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Koszul complexes, harmonic oscillators, and the Todd class. (English) Zbl 0702.58071

If \(0\to L\to M\to N\to 0\) is an exact sequence of holomorphic vector bundles on a complex manifold then the Todd polynomials satisfy \(Td(M)=Td(L)Td(N)\). Introducing hermitian metrics, one has explicit Chern-Weil representatives \(Td(g^ M)\) etc. for these classes, but the above relation holds only in cohomology. This paper calculates a form \({\mathfrak B}\), well-defined modulo \(\partial\) and \({\bar \partial}\) coboundaries such that \[ \partial {\bar \partial}{\mathfrak B}=Td(g^ L)- Td(g^ M)/Td(g^ N). \] The motivation for doing so is to understand better the relationship between Quillen metrics and determinants for the situation of an immersion of one complex manifold in another. It also forms part of a large program of work undertaken by the author in this direction. In the course of the proof, which uses a variety of techniques and results including an infinite-dimensional version of Quillen’s superconnections, a number of interesting formulae and concepts come to light. One of these is a 1-parameter deformation of the Hirzebruch \(\hat A\)-polynomial based on the function \[ \phi(x,u)= (4/u)\sinh[(x+\sqrt{x^ 2+4u})/4] \sinh[(-x+ \sqrt{x^ 2+4u})/4]. \] Another is the natural appearance of the power series \[ R(x)=\sum_{n\geq 1,n\quad odd}(\sum^{n}_{1}\frac{1}{j}+ 2\frac{\zeta'(-n)}{\zeta(-n)}) \zeta(-n)\frac{x^ n}{n!} \] which plays an important role in the arithmetic Riemann-Roch theorem of Gillet and Soulé.
The results of this paper were first announced by the author in C. R. Acad. Sci., Paris, Sér. I 309, No.2, 111-114 (1989; Zbl 0673.58043).
Reviewer: N.J.Hitchin

MSC:

58J10 Differential complexes
57R42 Immersions in differential topology
57R20 Characteristic classes and numbers in differential topology
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results

Citations:

Zbl 0673.58043
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