arXiv research

Study on deformations of $(p,q)$-forms and spectral sequence degenerations.

Given a holomorphic family of pairs $\{(X_t,E_t)\}$, where each $E_t$ is holomorphic vector bundle over compact complex manifold $X_t$. For small enough $t$, we get a correspondence between the Dolbeault complex of $E_t$-valued $(p,q)$-forms on $X_t$ and the one of $E_0$-valued $(p,q)$-forms on $X_0$.

We present a gauge invariant generalization of Maxwell's equations and p-form electromagnetism to Kaehler spacetimes.

This paper classifies quadratic form parameters over integers and computes their Witt groups.

Let $φ\in C^\infty(\Complex^n)$ be a given real valued function. We assume that $\pr\ddbarφ$ is non-degenerate of constant signature $(n_-,n_+)$ on $\Complex^n$. When $q=n_-$, it is well-known that the Bergman kernel for $(0,q)$ forms with respect to the $k$-th weight $e^{-2kφ}$, $k>0$, admits a full asymptotic expansi…

Constructs finite element spaces for $(p,q)$-forms, excluding one subspace.

Positive representations of surface groups in PO(p,q) form connected components of character varieties.

We prove a refined Kato inequality for closed and coclosed differential $(p,q)$ forms on a Kahler manifold.

New method for spectral and Bergman kernels under local spectral gap condition.

Method implements symmetries in TQFT for finite groups.

We give the extension formulae on almost complex manifolds and give decompositions of the extension formulae. As applications, we study $(n,0)$-forms, the $(n,0)$-Dolbeault cohomology group and $(n,q)$-forms on almost complex manifolds.

Study canonical deformations of complex forms and their cohomology properties.

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 comp…

Extended surgery theory proves diffeomorphism for simply-connected 4k-manifolds.

On an asymptotically conic manifold $(M,g)$, we analyze the asymptotics of the integral kernel of the resolvent $R_q(k):=(Δ_q+k^2)^{-1}$ of the Hodge Laplacian $Δ_q$ on $q$-forms as the spectral parameter $k$ approaches zero, assuming that 0 is not a resonance. The first application we give is an $L^p$ Sobolev estimate…

For $1\leq q \leq n-2$, we provide explicit examples to demonstrate non-compactness of the Neumann operator for the Kohn Laplacian acting on $L^2$ $(0,q)$-forms on the unit ball in $(2n+1)$-dimensional Heisenberg space.

Abstract: Expresses zeta-determinant of Dirichlet-to-Neumann operator on forms.

The paper computes inertia groups of certain high-dimensional manifolds.

The paper studies rigidity results for harmonic forms on Kähler manifolds.

We give an explicit description and calculate the dimension of the vector space of linear natural liftings of $p$-forms on $n$-dimensional manifolds $M$ to $q$-forms on $T^{{\mathbb D}^r_k}M$, where ${\mathbb D}^r_k$ is the Weil algebra of $r$-jets at 0 of smooth functions ${\mathbb R}^k\longrightarrow{\mathbb R}$, for…

The authors study the Hodge theory of the exterior differential operator $d$ acting on $q$-forms on a smoothly bounded domain in $\RR^{N+1}$, and on the half space $\rnp$. The novelty is that the topology used is not an $L^2$ topology but a Sobolev topology. This strikingly alters the problem as compared to the classic…

The paper investigates Liouville type theorems for various harmonic forms on Riemannian manifolds.

The abstract discusses embedding theorems for pseudo-Kähler manifolds.

We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in the high tensor powers of the line bund…

Algorithm computes eigenvalues and eigenforms on Calabi-Yau threefolds.

In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic $(0,q)$-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. As applications, we e…

We consider a general Hermitian holomorphic line bundle $L$ on a compact complex manifold $M$ and let ${\Box}^q_p$ be the Kodaira Laplacian on $(0,q)$ forms with values in $L^p$. The main result is a complete asymptotic expansion for the semi-classically scaled heat kernel $\exp(-u{\Box}^q_p/p)(x,x)$ along the diagonal…

In this paper, we establish a deformation theory for Dolbeault cohomology classes valued in holomorphic tensor bundles. We prove the extension equation which will play the role of Maurer-Cartan equation. Following the classical theory of Kodaira-Spencer-Kuranishi, we construct a canonical complete family of deformation…

The paper studies the Poisson transform of differential forms on hyperbolic spaces.

The aim of this paper is to present the construction, out of the Kohn-Rossi complex, of a new hypoelliptic operator $Q_{L}$ on almost CR manifolds equipped with a real structure. The operator acts on all (p,q)-forms, but when restricted to (p,0)-forms and (p,n)-forms it is a sum of squares up to sign factor and lower o…

Analyzes complex structure deformations using cohomology contraction methods.

Let $X$ be a compact connected CR manifold of dimension $2n-1, n\geq 2$. We assume that there is a transversal CR locally free $S^1$ action on $X$. Let $L^k$ be the $k$-th power of a rigid CR line bundle $L$ over $X$. Without any assumption on the Levi-form of $X$, we obtain a scaling upper-bound for the partial Szegő …

Study $L^{2}$-harmonic forms on almost Kähler manifolds, extending vanishing theorems.

We have performed detailed multifractal analysis on the minutely volatility of two indexes and 1139 stocks in the Chinese stock markets based on the partition function approach. The partition function $χ_q(s)$ scales as a power law with respect to box size $s$. The scaling exponents $τ(q)$ form a nonlinear function of …

Let $Ω$ be a pseudoconvex domain with $C^2$-smooth boundary in $\mathbb CP^n$. We prove that the $\bar\partial-Neumann operator$N$exists for$(p,q)$-forms on$Ω$. Furthermore, there exists a$t_0>0$such that the operators$N$,$\bar\partial^*N$,$\bar\partial N$ and the Bergman projection are regular in the Sobolev …

Generalizes Kodaira vanishing theorem to Kahler Lie algebroids.

The paper characterizes integrability of tensors on manifolds.

For a Kähler manifold endowed with a weighted measure $e^{-f}\,dv,$ the associated weighted Hodge Laplacian $Δ_{f}$ maps the space of $(p,q)$-forms to itself if and only if the $(1,0)$-part of the gradient vector field $\nabla f$ is holomorphic. We use this fact to prove that for such $f$, a finite energy $f$ harmonic …

We study the $\bar{\partial}_b$-Neumann problem for domains $Ω$ contained in a strictly pseudoconvex manifold M^{2n+1} whose boundaries are noncharacteristic and have defining functions depending solely on the real and imaginary parts of a single CR function w. When the Kohn Laplacian is a priori known to have closed r…

New geometric conditions ensure compactness of $\bar{\partial}$-Neumann problem.

Study Bergman and spectral kernels for non-compact complex manifolds.

We propose a method for explicit computation of the Chern character form of a holomorphic Hermitian vector bundle $(E,h)$ over a complex manifold $X$ in a local holomorphic frame. First, we use the descent equations arising in the double complex of $(p,q)$-forms on $X$ and find explicit degree decomposition of the Cher…

We present a new method to solve certain $\bar{\partial}$-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a $\bar{\partial}$-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness…

Study properties of holomorphic $p$-contact manifolds, including non-Kähler hyperbolicity and deformations.

We study isospectrality on p-forms of compact flat manifolds by using the equivariant spectrum of the Hodge-Laplacian on the torus. We give an explicit formula for the multiplicity of eigenvalues and a criterion for isospectrality. We construct a variety of new isospectral pairs, for instance, pairs of flat manifolds o…

Calculates Laplacian spectra on Calabi-Yau hypersurfaces.

This is the first of a series of articles in which we are going to study the regularized determinants of the Laplacians of Calabi Yau metrics acting on (0,q) forms on the moduli space of CY manifolds with a fixed polarization. It is well known that in case of the elliptic curves the Kronecker limit formula gives an exp…

The paper extends Hodge-de Rham and Lichnérowicz Laplacians to double forms and proves vanishing theorems.