In this paper, we prove that birational smooth minimal models over C have equal Hodge numbers in all dimensions by an arithmetic method. Our method is a refinement of the method of Batyrev on Betti numbers who used p-adic integration and the Weil conjecture. Our ingredient is to use further arithmetic results such as the Chebotarev density theorem and p-adic Hodge theory.

Whereas usual Hodge theory concerns mainly the usual or abelian cohomology of an algebraic variety—or eventually the rational homotopy theory or nilpotent completion of π1 which are in some sense obtained by extensions—nonabelian Hodge theory concerns the cohomology of a variety with nonabelian coefficients. Because of the basic fact that homotopy groups in higher dimensions are abelian, and si...

We study G-valued semi-stable Galois deformation rings where G is a reductive group. We develop a theory of Kisin modules with G-structure and use this to identify the connected components of crystalline deformation rings of minuscule p-adic Hodge type with the connected components of moduli of “finite flat models with G-structure.” The main ingredients are a construction in integral p-adic Hod...

In this note we compute the rational cohomology ring of Sp3(Z), or equivalently, of A3, the moduli space of principally polarized abelian 3-folds. Denote the Q-Hodge structure of dimension 1 and weight −2n by Q(n). (For those not interested in Hodge theory, just interpret this as one copy of Q.) Denote by λ the first Chern class in H(A3;Q) of the Hodge bundle detπ∗Ω 1 associated to the projecti...

We prove a vanishing theorem for the Hodge number h of projective toric varieties provided by a certain class of polytopes We explain how this Hodge number also gives information about the deformation theory of the toric Gorenstein singularity derived from the same polytope In particular the vanishing theorem for h implies that these deformations are unobstructed

This is a survey of results and conjectures on mirror symmetry phenomena in the nonAbelian Hodge theory of a curve. We start with the conjecture of Hausel–Thaddeus which claims that certain Hodge numbers of moduli spaces of flat SL(n, C) and PGL(n, C)connections on a smooth projective algebraic curve agree. We then change our point of view in the non-Abelian Hodge theory of the curve, and conce...

The purpose of the present manuscript is to continue the survey of the Hodge-Arakelov theory of elliptic curves (cf. [7, 8, 9, 10, 11]) that was begun in [12]. This theory is a sort of “Hodge theory of elliptic curves” analogous to the classical complex and p-adic Hodge theories, but which exists in the global arithmetic framework of Arakelov theory. In particular, in the present manuscript, we...

A class of elliptic-hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, which is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fiel...

Abstract Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge ...

We give an example of a projective smooth surface X over a p-adic field K such that for any prime l different from p, the l-primary torsion subgroup of CH0(X), the Chow group of 0-cycles on X, is infinite. A key step in the proof is disproving a variant of the Block-Kato conjecture which characterizes the image of an l-adic regulator map from a higher Chow group to a continuous étale cohomology...