Related papers: Homological Mirror Symmetry for Hypertoric Varieti…
We study the derived categories of coherent sheaves of weighted projective spaces and their noncommutative deformations, and the derived categories of Lagrangian vanishing cycles of their mirror Landau-Ginzburg models. In particular, we…
In this paper, we prove a homological mirror symmetry equivalence for pairs of multiplicative hypertoric varieties, and we calculate monodromy autoequivalences of these categories by promoting our result to an equivalence of perverse…
We work out the notion of mirror symmetry for abelian varieties and study its properties. Our construction are based on the correspondence between two $Q$--algebraic groups. One is the Hodge (or special Mumford--Tate) group. The second…
We propose a way of understanding homological mirror symmetry when a complex manifold is a smooth compact toric manifold. So far, in many example, the derived category $D^b(coh(X))$ of coherent sheaves on a toric manifold $X$ is compared…
Homological mirror symmetry is a conjecture that a category constructed in the A-model and a category constructed in the B-model are equivalent in some sense. We construct a cyclic differential graded (DG) category of holomorphic vector…
We construct an $A_{\infty}$-structure on the Ext-groups of hermitian holomorphic vector bundles on a compact complex manifold. We propose a generalization of the homological mirror conjecture due to Kontsevich. Namely, we conjecture that…
We study the homological mirror symmetry statement where A-side is the conic bundle Hori--Vafa mirror $\mathcal{Y} = \{uv = f(z)\} \subset \mathbb{C}^2 \times (\mathbb{C}^\ast)^n$ for a Laurent polynomial $f$ in $(\mathbb{C}^\ast)^n$, and…
We prove a homological mirror symmetry equivalence between an $A$-brane category for the pair of pants, computed as a wrapped microlocal sheaf category, and a $B$-brane category for a mirror LG model, understood as a category of matrix…
The derived category of coherent sheaves $\mathcal{T}_B$ associated to a birational cobordism which is either a weighted projective space, a stacky Atiyah flip, or a stacky blow-up of a point has a conjectural mirror Fukaya-Seidel category…
We study Homological Mirror Symmetry (HMS) for $A_n$-resolutions from the SYZ viewpoint. Let $X\to\bC^2/\bZ_{n+1}$ be the crepant resolution of the $A_n$-singularity. The mirror of $X$ is given by a smoothing $\check{X}$ of…
We argue that for a certain class of symplectic manifolds the category of A-branes (which includes the Fukaya category as a full subcategory) is equivalent to a noncommutative deformation of the category of B-branes (which is equivalent to…
In this article, we establish homological Berglund--H\"ubsch mirror symmetry for curve singularities where the A--model incorporates equivariance, otherwise known as homological Berglund--H\"ubsch--Henningson mirror symmetry, including for…
Let X be a smooth toric variety defined by the fan {\Sigma} . We consider {\Sigma} as a finite set with topology and define a natural sheaf of graded algebras A_{\Sigma} on {\Sigma} . The category of modules over A_{\Sigma} is studied…
The B-side of Kontsevich's Homological Mirror Symmetry Conjecture is discussed. We give first a self-contained study of derived categories and their homological algebra, and later restrict to the bounded derived category of schemes and…
Motivated by applications to the categorical and geometric local Langlands correspondences, we establish an equivalence between the category of filtered $\mathcal{D}$-modules on a smooth stack $X$ and the category of $S^1$-equivariant…
We leverage the results of the prequel in combination with a theorem of D. Orlov to yield some results in Hodge theory of derived categories of factorizations and derived categories of coherent sheaves on varieties. In particular, we…
We study hybrid models arising as homological projective duals (HPD) of certain projective embeddings $f:X\rightarrow\mathbb{P}(V)$ of Fano manifolds $X$. More precisely, the category of B-branes of such hybrid models corresponds to the HPD…
The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring $A = k[\Gamma]$, the symmetric homology is related to…
Let $X_\Sigma$ be a complete toric variety. The coherent-constructible correspondence $\kappa$ of \cite{FLTZ} equates $\Perf_T(X_\Sigma)$ with a subcategory $Sh_{cc}(M_\bR;\LS)$ of constructible sheaves on a vector space $M_\bR.$ The…
We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type $\operatorname{SL}_n$ and $\operatorname{PGL}_n$. More precisely, we establish an equality of stringy Hodge numbers for…