We develop a version of Hodge theory for a large class of smooth cohomologically proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant $K$-theory of $X$ with respect to a maximal compact subgroup $M subset G$. The result is a natural pure Hodge structure of weight $n$ on $K^n_M(X^{an})$. We also treat categories of matrix factorizations for equivariant Landau-Ginzburg models.
We discuss homological mirror symmetry for the conifold from the point of view of the Strominger-Yau-Zaslow conjecture.
A sequel to arXiv:1111.1460, this paper elaborates on some of the themes in the above paper. Connections to Symplectic Field Theory (SFT) and mirror symmetry are explored.
Given a simply connected manifold M such that its cochain algebra, C^star(M), is a pure Sullivan dga, this paper considers curved deformations of the algebra C_star({Omega}M) and consider when the category of curved modules over these algebras becomes fully dualizable. For simple manifolds, like products of spheres, we are able to give an explicit criterion for when the resulting category of curved modules is smooth, proper and CY and thus gives rise to a TQFT. We give Floer theoretic interpretations of these theories for projective spaces and their products, which involve defining a Fukaya category which counts holomorphic disks with prescribed tangencies to a divisor.
We study matrix factorization and curved module categories for Landau-Ginzburg models (X,W) with X a smooth variety, extending parts of the work of Dyckerhoff. Following Positselski, we equip these categories with model category structures. Using results of Rouquier and Orlov, we identify compact generators. Via Toens derived Morita theory, we identify Hochschild cohomology with derived endomorphisms of the diagonal curved module; we compute the latter and get the expected result. Finally, we show that our categories are smooth, proper when the singular locus of W is proper, and Calabi-Yau when the total space X is Calabi-Yau.
