Related papers: Holomorphic Approximation via Dolbeault Cohomology

We introduce the \verb|Macaulay2| package \verb|RepHomology| for the computations of representation homology of certain spaces. The main methods implement computing the representation homology of surfaces (with group coefficients, and…

We review the recent development of Hodge theory for almost complex manifolds. This includes the determination of whether the Hodge numbers defined by $\bar\partial$-Laplacian are almost complex, almost K\"ahler, or birational invariants in…

Let D be a domain in C^n with smooth boundary, of finite 1-type at a point p in the boundary and such that the closure of D has a basis of Stein Runge neighborhoods. Assume that there exists an analytic disc which intersects the closure of…

We show that any diffeomorphism of a compact manifold can be C1 approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.

In this article, we investigate the topological properties of complex manifolds by studying Dolbeault-Morse-Novikov cohomology. By establishing an integral inequality, we obtain two main results: (1) When a closed complex manifold admits a…

In this paper we initiate the study of submanifolds of almost hypercomplex manifolds with Hermitian and Norden metrics. Object of investigations are holomorphic submanifolds of the hypercomplex manifolds which are locally conformally…

Let $K$ be a closed polydisc or ball in $\C^n$, and let $Y$ be a quasi projective algebraic manifold which is Zariski locally equivalent to $\C^p$, or a complement of an algebraic subvariety of codimension $\ge 2$ in such manifold. If $r$…

We strengthen the classical approximation theorems of Weierstrass, Runge and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$…

We study almost complex structures with lower bounds on the rank of the Nijenhuis tensor. Namely, we show that they satisfy an $h$-principle. As a consequence, all parallelizable manifolds and all manifolds of dimension $2n\geq 10$…

In this paper, we study deformations of complex structures on Lie algebras and its associated deformations of Dolbeault cohomology classes. A complete deformation of complex structures is constructed in a way similar to the Kuranishi…

Motivated by orbifold string theory, we introduce orbifold cohomology group for any almost complex orbifold and orbifold Dolbeault cohomology for any complex orbifold. Then, we show that our new cohomology group satisfies Poincare duality…

We review properties of closed meromorphic $1$-forms and of the foliations defined by them. We present and explain classical results from foliation theory, like index theorems, the existence of separatrices, and resolution of singularities…

We view Dolbeault-Morse-Novikov cohomology H^{p,q}_\eta(X) as the cohomology of the sheaf \Omega_{X,\eta}^p of \eta-holomorphic p-forms and give several bimeromorphic invariants. Analogue to Dolbeault cohomology, we establish the…

We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts.

We present some results dealing with the local geometry of almost complex manifolds. We establish mainly the complete hyperbolicity of strictly pseudoconvex domains, the extension of plurisubharmonic functions through generic submanifolds…

Let $M$ be a closed complex submanifold in ${\mathbb C}^N$ with the complete K\"ahler metric induced by the Euclidean metric. Several finiteness theorems on the $L^p$ Bergman space of holomorphic sections of a given Hermitian line bundle…

In this talk I give an introduction and present some recent progress towards understanding the cohomology rings of character varieties of Riemann surfaces, such as the proof of the $P=W$ conjecture and the computation of the…

In this paper, we complete the nonabelian Hodge theory (NAHT) triangle of isomorphisms for stacks between the Borel-Moore homologies of the Dolbeault, Betti, and de Rham moduli stacks. We first explain how to realise the category of…

We study the homology groups of the complement of a complexified real line arrangement with coefficients in complex rank-one local systems. Using Borel--Moore homology, we establish an algorithm computing their dimensions via the real…

The purpose of this paper is to study deformation theory of Hom-associative algebra morphisms and Hom-Lie algebra morphisms. We introduce a suitable cohomology and discuss Infinitesimal deformations, equivalent deformations and…