The Hodge diamond of a smooth projective complex variety contains essential topological and analytic information, including fundamental symmetries provided by Poincaré and Serre duality. I will describe recent progress on understanding how much symmetry there is in the analogous Hodge-Du Bois diamond of a singular variety, and the concrete ways in which this symmetry reflects the singularity types. In the process, we will see how invariants from commutative algebra and birational geometry influence the topology of an algebraic variety, for instance by means of new weak Lefschetz theorems.
Hodge symmetries of singular varieties Sponsored by the Meyer Fund
Conservative graph-colouring problems, also known as list homomorphism problems, have been examined for various classes of finite digraphs. Each instance consists of a finite digraph along with a list of permitted values for each vertex. Lifting the scenario to infinite digraphs G, the 1-orbits of an oligomorphic subgroup of Aut(G) take the role of elements. A conservative template in this framework is enriched by a unary relation for all unions of 1-orbits of . Assuming only access to unions of pairs of 1-orbits, we obtain the first structural dichotomy for -categorical smooth digraphs of algebraic length 1, yielding a hardness criterion for the respective conservative graph-colouring problems. We thereby overcome previous obstacles to lifting structural results for digraphs in this context from finite to -categorical structures -- the strongest lifting results hitherto applying only to undirected graphs. This is joint work with Marcin Kozik, Tomas Nagy, and Michael Pinsker.
Hodge symmetries of singular varieties