Related papers: The Hodge theoretic fundamental group and its coho…

Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. Fix $n\geq 2$, and an integer $d$. A pair $(E,\phi)$ over $X$ consists of an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $X$ and a section…

If $X$ is a topological space then there is a natural homomorphism $\pi_1(X)\rightarrow K_1(X)$ from a fundamental group to a $K_1$-homology group. Covering projections depend of fundamental group. So $K_1$-homology groups are interrelated…

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas…

In this paper we investigate the Lie algebra structure of the first relative Hochschild cohomology and its relation with the relative notion of fundamental group. Let $A,B$ be finite-dimensional basic $k$-algebras over an algebraically…

Let K be a field of characteristic zero, f(x) be a polynomial with coefficients in K and without multiple roots. We consider the superelliptic curve C_{f,q} defined by y^q=f(x), where q=p^r is a power of a prime p. We determine the Hodge…

We classify pointed Hopf algebras of discrete corepresentation type over an algebraically closed field K with characteristic zero. For such algebras $H$, we explicitly determine the algebra structure up to isomorphism for the link…

We investigate models of algebraic theories in the category of cocommutative coalgebras over a field. We establish some of their categorical properties, similar to those of algebraic varieties. We introduce a class of categories of…

Motivated by our attempt to understand characteristic classes of Lie groupoids and geometric structures, we are brought back to the fundamentals of the cohomology theories of Lie groupoids and algebroids. One element that was missing in the…

This is a survey on formality results relying on weight structures. A weight structure is a naturally occurring grading on certain differential graded algebras. If this weight satisfies a purity property, one can deduce formality. Algebraic…

Given a scheme in characteristic p together with a lifting modulo p^2, we construct a functor from a category of suitably nilpotent modules with connection to the category of Higgs modules. We use this functor to generalize the…

In this paper we are interested in proving that the infinitesimal variations of Hodge structure of hypersurfaces of high enough degree lie in a proper subvariety of the variety of all infinitesimal variations. This is proved using a space…

We introduce the notion of mixed Hodge complex on an algebraic variety, improving Du Bois' filtered complex, and relate Deligne's theory of mixed Hodge structure with the theory of mixed Hodge module. This was supposed to be true, but is…

Let $V$ be a complex projective variety with isolated singularities. Let the smooth part be given the metric induced by a projective imbedding. Then we develop the $L_2$ harmonic theory and construct a pure Hodge structure on the…

We define an unstable equivariant motivic homotopy category for an algebraic group over a Noetherian base scheme. We show that equivariant algebraic $K$-theory is representable in the resulting homotopy category. Additionally, we establish…

We give a geometric proof of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber for the direct image of the intersection cohomology complex under a proper map of complex algebraic varieties. The method rests on new…

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that…

Collino proved that the fundamental group of a certain Zariski open set of the symmetric square of a hyperelliptic curve is isomorphic to the integral Heisenberg group. We compute the mixed Hodge structure on this fundamental group, and…

We introduce and study the category of Hodge microsheaves which is a Hodge-version of the category of microsheaves for a certain class of holomorphic exact symplectic manifolds. We then study Hodge-theoretic version of wrapped sheaves and…

We show that Hopf invariants, defined by evaluation in Harrison cohomology of the commutative cochains of a space, calculate the logarithm map from a fundamental group to its Malcev Lie algebra. They thus present the zeroth Harrison…

We study rank $1$ flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank $1$ flat bundles, we represent the structure theorem of K\"ahler solvmanifolds as…