The Hodge de Rham theory of relative Malcev completion

This is a significant revision of the early version of this paper which was posted last December. The speculative section has been removed in light of some recent results of Morita and Kawazumi. Numerous typos have been fixed. The companion paper "The Hodge de Rham Theory of Relative Malcev Completion" has just been posted.

Lecture notes of lectures delivered at the Hodge Theory Summer School in ICTP Trieste, June 2010.

These lecture notes in the De Rham-Hodge theory are designed for a 1-semester undergraduate course (in mathematics, physics, engineering, chemistry or biology). This landmark theory of the 20th Century mathematics gives a rigorous foundation to modern field and gauge theories in physics, engineering and physiology. The only necessary background for comprehensive reading of these notes is Green's theorem from multivariable calculus.

Restrictions are placed on the presentation and holonomy algebra of a K\"ahler group. Let $X$ be compact K\"ahler. It is known that the formality of $X$, its Albanese map and the Hodge structure of $H^{\ast}(X)$ impose restrictions on the holonomy algebra of $\pi_1 X$. We derive further restrictions from Sullivan's 1-minimal model. An effective algorithm is given to compute the holonomy algebra of a group from a presentation. Some very strong restrictions on K\"ahler groups w...

We prove that, generically on a smooth, connected variety in characteristic zero, local systems of geometric origin are stable under extension in the category of all local systems. As a consequence of this, we obtain a (generic) motivic strengthening of Hain's theorem on Malcev completions of monodromy representations. Our methods are Tannakian, and rely on an abstract criterion for ``Malcev completeness'', which is proved in the first part of the paper. A couple of seconda...

The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a novel way.

In this survey paper, we give a complete list of known results on the first and the second homology groups of surface mapping class groups. Some known results on higher (co)homology are also mentioned.

Two standard invariants used to study the fundamental group G of the complement X of a hyperplane arrangement are the Malcev completion of G and the cohomology groups of X with coefficients in rank one local systems. In this paper, we develop a tool that unifies these two approaches. This tool is the Malcev completion S_p of G relative to a homomorphism p from G into (C^*)^N. This is a prosolvable group that is tightly controlled by the cohomology groups of X with coefficient...

In this paper, we explore a notion of nonabelian Hodge structure on the fundamental group of an algebraic variety. This is approach is compared to some alternative approaches due to Morgan, Hain and others. We also give criteria for a variety to be a Hodge theoretic K(pi,1), which roughly means that the cohomology of variations of mixed Hodge structure can be determined from the group.

In this paper we study the proalgebraic completion of mapping class relative to their maps to the symplectic group. The main result is that the natural map from the unipotent (a.k.a. Malcev) completion of the Torelli group to the prounipotent radical of the Sp_g completion of the mapping class group is a non trivial central extension with kernel isomorphic to Q, at least when g \ge 8. The theorem is proved by relating the central extension to the line bundle associated to the...