Nonabelian Hodge theory in characteristic p

Nonabelian Hodge Theory in Characteristic p

Given a scheme in characteristic p together with a lifting modulo p2, 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 decomposition theorem of Deligne-Illusie to the case of de Rham cohomology with coefficients.

Algebraic aspects of higher nonabelian Hodge theory

Nonabelian mixed Hodge structures

We propose a definition of “nonabelian mixed Hodge structure” together with a construction associating to a smooth projective variety X and to a nonabelian mixed Hodge structure V , the “nonabelian cohomology of X with coefficients in V ” which is a (pre-)nonabelian mixed Hodge structure denoted H = Hom(XM , V ). We describe the basic definitions and then give some conjectures saying what is su...

Opers versus Nonabelian Hodge

For a complex simple simply connected Lie group G, and a compact Riemann surface C, we consider two sorts of families of flat G-connections over C. Each family is determined by a point u of the base of Hitchin’s integrable system for (G, C). One family ∇h̄,u consists of G-opers, and depends on h̄ ∈ C×. The other family ∇R,ζ,u is built from solutions of Hitchin’s equations, and depends on ζ ∈ C×, ...

Stringy Hodge Numbers and P-adic Hodge Theory

Stringy Hodge numbers are introduced by Batyrev for a mathematical formulation of mirror symmetry. However, since the stringy Hodge numbers of an algebraic variety are defined by choosing a resolution of singularities, the well-definedness is not clear from the definition. Batyrev proved the well-definedness by using the theory of motivic integration developed by Kontsevich, Denef-Loeser. The a...