Volume 2013, Issue 680

We formulate a conjecture which describes the Fukaya category of an exact Lefschetz fibration defined by a Laurent polynomial in two variables in terms of a pair consisting of a consistent dimer model and a perfect matching on it. We prove this conjecture in some cases, and obtain homological mirror symmetry for quotient stacks of toric del Pezzo surfaces by finite subgroups of the torus as a corollary.

We establish a global geometric lower bound for the second fundamental form of the level surfaces of solutions to F ( D 2 u ,  Du ,  u ,  x ) = 0 in convex ring domains, in terms of boundary geometry and the structure of the elliptic operator F . We also prove a microscopic constant rank theorem, under a general structural condition introduced by Bianchini–Longinetti–Salani in 2009.

For given non-zero integers a , b , q we investigate the density of solutions ( x ,  y ) ∈ ℤ 2 to the binary cubic congruence ax 2  +  by 3  ≡ 0 mod  q , and use it to establish the Manin conjecture for a singular del Pezzo surface of degree 2 defined over ℚ.

As a first step in a general verification of the Alperin–McKay conjecture, we prove a reduction of the conjecture to statements about simple groups. Furthermore we show in numerous cases, that simple groups satisfy the required conditions. The methods are also applied to obtain similar results for refinements due to Isaacs and Navarro.

Let ( M ,  Q ) be an n -dimensional connected Riemannian manifold and 1 ≦  k  ≦  n . Then ( M ,  Q ) is called k -fold symmetric if given any k tangent vectors ξ 1 , ξ 2 , … ,  ξ k at a point x  ∈  M , there exists an isometry σ such that σ ( x ) =  x and dσ ( ξ i ) = − ξ i , i  = 1, 2, … ,  k . This kind of manifolds with k  = 1, usually called weakly symmetric Riemannian manifolds, was introduced by A. Selberg as a weakening of the notion of n -fold symmetric ones, i.e., globally symmetric Riemannian manifolds. It is well known that there are many more weakly symmetric spaces than globally symmetric ones. In this paper, we prove that a connected simply connected 2-fold symmetric Riemannian manifold must be globally symmetric.