Related papers: Characteristic classes of mixed Hodge modules and …
Differential graded (DG) commutative algebra provides powerful techniques for proving theorems about modules over commutative rings. These notes are a somewhat colloquial introduction to these techniques. In order to provide some motivation…
The moduli spaces refered to are topological spaces whose path components parametrize homotopy types. Such objects have been studied in two separate contexts: rational homotopy types, in the work of several authors in the late 1970's; and…
In previous work, we constructed for a smooth complex variety $X$ and for a linear algebraic group $G$ a mixed Hodge structure on the complete local ring $\widehat{\mathcal{O}}_\rho$ to the moduli space of representations of the fundamental…
Soergel bimodules are certain bimodules over polynomial algebras, associated with Coxeter groups, and introduced by Soergel in the 1990's while studying the category O of complex semisimple Lie algebras. Even though their definition is…
We define a notion of Hodge modules with rational singularities. A variety has rational singularities in the usual sense, if it is normal and the Hodge module related to intersection cohomology has rational singularities in the present…
Given an affine toric variety $X$ embedded in a smooth variety, we prove a general result about the mixed Hodge module structure on the local cohomology sheaves of $X$. As a consequence, we prove that the singular cohomology of a proper…
We show that the dualizing sheaves of reduced simple normal crossings pairs have a canonical weight filtration in a compatible way with the one on the corresponding mixed Hodge modules by calculating the extension classes between the…
We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory of non-commutative Hodge structures, investigate…
We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. We investigate how various properties of a C-module are affected when considered in the module category of B. We give a complete…
Formal (mixed) Hodge structures FHS are introduced in such a way that the Hodge realization of Deligne's 1-motives extends to a realization from Laumon's 1-motives to formal Hodge structures of level 1, providing an equivalence of…
We give a generalization of the Hodge operator to spaces $(V,h)$ endowed with a hermitian or symmetric bilinear form $h$ over arbitrary fields, including the characteristic two case. Suitable exterior powers of $V$ become free modules over…
In this paper we investigate the relationship between twisted and untwisted character varieties via a specific instance of the Cohomological Hall algebra for moduli of objects in 3-Calabi-Yau categories introduced by Kontsevich and…
We give an overview of differential cohomology from a modern, homotopy-theoretic perspective in terms of sheaves on manifolds. Although modern techniques are used, we base our discussion in the classical precursors to this modern approach,…
These are the lecture notes from my course in the January 2011 School on Moduli Spaces at the Newton Institute. I give an introduction to Higgs bundles and their application to the study of character varieties for surface group…
In this article we are interested in morphisms without slope for mixed Hodge modules. We first show the commutativity of iterated nearby cycles and vanishing cycles applied to a mixed Hodge module in the case of a morphism without slope.…
We introduce a theory of modules over a representation of a small category taking values in entwining structures over a semiperfect coalgebra. This takes forward the aim of developing categories of entwined modules to the same extent as…
We elaborate on the recent proposal that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with punctures compute tree-level scattering amplitudes in quantum field theories. The relevant…
In representation theory, the double centraliser property is an important property for a module (bimodule). It plays a fundamental role in many theories. In this paper, we extend this property to complexes in derived categories of finite…
In this paper, we show that the tilting modules over a cluster-tilted algebra $A$ lift to tilting objects in the associated cluster category $\mathcal{C}_H$. As a first application, we describe the induced exchange relation for tilting…
We identify the category of real mixed Hodge structures with the category of vector bundles with connections (not necessarily flat) on C, equivariant with respect to C^*. Here C is the complex plane considered as a 2-dimensional real…