The compactification of the universal moduli space of principal G-bundles-2
Angel Luis Muñoz Castañeda (Universidad de León)
Abstract: These lectures aim to introduce the problem of the compactification of the universal moduli space of principal G-bundles over , G being a semisimple linear algebraic group. I will explain recent developments on the subject based on Schmitt’s works on singular principal G-bundles. After a brief introduction to the classical theory of principal G-bundles on smooth projective curves, I will introduce the notion of singular principal G-bundle. Such objects and their semistability condition can also be introduced over stable curves, and generalized by allowing the underlying vector bundle to be a torsion-free sheaf. When trying to construct a universal moduli space of singular principal G-bundles over , a problem regarding the behavior, along with , of certain numerical parameters (related to the objects and their semistability condition) show up. I will explain the recent results about this problem and state the Existence Theorem of a universal projective moduli space of semistable singular principal G-bundles over . This moduli space contains the universal moduli space of semistable principal G-bundles over M_g as an open subset. This condition makes the constructed space a good candidate for an analog of Pandharipande’s universal compactification of the universal moduli space of vector bundles. This is part of joint work with A. Schmitt. If time permits, I will speak about some open problems in the subject.
algebraic geometry
Audience: researchers in the topic
Algebraic Geometry at IIT Madras
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