Calabi-Yau structures, spherical functors, and shifted symplectic structures

Ludmil Katzarkov, Pranav Pandit, Theodore Spaide

Research output: Contribution to journalArticlepeer-review


A categorical formalism is introduced for studying various features of the symplectic geometry of Lefschetz fibrations and the algebraic geometry of Tyurin degenerations. This approach is informed by homological mirror symmetry, derived noncommutative geometry, and the theory of Fukaya categories with coefficients in a perverse Schober. The main technical results include (i) a comparison between the notion of relative Calabi-Yau structures and a certain refinement of the notion of a spherical functor, (ii) a local-to-global gluing principle for constructing Calabi-Yau structures, and (iii) the construction of shifted symplectic structures and Lagrangian structures on certain derived moduli spaces of branes. Potential applications to a theory of derived hyperkähler geometry are sketched.

Original languageEnglish (US)
Article number108037
JournalAdvances in Mathematics
StatePublished - Dec 3 2021


  • Calabi-Yau structures
  • Derived geometry
  • Fukaya-Seidel categories
  • Perverse Schobers
  • Picard-Lefschetz theory
  • Spherical functors

ASJC Scopus subject areas

  • Mathematics(all)


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