Giovanni Felder

Abstract: In 1956 Hirzebruch wrote a formula for the Euler characteristic of a holomorphic vector bundle on a compact complex manifold as the integral of a characteristic class, generalizing the classical Riemann-Roch formula for line bundles on a curve. In 1988, Feigin and Tsygan explained that the origin of this class lies in the Hochschild cohomology of the algebra of polynomial differential operators, which is one dimensional. Using ideas from formal geometry they constructed a map from this cohomology to de Rham cohomolgy of any complex manifold with a vector bundle and showed that the image of a canonically normalized cocycle is the Hirzebruch class. I will review these ideas and discuss recent developments, including an explicit formula for the cocycle and the extension of the Riemann-Roch-Hirzebruch formula to a formula for the alternating sum of traces of the action of a holomorphic differential operator on sheaf cohomology. This talk is based on joint papers with B. Feigin and B. Shoikhet, with M. Engeli and with X. Tang.