Vector?eldsonmanifoldsplaymajorrolesinmathematicsandothersciences. In particular, the Poincar e Hopf index theorem and its geometric count- part, the Gauss Bonnettheorem, giveriseto the theoryof Chernclasses, key invariants of manifolds in geometry and topology. One has often to face problems where the underlying space is no more a manifold but a singular variety. Thus it is natural to ask what is the good notionofindexofavector?eld, andofChernclasses, ifthespaceacquiress- gularities.Thequestionwasexploredbyseveralauthorswithvariousanswers, starting with the pioneering work of M.-H. Schwartz...

Complex Analytic Geometry is one of the most important fields of Mathematics. It has a long history that culminated in the Cauchy integral formula in the 19th century. The theory was vastly developed and closely related to many other fields of Mathematics as well as the Sciences, where numerous applications have been found. This book starts off with the basic material, introducing characteristic classes mainly via the Chern - Weil theory, explaining the idea of localization of characteristic classes and presenting the aforementioned invariants and relations in a unified way from this...