I-Hsun Tsai

Curvature of determinant bundles for degenerate families

We calculate the (1, 1) curvature of the Beilinson Schechtman connection for the determinant bundle associated to a family of Riemann surfaces with ordinary singularities. As consequences we obtain generalizations of theorems of Bismut and Bost.

An identification of the connections of Quillen and Beilinson-Schechtman

AbstractGiven a family of Riemann surfaces and a holomorphic vector bundle Beilinson and Schechtman construct a canonical connection on the associated determinant bundle. We prove the conjecture which states that their connection coincides with the Quillen connection. This is done by reducing to the case where

Determinant Bundle in a Family of Curves,¶after A. Beilinson and V. Schechtman

\bar \partial

Heat kernel asymptotics, local index theorem and trace integrals for CR manifolds with

along fibers are invertible. Both connection forms become more accessible in this case.

S^1

The SL(2,C) Yang-Mills instanton solutions constructed recently by the biquaternion method were shown to satisfy the complex version of the ADHM equations and the Monad construction. Moreover, we discover that, in addition to the holomorphic vector bundles on CP^3 similar to the case of SU(2) ADHM construction, the SL(2,C) instanton solutions can be used to explicitly construct instanton sheaves on CP^3. Presumably, the existence of these instanton sheaves is related to the singularities of the SL(2,C) instantons on S^4 which do not exist for SU(2) instantons.

action

Abstract:Let π:X→S be a smooth projective family of curves over a smooth base S over a field of characteristic 0, together with a bundle E on X. Then A. Beilinson and V. Schechtman define in [1] a beautiful “trace complex” on X, the 0th relative cohomology of which describes the Atiyah algebra of the determinant bundle of E on S. Their proof reduces the general case to the acyclic one. In particular, one needs a comparison of for F≡A and F≡E(D), where D is étale over S (see Theorem 2.3.1, reduction ii) in [1]). In this note, we analyze this reduction in more detail and correct a point.