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Equivariant short exact sequences of vector bundles and their analytic torsion forms. (English) Zbl 0817.32014

Let \(E : 0 \to L \to M \to N \to 0\) be a short exact sequence of holomorphic Hermitian vector bundles on a complete manifold \(B\), equipped with a holomorphic unitary chain map \(g\). The author tries to construct certain characteristic classes associated to the sequence \(E\) in order to generalize the author’s previous work [J. Am. Math. Soc. 3, No. 1, 159- 256 (1990; Zbl 0702.58071)], where the case \(g = 1\) was considered. Let \(h^ M\) be a Hermitian metric on \(M\) and let \(h^ L\), \(h^ N\) be the induced metrics on \(L,N\). Using these metrics the author first constructs the Levi-Civita superconnection \(B_ u\) for \(u > 0\). The author next defines the generalized supertrace, which is a smooth closed form on \(B\). After studying the asymptotic behavior of the generalized supertraces the author finally constructs generalized analytic torsion forms \(B_ g (L,M, h^ M)\) on \(B\). The main purpose of this paper is to calculate the form \(B_ g (L,M, h^ M)\) in terms of other invariants such as the Todd characteristic classes.

MSC:

32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
32Q99 Complex manifolds
57R20 Characteristic classes and numbers in differential topology

Citations:

Zbl 0702.58071
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References:

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