# A local index theorem for CR manifolds with $S^1$ action

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Jih-Hsin Cheng

created by malchiodi on 06 May 2016

10 may 2016
-- 15:00
[open in google calendar]

Scuola Normale Superiore, Aula Tonelli

**Abstract.**

Among those transversally elliptic operators initiated by Atiyah and Singer, Kohn's $\Box_b$ operator on CR manifolds with $S^1$ action is a natural one of geometric significance for complex analysts. Our main result computes a local index density, in terms of \emph{tangential} characteristic forms, on such manifolds including \emph{Sasakian manifolds} of interest in String Theory. As applications of our CR index theorem we can prove a CR version of Grauert-Riemenschneider criterion, and produce many CR functions on a weakly pseudoconvex CR manifold with transversal $S^1$ action and many CR sections on some class of CR manifolds, answering some long-standing questions raised by Kohn and Henkin respectively. Moreover in some cases, we can reinterpret Kawasaki's Hirzebruch-Riemann-Roch formula for a complex orbifold with an orbifold holomorphic line bundle as an index theorem obtained by an integral over a smooth CR manifold which is essentially the circle bundle of this line bundle. This is joint work with Chin-Yu Hsiao and I-Hsun Tsai