Algebraic cycles and coisotropic subvarieties on irreducible symplectic manifolds

• Applicant: Professor Dr. Christian Lehn
• Subject Area: Mathematics
• Term: from 2014 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 269045579

The research project is concerned with a central and very active research topic in algebraic geometry: algebraic cycles on irreducible (holomorphic) symplectic manifolds. It mainly pursues two closely related questions: on the one hand, the existence of algebraically coisotropic subvarieties in ISMs and on the other hand, the splitting of Bloch-Beilinson type filtrations the Chow ring of a projective irreducible symplectic manifold.Our project is inspired by a remarkable recent set of conjectures by Claire Voisin concerning the structure of the Chow ring of a projective irreducible symplectic manifold. Voisin presented a far reaching conjectural program where coisotropic subvarieties as well as zero-cycles play a special role. These are beautiful conjectures and their proof would be a cornerstone in the understanding of algebraic cycles on irreducible symplectic manifolds. Moreover, Voisin's framework is rather explicit and in recent years there have been many advances in hyperkähler theory suitable to applications of this kind, so the moment seems to be right to exploit these methods to make progress on the field of algebraic cycles. These conjectures have partially been verified in examples, but there are only very few general results available.The goal is to substantially contribute to the construction of algebraic cycles for hyperkähler varieties by general deformation theoretic methods and to tackle more far reaching problems concerning the splitting of the conjectural Bloch-Beilinson filtration for a concrete maximal algebraic family of irreducible symplectic manifolds.A complete proof of Voisin's conjectures is out of reach. So the long-term goal should be to prove Voisin's conjectures for the known deformation types.

• DFG Programme: Research Grants