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Project

First Steps in Mirror Symmetry for Generalized Complex Geometry

Generalized complex (GC) geometry unifies complex and symplectic geometry, two important research areas in modern pure mathematics. While general GC structures are not yet well-understood, a number of important results from complex or symplectic geometry have already been extended to these more general structures. Further, complex and symplectic geometry are intimately related via mirror symmetry, a conjectured duality between certain complex and symplectic manifolds discovered in theoretical physics in the context of string theory. This duality has been proven in special cases. For this project I propose an approach to extend homological mirror symmetry to certain subclasses and examples of GC manifolds, centred around three objectives: (O1) Quantify the effect of stable GC compactifications of LandauGinzburg mirrors of del Pezzo surfaces on their Fukaya category. (O2) Construct a Wrapped Fukaya category for oriented surfaces with log symplectic structures. (O3) Develop a notion of 'holomorphic families of Fukaya categories'. 
In the case of (O1) and (O3), the construction of a Fukaya-type category immediately suggests mirror partners for certain classes of examples, the first extension of mirror symmetry to the GC context. In the construction for (O1), I expect to use my results on Lagrangian-type submanifolds with boundary of stable GC manifolds, which naturally arise in examples and are candidates for objects of Fukaya-Seidel-type categories of stable GC manifolds.
 

Date:1 Nov 2020 →  Today
Keywords:Generalized complex geometry, Mirror symmetry, Symplectic geometry
Disciplines:Geometry, Geometry not elsewhere classified, Differential geometry