分野別セミナー

＊代数学・代数幾何学・幾何学合同セミナーです。
Local mirror symmetry is a variant of mirror symmetry derived from mirror symmetry of toric Calabi--Yau hypersurfaces.
Its statement is as follows. Take a 2-dimensional reflexive polyhedron (e.g the convex hull of (1,0),(0,1),(-1,-1) ).
On one side, one can associate to this a toric surface whose fan is generated by integral points of the polyhedron (e.g. P^2),
and its local Gromov--Witten invariants (local A-model).
On other side , one can associate an affine hypersurface C in 2-dimensional algebraic torus T^2 whose defining equation is the sum of Laurent monomials corresponding to integral points of the given polyhedron, and the relative cohomology group H^2(T^2, C) (local B-model).
The both of them are closely related to　a system of differential equations associated to the polyhedron　called the A-hypergeometric system due to Gel'fand, Kapranov, Zelevinsky.
As to the local B-model,the (V)MHS of H^2(T^2,C) has been studied by Batyrev and Stienstra.
In the joint work with Satoshi Minabe (arXiv:0907.4108),
we defined, using their results, an analogue of the Yukawa coupling whose direct definition was not known so far.
In this talk, I explain these mixed Hodge theoretic aspects of the local B-model.