and transverse Ricci flow for Riemannian foliations

In this paper, we introduce an entropy functional on Riemannian foliation, inspired by the work of , which is monotonically along the transverse Ricci flow. We relate their gradient flow, via diffeomorphism preserving the foliated structure of the manifold with Riemannian foliation, to the transverse Ricci flow. Moreover, inspired by the work of Fuquan Fang and Yuhao Zhang, we give a necessary condition for codimension 4 Riemannian foliation admitting the transverse Einstein metric.

math.DG↗
arXiv2020

Monopole Floer homology for codimension-3 Riemannian foliation

In this paper, we give a systematic study of Seiberg-Witten theory on closed oriented manifold $M$ with codimension-$3$ oriented Riemannian foliation $F$. Under a certain topological condition, we construct the basic Seiberg-Witten invariant and the monopole Floer homologies $\bar{HM}(M,F,\mfs;\Gamma),~\hat{HM}(M,F,\mfs;\Gamma), ~\widecheck{HM}(M,F,\mfs;\Gamma)$, for each transverse \spinc structure $\mfs$, where $\Gamma$ is a complete local system. We will show that these homologies are independent of the bundle-like metric and generic perturbation. The major difference between the basic monopole Floer homologies and the ones on manifolds is the necessity to use the Novikov ring on basic monopole Floer homologies.

math.DG↗
arXiv2019

Index of transverse Dirac operator and cohomotopy Seiberg-Witten invariant for codimension $4$ Riemannian foliation

For closed manifolds endowed with a Riemannian foliation of codimension $4$, one can define a transversal Seiberg-Witten map. We show that there is a finite dimensional approximation for such a map. By such a method and under the condition that $H^1_b(M)\cap H^1(M,\mathbb Z)$ is a lattice of $H^1_b(M)$, we can define a foliated version of Bauer-Furuta invariant. Moreover, if the basic cohomological group is of zero dimension, we can give an estimate for the index of transversal Dirac operator of a foliated spin structure. Furthermore, under the condition that $H^\pm_b(M)=1$, we show the vanishing of the index of the transverse Dirac operator. This gives a topological condition for the vanishing of the index of the transverse Dirac operator.

math.DG↗
arXiv2017

Relative Chern character number and super-connection

For two complex vector bundles admitting a homomorphism, whose singularity locates in the disjoint union of some odd--dimensional spheres, we give a formula to compute the relative Chern characteristic number of these two complex vector bundles. In particular, for a spin manifold admitting some sphere bundle structure, we give a formula to express the index of a special twisted Dirac operator.

math.DG↗