Hodge theory of lagrangian subvarieties and applications to deformation theory

• Applicants: Professor Dr. Christian Lehn; Professor Dr. Christian Sevenheck
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 550535392

This project is concerned with singular lagrangian subvarieties in holomorphic symplectic manifolds. Our special interest is to study the Hodge theory and the deformations of such subvarieties. In particular, we investigate how both are linked. The main philosophy is that deformations of lagrangian subvarieties are controlled by topological or Hodge theoretic data. We focus on local and global geometric situations, that is germs resp. affine varieties and compact resp. projective varieties. Our main technical tool is the lagrangian de Rham complex. We will study it building on previous work in the local case of germs and determine to which extend it is related to Hodge theory. Explicitly, in more technical terms, we ask whether it is the perverse sheaf underlying a mixed Hodge module. Another crucial part of the project is to investigate the obstruction theory of lagrangian subvarieties and study in detail the relation between their local and global deformation theory. We conjecture that the Hodge theoretic nature of lagrangian deformations leads to unobstructedness results (i.e. smoothness of local moduli spaces), at least under some restrictions on the singularities of the lagrangian subvarieties in question. Unobstructedness is the first step in proving results about smoothability of singular subvarieties, which is important in applications. We will also investigate deformations of lagrangian subvarieties in (possibly singular) symplectic varieties, allowing both the subvariety and the ambient variety to deform. This will be done by studying both lagrangian de Rham cohomology and Poisson cohomology, since the latter is well-known to control deformations of (singular) symplectic varieties. An important part of the project is to apply the abstract results on lagrangian de Rham cohomology and on deformation spaces to highly topical problems on irreducible holomorphic symplectic varieties involving lagrangian subvarieties. This includes O'Grady's resp. Voisin's conjecture on the existence of lagrangian (or, more generally, coisotropic) subvarieties. Finally, we will also contribute to finding examples of global lagrangian subvarieties with interesting properties with respect to deformation, e.g. subvarieties for which the lagrangian deformation theory is unobstructed, or, on the other hand, for which there are no lagrangian deformations at all (i.e. rigid examples). Beyond these concrete applications, we believe that the field of holomorphic symplectic geometry would greatly benefit from a general approach to lagrangian deformations. Recent results, e.g. by Bakker-Schnell, constitute a proof of concept that developments in Hodge theory place our goal within reach.

• DFG Programme: Research Grants