The Mukai pairing, I: a categorical approach

Details The Mukai pairing, I: a categorical approach The Mukai pairing, I: a categorical approach

2007-07-13

arxiv.org/abs/0707.2052v1

New York Journal of Mathematics 16 (2010) 61-98

Mathematics

Algebraic Geometry

39 pages, millions of figures; supersedes math.AG/0308079; complete rewriting in new categorical context

Scientific paper

We study the Hochschild homology of smooth spaces, emphasizing the importance of a pairing which generalizes Mukai's pairing on the cohomology of K3 surfaces. We show that integral transforms between derived categories of spaces induce, functorially, linear maps on homology. Adjoint functors induce adjoint linear maps with respect to the Mukai pairing. We define a Chern character with values in Hochschild homology, and we discuss analogues of the Hirzebruch-Riemann-Roch theorem and the Cardy Condition from physics. This is done in the context of a 2-category which has spaces as its objects and integral kernels as its 1-morphisms.

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