Current Developments in Mathematics, 2017
Versality in mirror symmetry
Nick Sheridan
Abstract. One of the attractions of homological mirror symmetry is that it not only implies the previous predictions of mirror symmetry (e.g., curve counts on the quintic), but it should in some sense be ‘less of a coincidence’ than they are and therefore easier to prove. In this survey we explain how Seidel’s approach to mirror symmetry via versality at the large volume/large complex structure limit makes this idea precise.
Contents 1. Introduction 38 1.1. Mirror symmetry for Calabi–Yaus 38 1.2. Moduli spaces 40 1.3. Closed-string mirror symmetry: Hodge theory 42 1.4. Open-string (a.k.a. homological) mirror symmetry: categories 45 1.5. Using versality 46 1.6. Versality at the large volume limit 47 2. Deformation theory via L∞ algebras 49 2.1. L∞ algebras 49 2.2. Maurer–Cartan elements 50 2.3. Versal deformation space 50 2.4. Versality criterion 51 3. Deformations of A∞ categories 52 3.1. A∞ categories 52 3.2. Curved A∞ categories 53 3.3. Curved deformations of A∞ categories 53 4. The large volume limit 55 4.1. Closed-string: Gromov–Witten invariants 55 4.2. Open-string: Fukaya category 57 5. Versality in the interior of the moduli space 63 5.1. Bogomolov–Tian–Todorov theorem 63 5.2. Floer homology 64 5.3. HMS and versality 66