## 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

Zbl 0702.58071
Full Text:

### References:

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