| 平成30年 12月19日（水）〜21日（金） |
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Song Sun 氏 （UC Berkeley）
Title:
Singularities of Hermitian-Yang-Mills connections
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Abstract
The Donaldson-Uhlenbeck-Yau theorem relates existence of Hermitian-Yang-Mills connection over a compact Kahler manifold with algebraic stability of a holomorphic vector bundle. This has been extended by Bando-Siu in 1994 to the case of reflexive sheaves, and the corresponding connection may have singularities in general. These singular objects also appear naturally in the study of compactifying moduli space of smooth Hermitian-Yang-Mills connections.
I will discuss recent study on the behavior of Hermitian-Yang-Mills connections near a singularity. Apart from being interesting in its own right, this also serves as a model problem towards understanding more general singularities in higher dimensional gauge theory. From the usual geometric analytic point of view, there is a notion of ``analytic tangent cones”, which arise as dilation limits at a singularity. We want to understand these in terms of the complex/algebraic geometry of the underlying reflexive sheaf. In particular, this motivated us to study the notion of an ``algebraic tangent cone”, which is a canonically defined torsion-free sheaf on the projective space. The conjectural picture then states that the analytic tangent cone is determined by the Harder-Narasimhan-Seshadri filtration of the algebraic tangent cone, and this is known to hold in important special cases. The goal of these lectures is to explain these developments.
Based on joint work with Xuemiao Chen.
講義時間
１９日：１３：００〜１４：００
２０日，２１日：１０：３０〜１１：３０
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