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

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 $E={\Omega }_{{\mathcal{X}}_t}^p$, 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.

arXiv comment21 pages; v3: minor modifications, to appear in Asian J. Math

Cited by in corpus (1)

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