A proof of the holomorphic Lefschetz formula for higher dimensional fixed point sets by the heat equation method. (English) Zbl 0538.58034
It is well known that the kernel of the heat equation gives a formula for the index of an elliptic operator and also for the Lefschetz fixed point formulas. For the de Rham, signature, and SPIN-c complexes this formula agrees with the classical formula of the Atiyah-Singer index theorem in terms of characteristic classes. For the Dolbeault complex, however, this formula does not agree in general with the formula of the index theorem. Toledo and Tong have given a formula for the Lefschetz number in the holomorphic case and it is of interest to what extent the heat equation formula agrees with this formula in the non-Kähler case (the two agree for Kähler manifolds and isometries). The author assumes that the fixed point set is Kähler and that the group action in question is semi- simple. He then constructs a metric in the normal direction so that the formulas of the heat equation agree with those of Toledo and Tong.
Reviewer: P.Gilkey
MSC:
58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
32Q99 | Complex manifolds |
57R20 | Characteristic classes and numbers in differential topology |
55N30 | Sheaf cohomology in algebraic topology |
58J20 | Index theory and related fixed-point theorems on manifolds |
35K99 | Parabolic equations and parabolic systems |