GSV-index for holomorphic Pfaff Systems

TL;DR

This paper introduces a GSV-type index for invariant varieties under holomorphic Pfaff systems on projective manifolds, establishing its non-negativity and using it to obstruct solutions to the Poincaré problem.

Contribution

It defines a new GSV index for holomorphic Pfaff systems and proves its non-negativity, providing a novel obstruction to the Poincaré problem in projective spaces.

Findings

GSV index for Pfaff systems is non-negative.

Non-negativity provides obstructions to Poincaré problem solutions.

Applicable to non-locally decomposable 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 non-negativity of the GSV-index gives us an obstruction to the solution of the Poincar\'e problem for Pfaff systems on projectives spaces.