Related papers: Mutation equivalence of toric Landau-Ginzburg mode…
Standard toric geometry methods used to construct Calabi-Yau varieties may be extended to complete intersections in non-Fano varieties encoded by star triangulating non-convex polytopes. Similarly, mirror symmetry is conjectured to hold in…
Landau-Ginzburg mirror symmetry predicts isomorphisms between graded Frobenius algebras (denoted $\mathcal{A}$ and $\mathcal{B}$) that are constructed from a nondegenerate quasihomogeneous polynomial $W$ and a related group of symmetries…
In this paper we calculate the elliptic genus of certain complete intersections in products of projective spaces. We show that it is equal to the elliptic genus of the Landau-Ginzburg models that are, according to Hori and Vafa, mirror…
We generalized the construction of deformations of affine toric varieties of K. Altmann and our previous construction of deformations of weak Fano toric varieties to the case of arbitrary toric varieties by introducing the notion of…
We introduce a duality of Landau-Ginzburg models based on the notion of the discrete Legendre transform given by Gross-Siebert. It generalizes the duality used to construct mirrors of complete intersections in toric varieties in a recent…
In a spirit of Givental's constructions Batyrev, Ciocan-Fontanine, Kim, and van Straten suggested Landau--Ginzburg models for smooth Fano complete intersections in Grassmannians and partial flag varieties as certain complete intersections…
We classify when the blowup of a complex Grassmannian $G(k, n)$ along a smooth Schubert subvariety $Z$ is Fano. We compute almost all the two-point, genus zero Gromov-Witten invariants of the blowup when $Z=G(k, n-1)$. We further prove a…
We give a new proof of Givental's mirror theorem for toric manifolds using shift operators of equivariant parameters. The proof is almost tautological: it gives an A-model construction of the I-function and the mirror map. It also works for…
We obtain results that relate Donaldson-Futaki type invariants (that is, the numerical invariants used to define K-stability for general polarised manifolds) for a toric polarised manifold and for a compactification of its mirror…
We generalize the combinatorial description of the orbifold (Chen--Ruan) cohomology and of the Grothendieck ring of a Deligne--Mumford toric stack and its associated stacky fan in a lattice $N$ in the presence of a deformation parameter…
Let X and Y be K-equivalent toric Deligne-Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant…
We show that every rank one smooth Fano threefold has a weak Landau-Ginzburg model coming from a toric degeneration. The fibers of these Landau-Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show…
The Landau-Ginzburg/Conformal Field Theory correspondence predicts tensor equivalences between categories of matrix factorisations of certain polynomials and categories associated to the $N=2$ supersymmetric conformal field theories. We…
We investigate Hodge-theoretic properties of Calabi-Yau complete intersections $V$ of $r$ semi-ample divisors in $d$-dimensional toric Fano varieties having at most Gorenstein singularities. Our main purpose is to show that the…
A typical large complex-structure limit for mirror symmetry consists of toric varieties glued to each other along their toric boundaries. Here we construct the mirror large volume limit space as a Weinstein symplectic manifold. We prove…
Hille and Perling associate to every cyclic full strongly exceptional sequence of line bundles on a toric weak Fano surface a toric system, which defines a new toric surface. In this note we interprete this construction as an instance of…
The mirror of a projective toric manifold $X_\Sigma$ is given by a Landau-Ginzburg model $(Y,W)$. We introduce a class of Lagrangian submanifolds in $(Y,W)$ and show that, under the SYZ mirror transformation, they can be transformed to…
Our earlier proof of mirror formulas for genus 0 Gromov -- Witten invariants of Fano and Calabi -- Yau toric complete intersections is illustrated in the example of quintic 3-folds.
We generalize the toric residue mirror conjecture of Batyrev and Materov to not necessarily reflexive polytopes. Using this generalization we prove the toric residue mirror conjecture for Calabi-Yau complete intersections in Gorenstein…
Using Cayley trick, we define the notions of mixed toric residues and mixed Hessians associated with $r$ Laurent polynomials $f_1,...,f_r$.We conjecture that the values of mixed toric residues on the mixed Hessians are determined by mixed…