Algebraic cycles and coisotropic subvarieties on irreducible symplectic manifolds
Final Report Abstract
The Chow ring of an algebraic variety, i.e., a geometric object defined by polynomial equations, is an important tool of algebraic geometry to study the variety, and it reflects the various facets of its geometry. On the one hand, it allows one to "measure" the geometry of the variety by intersecting its subvarieties; on the other hand, beyond intersection theory, it has an almost arithmetic dimension which often expresses itself in the interplay between algebraic cycles and Hodge theory. In any case, it is a complicated object, which many mathematicians dream of understanding. To understand the Chow ring of a general algebraic variety is probably a hopeless endeavor. In this project, we have studied a certain class of algebraic varieties and their Chow rings, the so- called irreducible symplectic varieties. For these, building on her studies of special subvarieties, Claire Voisin has described the structure of the Chow ring in a series of conjectures which, while still complicated, are much less abstract than the corresponding descriptions for general algebraic varieties. Many of these conjectures have been verified in examples, but in full generality they remain unsolved. Since a general solution of these conjectures seems currently out of reach, the aim of our project entitled Algebraic Cycles and Coisotropic Subvarieties of Irreducible Symplectic Manifolds was to develop geometric methods to be able to produce in large number examples of special subvarieties such as those considered by Voisin. These are the so-called coisotropic subvarieties and are in a certain sense extremal for the symplectic structure. This makes them amenable to deformation theoretic methods, which, together with coauthors, we have oftentimes used successfully in the past. So, roughly speaking, the idea is to generate general coisotropic subvarieties from special ones by small perturbations (or deformations) of the defining equations and to study their properties along the deformation process. We would like to mention two things which were most decisive for the success of the project. First, we succeeded in embedding the questions into the larger context of singular symplectic varieties. The treatment of singular objects is a priori more difficult, but also allows for a much greater degree of freedom, which, due to the recent progress on singular symplectic varieties, we were able to profitably apply to the goals of this project. Since there are many more singular symplectic varieties than non-singular ones, we thus expect an increase in meaningful examples here in the future, which in turn will allow us to better understand the Chow-theoretic properties of symplectic varieties and to approach Voisin's conjectures. Second, we made an observation that is somewhat surprising at first glance: in many situations, the subvarieties to be deformed avoid the singular locus of the singular variety, which is why in these cases we can pretend that the singularities are not present at all. This "avoidance behavior" may prove relevant in the future and deserves closer study. Both of these facts together led to a series of precise results and illuminating examples, which we hope will play an important role on the way to proving Vosin's conjectures.
Publications
- On the period of Lehn, Lehn, Sorger, and van Straten's symplectic eightfold. 2020
Nicolas Addington and Franco Giovenzana
(See online at https://doi.org/10.48550/arXiv.2003.10984) - On the uniruled Voisin divisor on the LLSvS variety. 2020
Franco Giovenzana
(See online at https://doi.org/10.48550/arXiv.2007.07570) - A global Torelli theorem for singular symplectic varieties. J. Eur. Math. Soc. (JEMS), 23(3):949-994, 2021
Benjamin Bakker and Christian Lehn
(See online at https://doi.org/10.4171/jems/1026) - Deformations of rational curves on primitive symplectic varieties and applications. 2021
Christian Lehn, Giovanni Mongardi, and Gianluca Pacienza
(See online at https://doi.org/10.48550/arXiv.2103.16356) - Algebraic approximation and the decomposition theorem for Kähler Calabi-Yau varieties. Invent. math. 2022
Benjamin Bakker, Henri Guenancia, and Christian Lehn
(See online at https://doi.org/10.1007/s00222-022-01096-y)
