Periodic Reporting for period 2 - RationAlgic (Rationality of varieties and algebraic cycles)
Reporting period: 2022-10-01 to 2024-03-31
The rationality problem in algebraic geometry is the question whether a given (system) of polynomial equations can be solved in terms of rational functions. This is a classical and important problem. Despite significant advances in the past decade, many fundamental problems in this area are still open. Another important branch of algebraic geometry is the study of algebraic cycles, which roughly speaking amounts to the study of subvarieties of a given variety. The rationality problem has various connections to questions in algebraic cycles and this project aims to make progress on both questions. An important ingredient in our approach is based on unramified cohomology, which is known to have powerful applications to both, to rationality problems as well as to the study of algebraic cycles.
In the first half of this project, Schreieder (PI of this project) introduced refined unramified cohomology groups, which are a generalizaton of unramified cohomology. He used this to tackle various open questions on algebraic cycles, such as the question on infinite torsion in Griffiths groups, or torsion analogues of Jannsen's conjecture and Green's conjecture. Some of these results were subsequently significantl improved by his student Theodosis Alexandrou.
In a different direction, this ERC-group used degeneration methods to attack problems on rationality and cycles in algebraic geometry. For instance, Pavic and Schreieder introduced a cycle-theoretic analogue of the motivic method of Nicaise--Shinder and Kontsevich--Tschinkel and used it to prove that very general quartic fourfolds in characteristic different from 2 are not retract rational, hence not rational. These results were subsequently generalized to some complete intersections by Jan Lange, who is a PhD student of Schreieder. Another PhD student of Schreieder, Matthias Paulsen, made a breakthrough on the famouse Griffiths--Harris conjecture which predicts that any curve on a very general 3-dimensional hypersurface of degree d at least 6 is divisible by d. Despite some progress by Kollàr in the 90s, this question was entirely open for any d. In a remarkable paper, Paulsen solved the conjecture for infinitely many degrees d (of positive density).
All members of the group regularly gave talks on their work at international conferences. In addition, Schreieder gave a lecture series on rationality at a summer school in Milano 2023 and is scientifically coordinating a week long summer school on rationality in France 2024.
In a different direction, this ERC-group used degeneration methods to attack problems on rationality and cycles in algebraic geometry. For instance, Pavic and Schreieder introduced a cycle-theoretic analogue of the motivic method of Nicaise--Shinder and Kontsevich--Tschinkel and used it to prove that very general quartic fourfolds in characteristic different from 2 are not retract rational, hence not rational. These results were subsequently generalized to some complete intersections by Jan Lange, who is a PhD student of Schreieder. Another PhD student of Schreieder, Matthias Paulsen, made a breakthrough on the famouse Griffiths--Harris conjecture which predicts that any curve on a very general 3-dimensional hypersurface of degree d at least 6 is divisible by d. Despite some progress by Kollàr in the 90s, this question was entirely open for any d. In a remarkable paper, Paulsen solved the conjecture for infinitely many degrees d (of positive density).
All members of the group regularly gave talks on their work at international conferences. In addition, Schreieder gave a lecture series on rationality at a summer school in Milano 2023 and is scientifically coordinating a week long summer school on rationality in France 2024.
The introduction of refined unramified cohomology together with its various applications, and the progress on degeneration methods mentioned above are the main progress beyond the state of the art performed in this project. The range of applications of these methods is not yet fully understood and explored and we hope to make more progress in the future. Some of the main challenges that we try to attack include: the integral Hodge conjecture for abelian varieties; relation between (refined) unramified cohomology and fundamental cycle conjectures such as the Tate conjecture; retract rationality of complete intersections.