Fixed point formula for characters of automorphism groups associated with Kähler classes (Q5953004)

In this paper the authors give a fixed-point formula for (lifting to an automorphism group of) a Lie algebra character on the complex Lie algebra of holomorphic fields on a closed Kähler manifold, the latter being an obstruction for Kähler classes to contain a metric with constant scalar curvature. NEWLINENEWLINENEWLINEIn order to put things in context, let us consider the following situation. Let \(M\) be a closed Kähler manifold of complex dimension \(m\), \(h(M)\) -- the Lie algebra of holomorphic vector fields, and let \(\omega\in \Omega\) be a Kähler form contained in the Kähler cohomology class \(\Omega\). Then there is a complex-valued Lie algebra character \(f_\Omega:h(M)\to C\), which is an obstruction for \(\Omega\) to contain a metric of constant scalar curvature. Under certain circumstances this Lie algebra character ``lifts'' to an additive group character defined on a subgroup \(G\) of a (Lie) group of biholomorphic automorphisms of \(M\). This is the case for a (finite) cyclic subgroup \(G\) (generated by an automorphism \(\sigma)\). In this case the character \(f_\Omega\) lifts to a group character \(\widehat f_\Omega\) on \(G\) defined as an algebraic expression in terms of Chern-Simons classes of the mapping torus of the natural lift of \(\sigma\) to an automorphism of some complex line bundle over the mapping torus of \(\sigma\) (Chern-Simons classes being computed with the help of the Bott connection related to a natural complex foliation of the mapping torus).NEWLINENEWLINENEWLINEThe main result of the paper is a formula, which expresses Chern-Simons classes (and therefore the character \(\widehat f_\Omega)\) in terms of the class \(\Omega\) and Chern classes of the normal bundles to the smooth components of the fixed-point set of the automorphism \(\sigma\). The proof of the formula is based on an interpretation of its entries as the eta-invariant of Dirac-type operators connected with the natural \(\text{Spin}^c\)-structure on the mapping torus, and the index theorem.