Richard Hain

The hodge de rham theory of relative malcev completion

Abstract Suppose that X is a smooth manifold and ρ : π1(X,x) → S is a representation of the fundamental group of X into a real reductive group with Zariski dense image. To such data one can associate the Malcev completion G of π1(X,x) relative to ρ. In this paper we generalize Chens iterated integrals and show that the Ho of a suitable complex of these iterated integrals is the coordinate ring of G . This is used to show that if X is a complex algebraic manifold and ρ is the monodromy representation of a variation of Hodge structure over X, then the coordinate ring of G has a canonical mixed Hodge structure.

Weighted Completion of Galois Groups and Galois Actions on the Fundamental Group of P1 - {0,1, ∞}

Fix a prime number ℓ. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro-ℓ completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goncharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted completion of a profinite group with respect to a reductive representation (and other appropriate data).

Locally Symmetric Families of Curves and Jacobians

The moduli space A g of principally polarized abelian varieties of dimension g is a locally symmetric variety. Denote the closure in A g of the locus of jacobians by J g . In this paper we make a preliminary investigation of locally symmetric subvarieties X of A g that are contained in J 9 and contain the moduli point of the jacobian of a smooth curve. Under certain hypotheses (X is “simple”, the corresponding family of abelian varieties can be lifted to a family of curves and a rank condition), we prove that such an X has to be a ball quotient. Our main tools are group cohomology and naive geometric considerations.

THE ABELIANIZATION OF THE JOHNSON KERNEL

We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a plane curve

Mixed Hodge structures on homotopy groups

C

Rational points of universal curves

and the stability of the sheaf of logarithmic vector fields along

A De Rham-Witt approach to crystalline rational homotopy theory

C

Twisting cochains and duality between minimal algebras and minimal Lie algebras

, the freeness of the divisor

On the fundamental groups of normal varieties

C

Remarks on non-abelian cohomology of proalgebraic groups

and the Torelli properties of