相关论文: A homology and cohomology theory for real projecti…

We study a natural Hodge theoretic generalization of rational (or $\mathbb{Q}$-)homology manifolds through an invariant ${\rm HRH(Z)}$ where $Z$ is a complex algebraic variety. The defining property of this notion encodes the difference…

Reflexive homology and dihedral homology are the homology theories associated to the reflexive and dihedral crossed simplicial groups respectively. The former has recently been shown to capture interesting information about…

Bivariant theory is a unified framework for cohomology and Borel-Moore homology theories. In this paper, we extract an $\infty$-enhanced bivariant homology theory from Gaitsgory-Rozenblyum's six functor formalism.

In this paper, we mainly build up the theory of sheaf-correspondence filtered spaces and stratified de Rham complexes for studying singular spaces. We prove the finiteness of a stratified de Rham cohomology and obtain its isomorphism to…

We explore and extend the application of homological algebra to describe quantum entanglement, initiated in arXiv:1901.02011, focusing on the Hodge-theoretic structure of entanglement cohomology in finite-dimensional quantum systems. We…

We revisit Spakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology. We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it…

In this expository article, we outline the theory of harmonic differential forms and its consequences. We provide self-contained proofs of the following important results in differential geometry: (1) Hodge theorem, which states that for a…

We prove equivariant versions of the Beilinson-Lichtenbaum conjecture for Bredon motivic cohomology of smooth complex and real varieties with an action of the group of order two. This identifies equivariant motivic and topological…

Let X be a complex projective variety of dimension n with only isolated normal singularities. In this paper we prove, using mixed Hodge theory, that if the link of each singular point of X is (n-2)-connected, then X is a formal topological…

We develop a theory of Cech-Bott-Chern cohomology and in this context we naturally come up with the relative Bott-Chern cohomology. In fact Bott-Chern cohomology has two relatives and they all arise from a single complex. Thus we study…

In this article we study the cohomological and homological (due to Jannsen) Hodge conjecture for singular varieties. The motivation for studying singular varieties comes from the fact that any smooth projective variety X is birational to a…

In this article, we pursue two main objectives. The first is to show that the fundamental results of Green-Lazarsfeld (1987, 1991) on generic vanishing theorems, and works of Budur-Wang (2015, 2020) on cohomology jumping loci, can be…

This note discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties tends to be quite large. In particular, any singular projective variety with at worst ordinary double points has…

We explore various aspects of 2-form topological gauge theories in (3+1)d. These theories can be constructed as sigma models with target space the second classifying space $B^2G$ of the symmetry group $G$, and they are classified by…

We provide a geometric approach to constructing Lefschetz collections and Landau-Ginzburg Homological Projective Duals from a variation of Geometric Invariant Theory quotients. This approach yields homological projective duals for Veronese…

We explicitly describe cohomology of the sheaf of differential forms with poles along a semiample divisor on a complete simplicial toric variety. As an application, we obtain a new vanishing theorem which is an analogue of the…

A real Bott manifold is the total space of iterated RP^1 bundles starting with a point, where each RP^1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their…

Let $G$ be a compact, connected, and simply-connected Lie group viewed as a $G$-space via the conjugation action. The Freed-Hopkins-Teleman Theorem (FHT) asserts a canonical link between the equivariant twisted $K$-homology of $G$ and its…

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 three previous papers, we introduce the notion of formal manifolds and study…

We introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an Eilenberg-MacLane space. Using these cocycles we develop a ``cohomology theory" for complex algebraic varieties. The theory is bigraded, functorial,…