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∂¯-tangential invariants of certain vector bundles over complex foliations

Research paper by Cristian Ida, Paul Popescu

Indexed on: 26 Nov '16Published on: 23 Nov '16Published in: Journal of Geometry and Physics



Abstract

The Dolbeault cohomology plays an important role in the study of some <img height="15" border="0" style="vertical-align:bottom" width="11" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0393044016302789-si1.gif">∂¯-invarants of complex and holomorphic vector bundles as <img height="15" border="0" style="vertical-align:bottom" width="11" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0393044016302789-si1.gif">∂¯-Chern classes and Atiyah classes. In this paper we generalize similar invariants and their properties in the tangential Dolbeault cohomology. More exactly, we introduce and we study tangential Atiyah classes for FF-holomorphic vector bundles and <img height="15" border="0" style="vertical-align:bottom" width="11" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0393044016302789-si1.gif">∂¯-tangential Chern classes for tangentially smooth vector bundles over a manifold MM endowed with a complex foliation FF. Also, <img height="15" border="0" style="vertical-align:bottom" width="11" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0393044016302789-si1.gif">∂¯-tangential secondary invariants are studied following similar constructions for Lie algebroids. The notions are introduced by a global formalism that is used in the tangential theory of foliated spaces.