arXiv subjects

Xuanming Ye

Publications and source records attributed to Xuanming Ye.

8 recordsLinked to original sources

arXiv2023

Dolbeault Type Cohomology Groups of Infinitesimal Deformations and Their Applications

In this paper, we establish a kind of Dolbeault type cohomology groups for the purpose of studying the varying of complex structure invariants in infinitesimal deformations of any order. We give a concrete description of the higher order Kodaria-Spencer maps by using these cohomology groups. We reformulate the obstruction formulas within the framework of these cohomology groups and give a new proof for the formulas.

math.AG↗

arXiv2022

Integrable Harmonic Higgs Bundles With Vanishing $\mathcal{U}$ And Eigenvalues of $\mathcal{Q}$

We study the tt*-geometry with vanishing endormorphism $\mathcal{U}$. Given an integrable harmonic Higgs bundle $(E, h, Φ, \mathcal{U},\mathcal{Q})$ on a complex manifold $M$, Firstly we prove that, under the \emph{IS} condition, vanishing $\mathcal{U}$ implies vanishing Higgs field $Φ$ and the Chern connection of the Hermitian Einstein metric $h$ is a holomorphic connection, so the metric $h$ and $\mathcal{Q}$ are invariant. Secondly, without the \emph{IS} condition, we show that vanishing $\mathcal{U}$ will imply vanishing Higgs field $Φ$ if we assume that the Chern connection of $h$ is a holomorphic connection. Finally, we add real structure $κ$. Given any \emph{CV}-structure, we prove that super-symmetric operator $\mathcal{Q}$ must have $0$ as an eigenvalue when the underlying bundle has odd rank.

math.DG↗

arXiv2018

Decomposition, purity and fibrations by normal crossing divisors

We give a simple geometric proof of the decomposition theorem in terms of Thom-Whitney stratifications by reduction to fibrations by normal crossings divisors over the strata and explain the relation with the local purity theorem an unpublished result of Deligne and Gabber.

math.AG↗

arXiv2014

On the cohomology groups of local systems over Hilbert modular varieties via Higgs bundles

Let $X$ be a Hilbert modular variety and $\mathbb{V}$ a non-trivial local system over $X$ with infinite monodromy. In this paper we study Saito's mixed Hodge structure (MHS) on the cohomology group $H^k(X,\mathbb{V})$ using the method of Higgs bundles. Among other results we prove the Eichler-Shimura isomorphism, give a dimension formula for the Hodge numbers and show that the mixed Hodge structure is split over $\mathbb{R}$. These results are analogous to Matsushima-Shimura [Annals of Mathematics 78, 1963] in the cocompact case and complement the results in Freitag [Book: Hilbert modular forms, Springer-Verlag, Berlin, 1990] for constant coefficients.

math.AG↗

arXiv2014

The jumping phenomenon of the dimensions of Bott-Chern cohomology groups and Aeppli cohomology groups

Let $X$ be a compact complex manifold, and let $π: \mathcal{X} \rightarrow B$ be a small deformation of $X$, the dimensions of the Bott-Chern cohomology groups $H_{\rm BC}^{p,q}(X(t))$ and Aeppli cohomology groups $H_{\rm A}^{p,q}(X(t))$ may vary under this deformation. In this paper, we will study the deformation obstructions of a $(p,q)$ class in the central fiber $X$. In particular, we obtain an explicit formula for the obstructions and apply this formula to the study of small deformations of the Iwasawa manifold.

math.AG↗

arXiv2012

$L^2$ and intersection cohomologies for the reductive representation of the fundamental groups of quasiprojective manifolds with unipotent local monodromy

Let $X$ be a projective manifold, and $D$ be a normal crossing divisor of $X$. By Jost-Zuo's theorem that if we have a reductive representation $ρ$ of the fundamental group $π_{1}(X^{*})$ with unipotent local monodromy, where $X^*=X-D$, then there exists a tame pluriharmonic metric $h$ on the flat bundle $\mathcal V$ associated to the local system $\mathbb V$ obtain from $ρ$ over $X^*$. Therefore, we get a harmonic bundle $(E, θ, h)$, where $θ$ is the Higgs field, i.e. a holomorphic section of $End(E)\otimesΩ^{1,0}_{X^*}$ satisfying $θ^2=0$. In this paper, we study the harmonic bundle $(E,θ,h)$ over $X^*$. We are going to prove that the intersection cohomology $IH^{k}(X; \mathbb V)$ is isomorphic to the $L^{2}$-cohomology $H^{k}(X, (\mathcal A_{(2)}^{\cdot}(X,\mathcal V), \mathbb D))$.

math.DG↗

arXiv2007

The Jumping Phenomenon of the Dimensions of Cohomology Groups of Tangent Sheaf

Let $X$ be a compact complex manifold, consider a small deformation $ϕ: \mathcal{X} \to B$ of $X$, the dimensions of the cohomology groups of tangent sheaf $H^q(X_t,\mathcal{T}_{X_t})$ may vary under this deformation. This paper will study such phenomenons by studying the obstructions to deform a class in $H^q(X,\mathcal{T}_X)$ with the parameter $t$ and get the formula for the obstructions.

math.AG↗

arXiv2007

The Jumping Phenomenon of Hodge Numbers

Let $X$ be a compact complex manifold, consider a small deformation $ϕ: \mathcal{X} \to B$ of $X$, the dimension of the Dolbeault cohomology groups $H^q(X_t,Ω_{X_t}^p)$ may vary under this defromation. This paper will study such phenomenons by studying the obstructions to deform a class in $H^q(X,Ω_X^p)$ with the parameter $t$ and get the formula for the obstructions.

math.AG↗