Theodore Spaide | Searcharxiv

arXiv2017

Calabi-Yau Structures, Spherical Functors, and Shifted Symplectic Structures

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.

math.AG↗

arXiv2016

Shifted Symplectic and Poisson Structures on Spaces of Framed Maps

We examine shifted symplectic and Poisson structures on spaces of framed maps. We prove some results about shifted Poisson structures analogous to those in existing ones about symplectic structures. Then, we consider the space Map(X,D,Y) of maps from X to Y framed along a divisor D. We give conditions under which this space has a shifted symplectic or Poisson structure. Classical examples of symplectic and Poisson structures are provided with this theorem.

math.AG↗