Legendrian Contact Homology (LCH) was among the first, and is still among the most important, non-classical invariants of Legendrian knots. In this talk, I will tell a story that builds up ever more sophisticated analogues of Poincare Duality in LCH. Despite the algebraic nature of the talk, I promise pictures and examples.
The duality long exact sequence relates linearised Legendrian contact homology and cohomology and was originally constructed by Sabloff in the case of Legendrian knots. We show how the duality long exact sequence can be generalised to a relative Calabi-Yau structure, as defined by Brav and Dyckerhoff. We also discuss the generalised notion of the fundamental class and give applications. The structure is established through the acyclicity of a version of Rabinowitz Floer Homology for Legendrian submanifolds with coefficiens in the Chekanov-Eliashberg DGA. This is joint work in progress with Legout.
Enumerative mirror symmetry is a correspondence between closed Gromov-Witten invariants of a space X, and period integrals of a family Y. One of the predictions of Homological Mirror Symmetry is that the closed Gromov-Witten invariants can be obtained from the Fukaya category. For Calabi-Yau varieties this has been demonstrated by Ganatra-Perutz-Sheridan. Recently, enumerative mirror symmetry has been extended, by including open Gromov-Witten invariants and extended period integrals. It is natural to expect that open Gromov-Witten invariants can be obtained from the Fukaya category. In this talk I will outline a construction which will demonstrate this for certain open Gromov-Witten invariants.
I will start by explaining the construction of a formal scheme starting with an integral affine manifold Q equipped with a decomposition into Delzant polytopes. This is a weaker and more elementary version of degenerations of abelian varieties originally constructed by Mumford. Then I will reinterpret this construction using the corresponding Lagrangian torus fibration X→Q and relative Floer theory of its canonical Lagrangian section. Finally, I will discuss a conjectural generalization of the story to decompositions of CY symplectic manifolds into symplectic log CY's whose boundaries are 'opened up'.
Given a log Calabi-Yau pair (X,D), consisting of a smooth projective variety X together with a normal crossings anti-canonical divisor D, we first provide a combinatorial algorithm for solving the enumerative problem of computing rational stable maps to (X,D) touching D at a single point. We then explain how to use the solution to write explicit equations for mirrors to such pairs at
arbitrary dimensions.
This will be an attempt to summarize what one might expect about Homological Mirror Symmetry in the presence of an anticanonical divisor... and the (much smaller, but more reliable) subset of things I can prove about that situation.
In this talk, I will first introduce the background of Bridgeland stability condition. Then I will mention some existence result of Bridgeland stability. Next I will prove the Bogomolov-Gieseker type inequality of X(2,4), Calabi-Yau threefold of complete intersection of quadratic and quartic hypersufaces, by proving the Clifford type inequality of the curve X(2,2,2,4). Then this will provide the existence of Bridgeland stability condition of X(2,4).
If a Calabi-Yau threefold varies in a one-parameter family and aquires some double points, a small resolution will produce a rigid space. The local monodromy at such a 'conifold transition' is of infinite order. In the talk I report on some work done with S. Cynk (Krakow), which shows similar transitions to rigid Calabi-Yaus are possible with monodromy of finite order, in sharp distinction to what can happen for K3 surfaces.
I will describe my recent work, joint with Shaoyun Bai, which studies a class of bifurcations of moduli spaces of embedded pseudo-holomorphic curves in symplectic Calabi-Yau 3-folds and their associated obstruction bundles. As an application, we are able to give a direct definition of the Gopakumar-Vafa invariant in a special case.
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