GSV-Index for Holomorphic Pfaff Systems

Abstract

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 nonnegativity of the GSV-index gives us an obstruction to the solution of the Poincaré problem for Pfaff systems on projectives spaces.

A brief introduction on residue theory of holomorphic foliations

“…It is well known that such a bound does not exist in general, see [21]. However, under certain hypotheses, there are several partial answers and generalizations even for flags and for Pfaff systems; see for instance [15,19,21,23,24,28,33,36,37,49,51]. The residue theory, and especially the Baum-Bott Theorem, are powerful tools and obstructions for several problems related to foliations with singularities.…”

“…In [13] Brunella says that the GSV-index is the obstruction to a positive solution to Poincaré problem and gives a simple condition that implies the non-negativity of this index. Motivated by Brunella's work, Corrêa and Machado in [28] introduced a GSV type index for invariant varieties by holomorphic Pfaff systems on projective manifolds. The authors proved, with certain hypotheses, a non-negativity property for this index.…”

“…It is worth remark that there are other types of residues and invariants associated with a foliations and distributions. For instance, residues of logarithmic vector fields in [31], Camacho-Sad index in [20], GSV-index of foliations and Pfaff systems in [28,39]. Another interesting topic that has been studied recently is the residue theory for flags of foliations and distributions.…”

A brief survey on residue theory of holomorphic foliations

“…This question is known as the Poincaré problem. Although it is well known that such a bound does not exist in general, see [19], under certain hypotheses, there are several works about this problem which answer it partially and there are several generalizations even for flags and for Pfaff systems; see for instance [14,17,19,30,33,43,46,26,27,32,25]. The residue theory, in special the Baum-Bott Theorem, are powerful tool and obstructions of several problems related to foliations with singularities.…”

“…It is worth remark that there are other types of residues and invariants associated to a foliation, as residues of logarithmic vector fields in [24], Camacho-Sad index in [16] and GSV-index of foliations and Pfaff systems in [52,25].…”