Combinatorics of binomial decompositions of the simplest Hodge integrals

abelian varieties的退化组合学与Hodge理论：Mumford构造的综述

Combinatorics and Hodge theory of degenerations of abelian varieties: A survey of the Mumford construction

综述Mumford构造退化的abelian varieties，分析其与toric几何的关系，并研究与regular matroids相关的多变量退化几何与Hodge理论，扩展Clemens关于1参数半稳定退化的基本结果。

Hodge theory in combinatorics

George Birkhoff proved in 1912 that the number of proper colorings of a finite graph G with n colors is a polynomial in n, called the chromatic polynomial of G. Read conjectured in 1968 that for any graph G, the sequence of absolute values of coefficients of the chromatic polynomial is unimodal: it goes up, hits a peak, and then goes down. Read's conjecture was proved by June Huh in a 2012 paper making heavy use of methods from algebraic geometry. Huh's result was subsequently refined and generalized by Huh and Katz, again using substantial doses of algebraic geometry. Both papers in fact esta…

Notes on the Hodge Conjecture for Fermat Varieties

We review a combinatoric approach to the Hodge Conjecture for Fermat Varieties and announce new cases where the conjecture is true.

Coxeter combinatorics and spherical Schubert geometry

For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are spherical for the action of a Levi subgroup. We evidence the conjecture, employing the combinatorics of Demazure modules, and work of R. Avdeev-A. Petukhov, M. Can-R. Hodges, R. Hodges-V. Lakshmibai, P. Karuppuchamy, P. Magyar-J. Weyman-A. Zelevinsky, N. Perrin, J. Stembridge, and B. Tenner. In type A, we establish connections with the key polynomials of A. …

The Hodge theory of the Decomposition Theorem (after de Cataldo and Migliorini)

In its simplest form the Decomposition Theorem asserts that the rational intersection cohomology of a complex projective variety occurs as a summand of the cohomology of any resolution. This deep theorem has found important applications in algebraic geometry, representation theory, number theory and combinatorics. It was originally proved in 1981 by Beilinson, Bernstein, Deligne and Gabber as a consequence of Deligne's proof of the Weil conjectures. A different proof was given by Saito in 1988, as a consequence of his theory of mixed Hodge modules. More recently, de Cataldo and Migliorini foun…

Rational Mixed Tate Motivic Graphs

In this paper, we study the combinatorics of a subcomplex of the Bloch-Kriz cycle complex [4] used to construct the category of mixed Tate motives. The algebraic cycles we consider properly contain the subalgebra of cycles that correspond to multiple logarithms (as defined in [12]). We associate an algebra of graphs to our subalgebra of algebraic cycles. We give a purely graphical criterion for admissibilty. We show that sums of bivalent graphs correspond to coboundary elements of the algebraic cycle complex. Finally, we compute the Hodge realization for an infinite family of algebraic cycles …