Double transgressions and Bott-Chern duality

Daniel Cibotaru; Vincent Grandjean; Blaine Lawson, Jr

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arXiv · 2003.14326

Double transgressions and Bott-Chern duality

Daniel Cibotaru·Vincent Grandjean·Blaine Lawson, Jr

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Abstract

We present a general framework for obtaining currential double transgression formulas on complex manifolds which can be seen as manifestations of Bott-Chern Duality. These results complement on one hand the simple transgression formulas obtained by Harvey -Lawson and on the other hand the double transgression formulas of Bismut-Gillet-Soul\'e. Among the applications we mention a Gysin isomorphism for Bott-Chern cohomology, an abstract Poincar\'e-Lelong formula for sections of holomorphic and Hermitian vector bundles implying Andersson's generalization of the standard Poincar\'e-Lelong, a Bott-Chern duality formula for the Chern-Fulton classes of singular varieties or a refinement of the first author's simple transgression formula for the Chern character of a Quillen superconnection associated to a self-adjoint, odd endomorphism. The existence of a Bismut-Gillet-Soul\'e double transgression without the hypothesis of degeneration along a submanifold stands out and is based on an extension to linear correspondences of the operation of morphism addition. Finally, as a by-product we also obtain a statement about the {pointwise localization} of the Samuel multiplicity of an analytic subvariety of a complex manifold along an irreducible component.

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Daniel Cibotaru, Vincent Grandjean, Blaine Lawson, Jr. 2020-03-31. Double transgressions and Bott-Chern duality. https://arxiv.org/abs/2003.14326

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