Abigail Ward: Homological mirror symmetry for elliptic Hopf surfaces

Abigail Ward
Title: Homological mirror symmetry for elliptic Hopf surfaces
Abstract: One can produce non-Kähler complex surfaces by performing logarithmic transformations on projective elliptic surfaces; for example, elliptic Hopf surfaces (including the classical Hopf surface S^1 x S^3) can be obtained by performing such operations to the product of the projective plane with an elliptic curve. In situations where the original surface has a mirror symplectic space, one can ask if there is a "mirror operation" to the logarithmic transformation, i.e. a way of producing a mirror to the logarithmically transformed surface from the original mirror space. We will discuss an answer to this question in the case of elliptic Hopf surfaces. For each such surface S, we will produce a mirror "non-algebraic Landau-Ginzburg model" with an associated Fukaya category. We will relate objects of this Fukaya category to coherent analytic sheaves on S.

Questions (time stamps ~15 minutes ahead):

12:33:17 From Agustin’s iPhone : can you briefly remind me of what the log transformation does?

12:49:00 From Hiro Lee Tanaka : When you complete, do you have to worry about what kinds of tensor products of CF* you’re actually taking when defining m^k operations?

12:49:42 From Sheel Ganatra : Can one think of the “admissibility” conditions on Lagrangians as arg W|_{L} = 0 for some globally defined (non-algebraic) LG model W: Y to C?

12:50:40 From abouzaid : The inclusion of the non-completed limit tends to be dense in the completion, so the operations are determined by their value on this dense submodule

13:08:15 From Roman Krutowski : may this approach be extended to understand mirror symmetry of Hopf manifolds(or even Calabi-Eckmann manifolds)?

13:12:35 From Marco Castronovo : Is there a guess for what a LG mirror to a Hopf surface that is not elliptically fibered could be?

13:14:16 From Zack Sylvan : Is there a nice relation between your construction in the Hopf surface case and the FLTZ mirror to (C^2)\0?

13:18:39 From Hiro Lee Tanaka : 2. Have you thought about using a holomorphic volume form on A compatible with (m,k) to define a stability condition, and see whether there’s a mirror “Hermitian Yang-Mills” type differential equation on the Hopf surface that picks out the semistable analytic coherent sheaves of rank 1?

3. So it sounds like we need a formalism for localizing Banach-space-enriched A-infinity categories to even write the localization sequence… Is that correct?

4. You have an analytically constructible sheaf on Pic(S) x Pic(S) = C* x C*, called the Ext sheaf. (It’s zero unless the ratio of the lambdas is well behaved.) Does your fully faithful functor actually give not just a functor, but a functor that’s “analytic” on the moduli of objects, where the wrapped complexes form an equivalent constructible sheaf?

13:18:42 From Hiro Lee Tanaka : 5. Is there a way to “expand” morphisms in your Fukaya category of A so that the mirror morphisms on the B side are not necessarily convergent, and use a symplectic (A model) criterion to detect the analytic/convergent functions on the Hopf surface (B model)?

6. It sounds like the H* level isomorphisms constructed from actual maps between analytic Hom-sheaves. (I.e., Ext is global sections of a hom sheaf; can we find a dg-equivalence of hom sheaves, where hom sheaves “have” topology.) What extra information do we get by recognizing this?