Related papers: Hodge theory of the Goldman bracket

In this paper we show that, after completing in the $I$-adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve $X$ with an algebraic framing is a morphism of…

For an oriented 2-dimensional manifold $\Sigma$ of genus $g$ with $n$ boundary components the space $\mathbb{C}\pi_1(\Sigma)/[\mathbb{C}\pi_1(\Sigma), \mathbb{C}\pi_1(\Sigma)]$ carries the Goldman-Turaev Lie bialgebra structure defined in…

In the mid eighties Goldman proved an embedded curve could be isotoped to not intersect a closed geodesic if and only if their Lie bracket (as defined in that work) vanished. Goldman asked for a topological proof and about extensions of the…

The Turaev cobracket, a loop operation introduced by V. Turaev, which measures self-intersection of a loop on a surface, is a modification of a path operation introduced earlier by Turaev himself, as well as a counterpart of the Goldman…

We discuss a new approach to computing the standard algebraic operations on homotopy classes of loops in surfaces: the homological intersection number, Goldman's Lie bracket, and the author's Lie cobracket. Our approach uses fillings of the…

We characterize in terms of the Goldman Lie algebra which conjugacy classes in the fundamental group of a surface with non empty boundary are represented by simple closed curves. We prove the following: A non power conjugacy class X…

W. Goldman and V. Turaev defined a Lie bialgebra structure on the $\mathbb Z$-module generated by free homotopy classes of loops of an oriented surface (i.e. the conjugacy classes of its fundamental group). We develop a generalization of…

We introduce and study certain topological spaces associated with connected rooted graphs. These spaces reflect combinatorial and order theoretic properties of the underlying graph and relate in the case of hyperbolic graphs to Gromov's…

We show that the category of mixed Hodge complexes admits a Cartan-Eilenberg structure, a notion introduced in [GNPR10] leading to a good calculation of the homotopy category in terms of (co)fibrant objects. This result provides a…

In this paper, we establish an innovative framework in logarithmic Hodge theory for toroidal varieties, introducing weighted toroidal structures and developing a systematic obstruction theory for Hodge classes. Building upon recent advances…

We relate the author's Lie cobracket in the module additively generated by loops on a surface with the Connes-Kreimer Lie bracket in the module additively generated by trees. To this end we introduce a pre-Lie coalgebra and a (commutative)…

The aim of this paper is to study the behavior of Hodge-theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized…

Let $X$ be a compact Riemann surface, $\Sigma$ a finite set of points and $M = X\setminus \Sigma$. We study the $L^2$ cohomology of a polarized complex variation of Hodge structure on a Galois covering of the Riemann surface of finite type…

The free $\mathbb{Z}$-module generated from the set of non-trivial homotopy classes of closed curves on an oriented surface has the structure of Lie bialgebra by two operations, the Goldman bracket and Turaev cobracket. M. Chas gave a…

We show that a homotopy equivalence between compact, connected, oriented surfaces with non-empty boundary is homotopic to a homeomorphism if and only if it commutes with the Goldman bracket.

In 1986 Goldman introduced a Lie algebra structure on the linear span L of conjugacy classes of the fundamental group of a closed oriented surface. It is easy to see that the class e_1 of the trivial loop is a central element in L. We prove…

By introducing a refinement of the Goldman-Turaev Lie bialgebra, we interpret the divergence cocycle in the Kashiwara-Vergne problem and the Enomoto-Satoh obstructions for the surjectivity of the Johnson homomorphisms as some part of a…

In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important…

Let $U$ be a smooth connected complex algebraic variety, and let $f\colon U\to \mathbb C^*$ be an algebraic map. To the pair $(U,f)$ one can associate an infinite cyclic cover $U^f$, and (homology) Alexander modules are defined as the…

We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and…