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.

中文翻译：

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关于$\operatorname{dim}_{\mathbb{C}} H^q (\mathcal{X}_t,\mathcal{E}_t)$的跳跃现象

令$X$ 是一个紧复流形，$E$ 是$X$ 上的一个全纯向量丛。给定一对 $(X,E)$ 在以原点为中心的小多盘 $B$ 上的变形 $(\mathcal{X},\mathcal{E})$，我们研究了上同调群 $ 的跳跃现象\dim_{\mathbb{C}}H^q(\mathcal{X}_t,\mathcal{E}_t)$ 接近 $t = 0$。对切​​丛 $E = T_{\mathcal{X}_t}$ 和余切丛 $E = \Omega^p_{\mathcal{X}_t}$ 的外幂推广 X. Ye 的先前结果，我们显示$\dim_{\mathbb{C}}H^q(\mathcal{X}_t,\mathcal{E}_t)$ 的稳定性恰好有两个上同调障碍，可以明确表示为与变形 $(\mathcal{X},\mathcal{E})$ 相关的 Maurer-Cartan 元素。作为应用程序，