A Chern-Weil approach to deformations of pairs and its applications
A Chern-Weil approach to deformations of pairs and its applications
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发表时间：
2014-06
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通讯作者：
Kwokwai Chan;Yat-Hin Suen
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作者：
Kwokwai Chan;Yat-Hin Suen
本文从分析的角度重新讨论了对$（X，E）$的形变理论，其中$X$是紧致复流形，$E$是$X$上的全纯向量丛.通过引入和利用一个辅助微分算子，我们导出了控制变形问题的Maurer-Cartan方程和DGLA，并将它们用微分几何的概念表示，如E的连通和曲率，得到一个陈-魏二氏对模问题的切空间和障碍空间分别由模问题的第一和第二上同调群给出的经典结果进行了改进，$E$在$X$上的Atiyah扩展。我们还调查的情况下，变形对是通畅的使用我们的分析方法。
We revisit the theory of deformations of pairs $(X, E)$, where $X$ is a compact complex manifold and $E$ is a holomorphic vector bundle over $X$, from an analytic viewpoint \`{a} la Kodaira-Spencer. By introducing and exploiting an auxiliary differential operator, we derive the Maurer-Cartan equation and DGLA governing the deformation problem, and express them in terms of differential-geometric notions such as the connection and curvature of $E$, obtaining a Chern-Weil--type refinement of the classical results that the tangent space and obstruction space of the moduli problem are respectively given by the first and second cohomology groups of the Atiyah extension of $E$ over $X$. We also investigate circumstances where deformations of pairs are unobstructed using our analytic approach.