The Harder-Narasimhan trace and unitarity of the KZ/Hitchin connection: genus 0

• 成果类型: Article
• 署名作者: Ramadas, T. R.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2009.169.1
• 发表日期: 2009
• 页码: 1-39
• 关键词: local systems; factorization

摘要

Let a reductive group G act oil a projective variety chi(+), and suppose given a lift of the action to an ample line bundle (theta) over cap. By definition, all G-invariant sections of (theta) over cap vanish on the nonsemistable locus chi(nss)(+). Taking an appropriate normal derivative defines a map H-0(chi(+),(theta) over cap)(G) -> H-0(S-mu,V-mu)(G), where V-mu is a G-vector bundle on a G-variety S-mu. We call this the Harder-Narasimhan trace. Applying this to the Geometric Invariant Theory construction of the moduli space of parabolic bundles oil a curve, we discover generalisations of Coulomb-gas representations, which map conformal blocks to hypergeometric local systems. In this paper we prove the unitarity of the KZ/Hitchin connection (in the genus zero, rank two, case) by proving that the above map lands in a unitary factor of the hypergeometric system. (All ingredient in the proof is a lower bound on the degree of polynomials vanishing on partial diagonals.) This elucidates the work of K. Gawedzki.