Hodge theory of toroidal compactifications and Torelli theorems — K3 surfaces, abelian varieties, and IHSM

• Applicants: Professor Dr. Klaus Hulek; Professor Dr. Christian Lehn
• Subject Area: Mathematics
• Term: from 2016 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 315548262

• Final Report Year: 2025

Final Report Abstract

This project explored compactications of moduli spaces of polarized K3 surfaces coming from mirror symmetry. The starting point was a construction proposed by Gross, Hacking, Keel, and Siebert (GHKS). Our aim was to develop a better understanding of the geometric and combinatorial structures underlying these compactications and the relation to degenerations of K3 surfaces and their higher-dimensional analogs, irreducible holomorphic symplectic manifolds (IHSM). A key achievement was a complete and explicit description of the Mori fan associated to the mirror family for degree 2 polarized K3 surfaces, based on a combinatorial invariant called "curve structures" introduced in the rst funding period. In the second funding period, this approach was developed further, leading to the notion of a "combinatorial K3 surface". These objects encode geometric data of degenerations of K3 surfaces via discrete invariants, and this allows a more effective algorithmic treatment. Thanks to this, the implementation of the project's ndings on a computer has become possible and is envisaged in SageMath as a promising follow-up project. The project also advanced our understanding of various other topics important to moduli spaces of K3 surfaces and their compactications. These include degenerations of IHSM, where the results of our project include the most comprehensive study of type II degenerations of IHSM to date. Further signicant contributions concern Torelli theorems for singular symplectic varieties, thus expanding the class of moduli spaces that can be constructed using period maps. Further results within the project concern moduli spaces of Enriques surfaces, the singularities of the perfect cone compactication, and the deformation theory of degenerate varieties. Although unforeseen challenges and promising new directions led us to deviate from our original plan, we were able to substantially improve our understanding of the foundational problems concerning these moduli spaces. The project opened up avenues for future research, including computational, combinatorial, and higher-dimensional geometric studies. Last but not least, several young scientists were involved and both beneted from and signicantly contributed to the project.

Publications

• A GIT Construction of Degenerations of Hilbert Schemes of Points. Documenta Mathematica, 24, 421-472.
  Gulbrandsen, Martin G.; Halle, Lars H. & Hulek, Klaus
  
• On the GHKS compactication of the moduli space of K3 surfaces of degree two
  Klaus Hulek, Christian Lehn & Carsten Liese
  
• The geometry of degenerations of Hilbert schemes of points. Journal of Algebraic Geometry, 30(1), 1-56.
  Gulbrandsen, Martin; Halle, Lars; Hulek, Klaus & Zhang, Ziyu
  
• The Kodaira dimension of some moduli spaces of elliptic K3 surfaces. Journal of the London Mathematical Society, 104(1), 269-294.
  Fortuna, Mauro & Mezzedimi, Giacomo
  
• The perfect cone compactification of quotients of type IV domains. manuscripta mathematica, 170(1-2), 49-61.
  Giovenzana, Luca
  
• The global moduli theory of symplectic varieties. Journal für die reine und angewandte Mathematik (Crelles Journal), 2022(790), 223-265.
  Bakker, Benjamin & Lehn, Christian
  
• The Mori fan of the Dolgachev-Nikulin-Voisin family in genus 2. Épijournal de Géométrie Algébrique, Volume 6.
  Hulek, Klaus & Liese, Carsten
  
• Deformations of rational curves on primitive symplectic varieties and applications. Algebraic Geometry, 199-227.
  Lehn, Christian; Mongardi, Giovanni & Pacienza, Gianluca
  
• Moduli of polarised Enriques surfaces — Computational aspects. Journal of the London Mathematical Society, 109(1).
  Sikirić, Mathieu Dutour & Hulek, Klaus
  
• Rational curves on primitive symplectic varieties of OGs - 6 type. Mathematische Zeitschrift, 304(2).
  Bertini, Valeria & Grossi, Annalisa
  
• Symplectic rigidity of O’Grady’s tenfolds. Proceedings of the American Mathematical Society.
  Giovenzana, Luca; Grossi, Annalisa; Onorati, Claudio & Veniani, Davide
  
• Corrigendum to the paper "The Mori fan of the Dolgachev-Nikulin-Voisin family in genus 2" by K. Hulek and C. Liese
  Mathieu Dutour Sikiri¢, Klaus Hulek & Christian Lehn
  
• Unobstructedness of deformations for a d-semistable central corank one boundary point. Geometriae Dedicata, 219(5).
  Marie, Emeryck
  
• Combinatorial K3 surfaces and the Mori fan of the Dolgachev–Nikulin–Voisin family in degree 2. Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry.
  Hulek, Klaus & Lehn, Christian