Revista Matemática Iberoamericana

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Volume 29, Issue 2, 2013, pp. 547–578
DOI: 10.4171/RMI/730

Published online: 2013-04-22

Localization of Atiyah classes

Marco Abate[1], Filippo Bracci[2], Tatsuo Suwa[3] and Francesca Tovena[4]

(1) Università di Pisa, Italy
(2) Università di Roma 'Tor Vergata', Italy
(3) Hokkaido University, Sapporo, Japan
(4) Università di Roma “Tor Vergata”, Italy

We construct the Atiyah classes of holomorphic vector bundles using (1,0)-connections and developing a Chern–Weil type theory, allowing us to effectively compare Chern and Atiyah forms. Combining this point of view with the Čech–Dolbeault cohomology, we prove several results about vanishing and localization of Atiyah classes, and give some applications. In particular, we prove a Bott type vanishing theorem for (not necessarily involutive) holomorphic distributions. As an example we also present an explicit computation of the residue of a singular distribution on the normal bundle of an invariant submanifold that arises from the Camacho–Sad type localization.

Keywords: Atiyah forms and classes, connections of type (1,0), Čech–Dolbeault cohomology, localization and residues, Bott type vanishing, singular holomorphic distributions

Abate Marco, Bracci Filippo, Suwa Tatsuo, Tovena Francesca: Localization of Atiyah classes. Rev. Mat. Iberoamericana 29 (2013), 547-578. doi: 10.4171/RMI/730