Universal classes and the Lefschetz formula for holomorphic differential operators

Klaus Willi Hummelsheim, Purdue University

Abstract

Let X be a complex, compact manifold of dimension n and let $F\rightarrow X$ be a holomorphic vector bundle with typical fibre V and structure group G. Let $G\sb{k}$ denote the group of k-jets (at $0\in\doubc\sp{n}$) of holomorphic maps $f:\doubc\sp{n}\rightarrow G$. We call a class in $\breve H\sp{q}(X,F$) universal, if it can be represented by a $\breve C$ech-cochain, which is locally a holomorphic function of finitely many derivative of the transition functions $\{\phi\sb{\alpha\beta}\}$ of F. More precisely, the maps $\phi\sb{\alpha\sb0\cdots\alpha\sb{q}} : U\sb{\alpha\sb0}\cap\cdots\cap U\sb{\alpha\sb{q}}\rightarrow G\sb{k}\times\cdots\times G\sb{K}$ defined by $\phi\sb{\alpha\sb0\cdots\alpha\sb{q}}(p) = (j\sb{k}(\phi\sb{\alpha\sb0\alpha\sb1}) (p),\cdots,j\sb{k}(\phi\sb{\alpha\sb{q-1}\alpha\sb{q}})(p$)) give rise to a map $\phi\sp\cdot:H\sbsp{hol}{\cdot}(G\sb{k},V)\rightarrow\breve H\sp\cdot(X,F$) from holomorphic group cohomology to $\breve C$ech-cohomology. We say a class is universal if it lies in the image of some $\phi\sp\cdot$. Let $E\rightarrow X$ be holomorphic vector bundle and let ${\cal D}\sbsp{E}{\leq k}$ be the sheaf of holomorphic differential operators acting on E. The Lefschetz number ${\cal L}\sb{k}(D)=\Sigma(-1)\sp{i}trace(D:H\sp{i}(X,E)\rightarrow H\sp{i}(X,E$)) defines a linear form on $\Gamma({\cal D}\sbsp{E}{\leq k}$) and consequently by Serre duality a class, ${\cal L}\sb{k}\in H\sp{n}(X,J\sb{k}E\otimes E\sp\vee\otimes\Omega\sbsp{X}{n}$), called the Lefschetz class. We prove that the Lefschetz class ${\cal L}\sb{k}$ is universal and depends in the case of trivial coefficients on at most k + 2 derivatives. To illustrate the methods used in the proof we give an explicit formula for differential operators of order three on a compact Riemann surface. We give a short proof of the Hirzebruch-Riemann-Roch formula as another application of this result. Finally we show, using Lie algebra cohomology, that in dimensions one and two, universal classes for holomorphic differential operators are completely determined by their values on first order operators.

Degree

Ph.D.

Advisors

Tong, Purdue University.

Subject Area

Mathematics|Physics

COinS