According to Deligne, the cohomology groups of a complex algebraic variety carry a generalized Hodge structure, or, in precise terms, a mixed Hodge structure [2]. The purpose of this paper is to introduce an absract theory of extensions of mixed Hodge structures which has proved useful in the study of low-dimensional varieties [1]. To give the theory meaning, we will give one simple, but illust...

If G is a finite graph, a proper coloring of G is a way to color the vertices of the graph using n colors so that no two vertices connected by an edge have the same color. (The celebrated four-color theorem asserts that if G is planar, then there is at least one proper coloring of G with four colors.) By a classical result of Birkhoff, the number of proper colorings of G with n colors is a poly...

Recall that a Mumford–Tate domain DM is an open set in its compact dual ĎM — the latter is a rational homogeneous variety defined over Q (think of the upper half plane H ⊂ P1). Thus a polarized Hodge structure (PHS) φ has an “upstairs” field of definition k(φ). If H•,• ⊂ T •,• is a subalgebra then the Noether-Lefschetz locus NLH = {φ : Hg•,• φ ⊇ H} is defined over Q. Its components are defined ...

This paper is an extended version of an expository talk given at the workshop “Topology of Stratified Spaces” at MSRI in September 2008. It gives an introduction and overview about recent developments on the interaction of the theories of characteristic classes and mixed Hodge theory for singular spaces in the complex algebraic context. It uses M. Saito’s deep theory of mixed Hodge modules as a...

The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar constraints on algebraic maps? Let f : X → Y a proper morphism. Our goal is going to be to linearize the problem. For this lecture, we will assume f is projective and smooth, to simplify the problem. In fact, we will assume X and Y are nonsingular. So our map factors X → P × Y → Y and df is s...

Introduction Logarithmic concavity is a property of a sequence of real numbers, occurring throughout algebraic geometry, convex geometry, and combinatorics. A sequence of positive numbers a0,... ,ad is log-concave if a2 i ≥ ai−1ai+1 for all i. This means that the logarithms, log(ai), form a concave sequence. The condition implies unimodality of the sequence (ai), a property easier to visualize:...

We discuss some basic applications of higher dimensional residues as presented in [7] and [8, Chap. V].