Index theorems for holomorphic maps and foliations

Abstract

We describe a general construction providing index theorems localizing the Chern classes of the normal bundle of a subvariety inside a complex manifold. As particular instances of our construction we recover both Lehmann-Suwa's generalization of the classical Camacho-Sad index theorem for holomorphic foliations and our index theorem for holomorphic maps with positive dimensional fixed point set. Furthermore, we also obtain generalizations of recent index theorems of Camacho-Movasati-Sad and Camacho-Lehmann for holomorphic foliations transversal to a subvariety.

Extendable cohomologies for complex analytic varieties

“…So, (4.7) and (4.8) are proved. Furthermore, a similar argument shows that (1,2) ) ) = dω 1 • F (1,2) . (4.12)…”

“…Then F A ( )ω 1 = F A ( )ω 2 and, in this case, the definition of ω is independent of the extension of ω. (1,2) \Sing (X ) (see Sect. 1).…”

“…(4.8) In order to prove this, let O 1 be an open subset of U 1 such that (1,2) ) and F (2,1) : O 1 → F (2,1) (O 1 ) a holomorphic extension of the biholomorphism F (2,1) : F 1 (A (1,2) ) → F 2 (A (1,2) ) (cp. (1.2)).…”

“…(1.2)). Let O 2 be an open subset of (1,2) ). Shrinking O 1 , if necessary, we can assume without loss of generality that F (2,1) (A (1,2) ) .…”

“…On the other hand,ω 1 | F 1 (A (1,2) (1,2) ) coincide on the dense subset F 1 (A (1,2) ) of F 1 (A (1,2) ), because of (4.3) and (4.4). Then, it resultsω 1 | F 1 (A (1,2) (A (1,2) ) .…”