On Gauss–Bonnet and Poincaré–Hopf Type Theorems for Complex  ∂ -Manifolds

Corrêa, Maurício; Lourenço, Fernando; Machado, Diogo da Silva; Ferreira, Antonio M.

doi:10.17323/1609-4514-2021-21-3-493-506

Cited by 2 publications

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“…This section is dedicated to present some results about residues associated with logarithmic holomorphic foliations of dimension one. For this we refer the readers to [1,25,31,38,40] and the references therein.…”

Section: 4mentioning

confidence: 99%

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A brief introduction on residue theory of holomorphic foliations

Lourenço¹,

Reis²

2023

Preprint

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This is a survey paper dealing with holomorphic foliations, with emphasis on residue theory and its applications. We start recalling the definition of holomorphic foliations as a subsheaf of the tangent sheaf of a manifold. The theory of Characteristic Classes of vector bundles is approached from this perspective. We define Chern classes of holomorphic foliations using the Chern-Weil theory and we remark that the Baum-Bott residue is a great tool that help us to classify some foliations. We present throughout the survey several recent results and advances in residue theory. We finish by presenting some applications of residues to solve for example the Poincaré problem and the existence of minimal sets for foliations.

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Section: 4mentioning

confidence: 99%

“…In [25] the authors prove new versions of Gauss-Bonnet and Poincaré-Hopf theorems for complex ∂-manifolds of the type X = X − D, where dimX = n ≥ 3 and D is a reduced divisor. More precisely, Theorem 2.18.…”

Section: 4mentioning

confidence: 99%

A brief introduction on residue theory of holomorphic foliations

Lourenço¹,

Reis²

2023

Preprint

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The Poincaré Index on Singular Varieties

Aleksandrov

2022

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In this paper, we discuss a few simple methods for computing the local topological index and its various analogs for vector fields and differential forms given on complex varieties with singularities of different types. They are based on properties of regular meromorphic and logarithmic differential forms, of the dualizing (canonical) module and related constructions. In particular, we show how to compute the index on Cohen–Macaulay, Gorenstein and monomial curves, on normal and non-normal surfaces and some others. In contrast with known traditional approaches, we use neither computers, nor integration, perturbations, deformations, resolution of singularities, spectral sequences or other related standard tools of pure mathematics.

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On Gauss–Bonnet and Poincaré–Hopf Type Theorems for Complex ∂ -Manifolds

Cited by 2 publications

References 0 publications

A brief introduction on residue theory of holomorphic foliations

A brief introduction on residue theory of holomorphic foliations

The Poincaré Index on Singular Varieties

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