Related papers: Sectorial descent for wrapped Fukaya categories
This (partially expository) paper discusses Lagrangian Floer cohomology in the context of Lefschetz fibrations, with emphasis on the algebraic structures encountered there. In addition to the well-known directed A_infinity algebras which…
In this paper we construct, in all odd complex dimensions, pairs of Liouville domains W_0 and W_1 which are diffeomorphic to the cotangent bundle of the sphere with one extra subcritical handle, but are not symplectomorphic. While W_0 is…
Given a (meromorphic) fibration $f:X\to Y$ where $X$ and $Y$ are compact complex manifolds of dimensions $n$ and $m$, we define $L_f$ to be the invertible subsheaf of the sheaf of holomorphic $m$-forms of $X$ given by the saturation of…
We develop a model structure on presheaves of small simplicially enriched categories on a site $\mathscr{C}$, for which the weak equivalences are 'stalkwise' weak equivalences for the Bergner model structure. This model structure is right…
We prove that the wrapped Fukaya category of a punctured sphere ($S^2$ with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the…
Let $X$ be a real analytic manifold, and let $T^*X$ be its cotangent bundle. In a recent paper with E. Zaslow \cite{NZ}, we showed that the dg category $Sh_c(X)$ of constructible sheaves on $X$ quasi-embeds into the triangulated envelope…
Consider a pair $(X,L)$, of a Weinstein manifold $X$ with an exact Lagrangian submanifold $L$, with ideal contact boundary $(Y,\Lambda)$, where $Y$ is a contact manifold and $\Lambda\subset Y$ is a Legendrian submanifold. We introduce the…
We give criteria for the existence of a Serre functor on the derived category of a gauged Landau-Ginzburg model. This is used to provide a general theorem on the existence of an admissible (fractional) Calabi-Yau subcategory of a gauged…
Mirror symmetry predicts that bounded derived category of a smooth Fano variety is equivalent to Fukaya-Seidel category of its Landau-Ginzburg model. It is expected that fibers of Landau-Ginzburg model with ordinary double points correspond…
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…
We associate an exact Lefschetz fibration with a pair of a consistent dimer model and an internal perfect matching on it, whose Fukaya category is derived-equivalent to the category of representations of the directed quiver with relations…
Let $\{\Lambda^\infty_t\}$ be an isotopy of Legendrians (possibly singular) in a unit cosphere bundle $S^*M$. Let $Sh(M, \Lambda^\infty_t)$ be the differential graded (dg) derived category of constructible sheaves on $M$ with singular…
Given a symplectic manifold $(M^{2n},\omega)$ we study Lagrangian cobordisms $V\subset E$ where $E$ is the total space of a Lefschetz fibration having $M$ as generic fiber. We prove a generation result for these cobordisms in the…
We introduce an idea of constructing Lefschetz fibrations of Weinstein manifolds from Weinstein handle decompositions on them. We prove theorems that formulate the idea for the cases of cotangent bundles and some plumbings. As a corollary,…
We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use "algebraic" characterizations of fibrations to produce factorizations…
We prove that for any element $L$ in the completion of the space of smooth compact exact Lagrangian submanifolds of a cotangent bundle equipped with the spectral distance, the $\gamma$-support of $L$ coincides with the reduced micro-support…
We prove homological mirror symmetry for projective hypersurfaces of sufficiently high degree using a functor from the wrapped Fukaya category of an affine hypersurface to the Fukaya category of its boundary at infinity.
We study categorical primitive forms for Calabi--Yau $A_\infty$ categories with semi-simple Hochschild cohomology. We classify these primitive forms in terms of certain grading operators on the Hochschild homology. We use this result to…
For any suitable base category $\mathcal{V} $, we find that $\mathcal{V} $-fully faithful lax epimorphisms in $\mathcal{V} $-$\mathsf{Cat} $ are precisely those $\mathcal{V}$-functors $F \colon \mathcal{A} \to \mathcal{B}$ whose induced…
We extend the notion of rational points and cohomological obstructions on varieties to categories fibred in groupoids. We also establish the generalized theory of descent by torsors. Then we interpret the obstruction given by the second…