Related papers: Hypercohomologies of truncated twisted holomorphic…
This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In two previous papers, we develop the basic theory of formal manifolds,…
The purpose of this article is to show a second main theorem with the explicit truncation level for holomorphic mappings of $ \mathbb{C} $ (or of a compact Riemann surface) into a compact complex manifold sharing divisors in subgeneral…
We use the Dolbeault cohomology to investigate the Koszul-Brylinski homology on holomorphic Poisson manifolds. We obtain the Leray-Hirsch theorem for Hochschild homology and the Mayer-Vietoris sequence, K\"{u}nneth theorem for holomorphic…
Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential…
Degree one twisting of Deligne cohomology, as a differential refinement of integral cohomology, was established in previous work. Here we consider higher degree twists. The Rham complex, hence de Rham cohomology, admits twists of any odd…
We prove the K\"unneth formula for the irregular Hodge filtrations on the exponentially twisted de Rham and the Higgs cohomologies of smooth quasi-projective complex varieties. The method involves a careful comparison of the underlying…
We use a sheaf-theoretic approach to obtain a blow-up formula for Dolbeault cohomology groups with values in the holomorphic vector bundle over a compact complex manifold. As applications, we present several positive (or negative) examples…
Let $X$ be a smooth projective $R$-scheme, where $R$ is a smooth $\Z$-algebra. As constructed by Hesselholt, we have the absolute big de Rham-Witt complex $\W\Omega^*_X$ of $X$ at our disposal. There is also a relative version…
We extend the results of Deligne and Illusie on liftings modulo $p^2$ and decompositions of the de Rham complex in several ways. We show that for a smooth scheme $X$ over a perfect field $k$ of characteristic $p>0$, the truncations of the…
We introduce a concept of blown-up \v{C}ech cohomology for coherent sheaves of homological dimension $\leq 1$ and some quasi-coherent sheaves on a non-singular real affine variety. Its construction involves a directed set of multi-blowups.…
We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences…
We develop the theory of twisted L^2-cohomology and twisted spectral invariants for flat Hilbertian bundles over compact manifolds. They can be viewed as functions on the first de Rham cohomology of M and they generalize the standard…
In a series of papers the authors introduced the so-called blown-up intersection cochains. These cochains are suitable to study products and cohomology operations of intersection cohomology of stratified spaces. The aim of this paper is to…
We prove a vanishing theorem for the twisted de Rham cohomology of a compact manifold.
In this thesis we examine a set of foundational questions concerning closed forms in superspace. By reformulating a number of definitions through the use of a new ring of (anti-)commuting variables and the concept of an exact Bianchi form,…
We study the cohomology of broken toric varieties via the derived push-forward of the constant sheaf to a complex of polytopes, proving a Deligne-type decomposition theorem, degeneration of the associated Leray-Serre spectral sequence, and…
We discuss twisted cohomology, not just for ordinary cohomology but also for $K$-theory and other exceptional cohomology theories, and discuss several of the applications of these in mathematical physics. Our list of applications is by no…
For having a Poincar\'e duality via a cap product between the intersection homology of a paracompact oriented pseudomanifold and the cohomology given by the dual complex, G. Friedman and J. E. McClure need a coefficient field or an…
Higher twisted $K$-theory is an extension of twisted $K$-theory introduced by Ulrich Pennig which captures all of the homotopy-theoretic twists of topological $K$-theory in a geometric way. We give an overview of his formulation and key…
In this paper, we develop a new approach to the deformation theory of restricted Lie-Rinehart algebras in positive characteristic, based on the deformation theory of restricted morphisms introduced in our earlier work. We provide a full…