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# Asian Journal of Mathematics

## Volume 23 (2019)

### Number 4

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

Pages: 681 – 702

DOI: https://dx.doi.org/10.4310/AJM.2019.v23.n4.a7

#### Authors

#### 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 $\operatorname{dim}_{\mathbb{C}} H^q (\mathcal{X}_t,\mathcal{E}_t)$ near $t = 0$. Generalizing previous results of X. Ye [8, 9] (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 $\operatorname{dim}_{\mathbb{C}} H^q (\mathcal{X}_t,\mathcal{E}_t)$, 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 $H^1 (\mathcal{X}_t , \operatorname{End}(T_{\mathcal{X}_t}))$, which is related to a question raised by physicists [5].

#### Keywords

deformation, obstruction, jumping phenomenon, cohomology group

#### 2010 Mathematics Subject Classification

14B10, 32G05, 32G07, 58H15

Received 7 November 2017

Accepted 8 June 2018

Published 7 January 2020