On the Fukaya categories of higher genus surfaces

Abstract

We construct the Fukaya category of a closed surface equipped with an area form using only elementary (essentially combinatorial) methods. We also compute the Grothendieck group of its derived category.

Homological Mirror Symmetry for manifolds of general type

“…This is not surprising since the map from torus equivariant Kähler classes on P 1 × P 1 has a one dimensional kernel and so there is a non-trivial relation between 1 …”

“…According to definition 2.1 the Fukaya-Seidel category is the A ∞ -category of modules over the A ∞ -version of the path algebra of this quiver where the higher products are given by disk instantons. With additional work, one can check that 1 …”

“…Since the compatification is defined modulo birational transformations, we can start with any compactification of C * ×C * . We will use the four-point blowup of P 1 × P 1 where the points of the blow ups are (…”

“…Let (Y ω) be an open symplectic manifold, and let w : Y → C be a symplectic Lefschetz fibration, i.e. a C ∞ complexvalued function with isolated non-degenerate critical points 1 . This means that the smooth parts of the fibers of w are symplectic submanifolds of (Y ω), and that near each we can find a ω-adapted almost complex structure on Y so that in almost-complex local coordinates w is given by w( 1 ) = w( )+ 2 1 +· · ·+ 2 .…”

“…To learn about X by studying the GLSM, it is important to be able to recognize the extra vacua. Let Z be a toric variety defined as a symplectic quotient of C N by a linear action of the gauge group G U (1) . The weights of this action will be denoted Q , where = 1 N and = 1 .…”