On the jumping phenomenon of $\dim_{\mathbb{C}}H^q(\mathcal{X}_t,\mathcal{E}_t)$

TL;DR

This paper investigates the conditions under which the dimensions of certain cohomology groups of a holomorphic vector bundle on a complex manifold change during deformation, identifying explicit obstructions to this jumping phenomenon.

Contribution

It generalizes previous results by explicitly characterizing the two cohomological obstructions to the stability of cohomology dimensions in deformations, using Maurer-Cartan elements.

Findings

Identifies two explicit obstructions to cohomology dimension stability.

Expresses obstructions in terms of Maurer-Cartan elements.

Applies results to the case of $H^1( ext{End}(T_X))$, relevant to physics.

Abstract

Let $X$ be a compact complex manifold and $E$ be a holomorphic vector bundle on $X$. Given a deformation $(\mathcal{X},\mathcal{E})$ of the pair $(X,E)$ over a small polydisk $B$ centered at the origin, we study the jumping phenomenon of the cohomology groups $\dim_{\mathbb{C}}H^q(\mathcal{X}_t,\mathcal{E}_t)$ near $t = 0$. Generalizing previous results of X. Ye for the tangent bundle $E = T_{\mathcal{X}_t}$ and exterior powers of the cotangent bundle $E = \Omega^p_{\mathcal{X}_t}$, we show that there are precisely two cohomological obstructions to the stability of $\dim_{\mathbb{C}}H^q(\mathcal{X}_t,\mathcal{E}_t)$, which can be expressed explicitly in terms of the Maurer-Cartan element associated to the deformation $(\mathcal{X},\mathcal{E})$. As an application, we study the jumping phenomenon of the dimension of the cohomology group $H^1(\mathcal{X}_t,\text{End}(T_{\mathcal{X}_t}))$ which is related to a question raised by physicists.