Hodge theory of classifying stacks

Motivic classes of some classifying stacks

We prove that the class of the classifying stack BPGLn is the multiplicative inverse of the class of the projective linear group PGLn in the Grothendieck ring of stacks K0(Stackk) for n = 2 and n = 3 under mild conditions on the base field k. In particular, although it is known that the multiplicativity relation {T} = {S} · {PGLn} does not hold for all PGLntorsors T → S, it holds for the univer...

Derived Equivalences of Smooth Stacks and Orbifold Hodge Numbers

An important step in the development of the parallelism between derived equivalences and the minimal model program, as emphasized especially in the work of Kawamata (see [Kaw09] for a survey), is to extend results about smooth projective Fourier-Mukai partners to the singular case. While in general there are foundational issues still to be resolved, good progress has been made in the case of va...

Hodge Theory of Maps

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...

Hodge Theory of Matroids

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:...

Hodge Theory and Representation Theory