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A differential-geometric approach to deformations of pairs (X, E)

Kwokwai Chan, Yat-Hin Suen (2016)

Complex Manifolds

This article gives an exposition of the deformation theory for pairs (X, E), where X is a compact complex manifold and E is a holomorphic vector bundle over X, adapting an analytic viewpoint `a la Kodaira- Spencer. By introducing and exploiting an auxiliary differential operator, we derive the Maurer–Cartan equation and differential graded Lie algebra (DGLA) governing the deformation problem, and express them in terms of differential-geometric notions such as the connection and curvature of E, obtaining...

A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

Ursula Ludwig (2011)

Annales de l’institut Fourier

The Witten deformation is an analytic method proposed by Witten which, given a Morse function f : M → R on a smooth compact manifold M , allows to prove the Morse inequalities. The aim of this article is to generalise the Witten deformation to stratified Morse functions (in the sense of stratified Morse theory as developed by Goresky and MacPherson) on a singular complex algebraic curve. In a previous article the author developed the Witten deformation for the model of an algebraic curve with cone-like singularities...

Classification of singular germs of mappings and deformations of compact surfaces of class VII₀

Georges Dloussky, Franz Kohler (1998)

Annales Polonici Mathematici

We classify generic germs of contracting holomorphic mappings which factorize through blowing-ups, under the relation of conjugation by invertible germs of mappings. As for Hopf surfaces, this is the key to the study of compact complex surfaces with b 1 = 1 and b ₂ > 0 which contain a global spherical shell. We study automorphisms and deformations and we show that these generic surfaces are endowed with a holomorphic foliation which is unique and stable under any deformation.

Cohomologie de dolbeault le long des feuilles de certains feuilletages complexes

Aziz El Kacimi Alaoui, Jihène Slimène (2010)

Annales de l’institut Fourier

La cohomologie de Dolbeault feuilletée mesure l’obstruction à résoudre le problème de Cauchy-Riemann le long des feuilles d’un feuilletage complexe. En utilisant des méthodes de cohomologie des groupes, nous calculons cette cohomologie pour deux classes de feuilletages : i) le feuilletage complexe affine de Reeb de dimension (complexe) 2 sur la variété de Hopf de dimension 5 ; ii) les feuilletages complexes sur le tore hyperbolique (fibration en tores de dimension n au-dessus d’un cercle et de monodromie...

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