Spaces of quasi-maps into the flag varieties and their applications

Details Spaces of quasi-maps into the flag varieties and their applications Spaces of quasi-maps into the flag varieties and their applications

2006-03-18

arxiv.org/abs/math/0603454v2

Mathematics

Algebraic Geometry

some references are corrected; 26 pages; to appear in the Proceedings of ICM 2006

Scientific paper

Given a projective variety X and a smooth projective curve C one may consider the moduli space of maps C --> X. This space admits certain compactification whose points are called quasi-maps. In the last decade it has been discovered that in the case when X is a (partial) flag variety of a semi-simple algebraic group G (or, more generally, of any symmetrizable Kac-Moody Lie algebra) these compactifications play an important role in such fields as geometric representation theory, geometric Langlands correspondence, geometry and topology of moduli spaces of G-bundles on algebraic surfaces, 4-dimensional super-symmetric gauge theory (and probably many others). This paper is a survey of the recent results about quasi-maps as well as their applications in different branches of representation theory and algebraic geometry.

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