Differential Geometry and Topology

course: Affine structures and mirror symmetry

speaker: Mark Gross (University of Warwick)

abstract: The goal of these three lectures is to survey the approach to mirror symmetry using affine manifolds. Given a real manifold whose transition maps are integral affine linear maps, one can construct either a symplectic manifold fibred in tori over the real manifold, or a complex manifold fibred in tori. These torus fibrations are naturally dual to each other, and exhibit a simple form of mirror symmetry (as predicted by the Strominger-Yau-Zaslow conjecture). In general, one does not get interesting examples because of the absence of singular fibres in the torus fibrations obtained in this way. In order to obtain interesting examples, such as mirror symmetry for K3 surfaces or complete intersections in toric varieties, one must study affine manifolds with singularities. In this case, it is much more difficult to construct symplectic or complex manifold structures on the torus fibrations over these affine manifolds with singularities.

My intention is to spend the first lecture on the basic picture of mirror symmetry from the point of view of affine manifolds, and the second and third lectures will be devoted to the problem of constructing complex manifolds from affine manifolds with singularities, using methods introduced by myself and Bernd Siebert. In the third lecture, I hope to explain how to carry out our procedure in the case of K3 surfaces, and relate our method to the recent paper of Kontsevich and Soibelman on the same topic.

Relevant papers are - Affine manifolds, log structures, and mirror symmetry (http:/it.arxiv.orgabsmath.AG0211094)

- Mirror Symmetry via Logarithmic Degeneration Data I (http:/it.arxiv.orgabsmath.AG0309070)

- Toric Degenerations and Batyrev-Borisov Duality (http:/it.arxiv.orgabsmath.AG0406171) of Bernd Siebert andor myself, and Homological mirror symmetry and torus fibrations (http:/it.arxiv.orgabsmath.SG0011041) Affine structures and non-archimedean analytic spaces (http:/it.arxiv.orgabsmath.AG0406564) by Kontsevich and Soibelman.

timetable:
Tue 14 Sep, 9:45 - 11:15, Sala Conferenze Centro De Giorgi
Wed 15 Sep, 11:15 - 12:15, Sala Conferenze Centro De Giorgi
Fri 17 Sep, 17:00 - 18:15, Sala Conferenze Centro De Giorgi
<< Go back