Periodic Reporting for period 2 - HyperK (Modern Aspects of Geometry: Categories, Cycles and Cohomology of Hyperkähler Varieties)
Berichtszeitraum: 2022-03-01 bis 2023-08-31
Hyperkähler geometry is an active area of algebraic geometry with many connections to other research directions in mathematics and mathematical physics. Among all geometries, the hyperkähler world exhibits some of the most fascinating phenomena. The special form of their curvature makes hyperkähler manifolds beautifully (super-)symmetric and the interplay of algebraic and transcendental aspects secures them a special place in modern mathematics. Hyperkähler geometry also serves as a testing ground for fundamental open questions and conjectures in algebraic geometry. The ultimate goal of the project is to draw a clear and distinctive picture of the hyperkähler landscape as a central part of mathematics.
The PIs of the project and the members of the team have made progress on a number of questions outlined in the proposal. One major advance concerns the topological classification of a large class of hyperkähler manifolds in dimension four, but progress has been made in various directions of the project (moduli spaces of hyperkähler manifolds, Chow groups, geometric and categorical aspects). In various constellations and partially with coauthors not funded by the project, members of the project have published approx. 30 research articles, all submitted for publication in peer-reviewed journals or already accepted or published. A collection of more expository articles, outlining the current state of the art of certain aspects has been published.
In the second period, Marcì and coauthors have continued their investigations antisymplectic involution, Voisin studied Chern classes and Chow groups of Lagrangian fibration, Huybrechst and Mattei introduced the Tate-Shafarevich group of an arbitrary linear system on a K3 surface and showed how to split Brauer classes by abelian torsors.
In the second period, Marcì and coauthors have continued their investigations antisymplectic involution, Voisin studied Chern classes and Chow groups of Lagrangian fibration, Huybrechst and Mattei introduced the Tate-Shafarevich group of an arbitrary linear system on a K3 surface and showed how to split Brauer classes by abelian torsors.
The topological characterization of a large class of hyperkähler manifolds is seen as a breakthrough result. It was not anticipated as such in the proposal and goes beyond the start of the art. There are versions of the result that should be within reach and will be pursued by members of the team. The majority of problems outlined in the proposal continue to be pursued actively by the members of the team with partial progress achieved already and with more results on the horizon. We expect further cross-fertilization within the project and with other modern developments in mathematical research.