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GSV-index for holomorphic Pfaff systems. (English) Zbl 1447.58001

Summary: In this work we introduce a GSV type index for varieties invariant by holomorphic Pfaff systems (possibly non locally decomposables) on projective manifolds. We prove a non-negativity property for the index. As an application, we prove that the non-negativity of the GSV-index gives us an obstruction to the solution of the Poincaré problem for Pfaff systems on projectives spaces.

MSC:

58A17 Pfaffian systems
32S65 Singularities of holomorphic vector fields and foliations
32M25 Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions
37F75 Dynamical aspects of holomorphic foliations and vector fields
37B30 Index theory for dynamical systems, Morse-Conley indices
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