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We give an introductory review of Fourier-Mukai transforms and their application to various aspects of moduli problems, string theory and mirror symmetry. We develop the necessary mathematical background for Fourier-Mukai transforms such as…

This is the first of a sequence of two papers. Here, a simple algebraic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold G^C/P is provided in terms of symplectic data. The result of this…

Given a smooth projective toric variety X, we construct an A-infinity category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line…

We give a necessary and sufficient condition for the nonsingular projective toric variety associated to the graph cubeahedron of a finite simple graph to be Fano or weak Fano in terms of the graph.

In this paper we prove that the linear Koszul duality isomorphism for convolution algebras in K-homology defined in a previous paper and the Fourier transform isomorphism for convolution algebras in Borel-Moore homology are related by the…

Let $X_\Sigma$ be a smooth, not necessarily compact toric variety. We show that a certain complex, defined in terms of the fan $\Sigma$, computes the integral cohomology of $X_\Sigma$, including the module structure over the homology of the…

Drinfeld twists, and the twists of Giaquinto and Zhang, allow for algebras and their modules to be deformed by a cocycle. We prove general results about cocycle twists of algebra factorisations and induced representations and apply them to…

We characterize the seminormality of an affine semigroup ring in terms of the dualizing complex, and the normality of a Cohen-Macaulay semigroup ring by the "shape" of the canonical module. We also characterize the seminormality of a toric…

For a smooth Deligne-Mumford stack over $\CC$, we define its associated Kodaira-Spencer differential graded Lie algebra and show that the deformation functor of the stack is isomorphic to the deformation functor of the Kodaira-Spencer…

We study the cohomology of broken toric varieties via the derived push-forward of the constant sheaf to a complex of polytopes, proving a Deligne-type decomposition theorem, degeneration of the associated Leray-Serre spectral sequence, and…

In this paper, we study wall elements of rank 2 cluster scattering diagrams based on dilogarithm elements. We derive two major results. First, we give a method to calculate wall elements in lower degrees. By this method, we may see the…

This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection…

We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of $A_\infty$-structures on the trivial extension algebras of the endomorphism algebras of the…

Given a smooth projective variety $X$ with a simple normal crossing divisor $D:=D_1+D_2+...+D_n$, where $D_i\subset X$ are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks $X_{D,\vec r}$ by constructing an…

We establish some properties of the derived category of torus-equivariant coherent sheaves on a split toric stack bundle. Our main result is a semi-orthogonal decomposition of such a category.

In this paper we study intertwining functors (Radon transforms) for twisted D-modules on partial flag varieties and their relation to the representations of semisimple Lie algebras. We show that certain intertwining functors give…

We prove the generalised Mukai conjecture for $\mathbb{Q}$-factorial spherical Fano varieties. In this case, a stronger inequality holds featuring an extra term - the minimum absolute complexity of a log Calabi-Yau pair - which measures how…

We generalise Dwork's theory of $p$-adic formal congruences from the univariate to a multi-variate setting. We apply our results to prove integrality assertions on the Taylor coefficients of (multi-variable) mirror maps. More precisely,…

We show that any $(\C ^*)^n$-invariant stably complex structure on a topological toric manifold of dimension $2n$ is integrable. We also show that such a manifold is weakly $(\C ^*)^n$-equivariantly isomorphic to a toric manifold.

We study the generalized hypergeometric system introduced by Gelfand, Kapranov and Zelevinsky and its relationship with the toric Deligne-Mumford (DM) stacks recently studied by Borisov, Chen and Smith. We construct series solutions with…