It is known that a given smooth del Pezzo surface or Fano threefold $X$ admits a choice of log Calabi-Yau compactified mirror toric Landau-Ginzburg model (with respect to certain fixed Kähler classes and Gorenstein toric degenerations). Here we consider the problem of constructing a corresponding map $Θ$ from a domain in the complexified Kähler cone of $X$ to a well-defined, separated moduli space $\mathfrak{M}$ of polarised manifolds endowed with a canonical metric. We prove a complete result for del Pezzos and a partial result for some special Fano threefolds. The construction uses some fundamental results in the theory of constant scalar curvature Kähler metrics. As a consequence $\mathfrak{M}$ parametrises $K$-stable manifolds and the domain of $Θ$ is endowed with the pullback of a Weil-Petersson form.
We consider the conjectures from the paper by Katzarkov, Kontsevich, and Pantev about Landau-Ginzburg Hodge numbers associated to tamely compactifiable Landau-Ginzburg models. We test these conjectures in case of dimension two, verifying some and giving a counterexample to the other.
We conjecture that the number of components of the fiber over infinity of Landau--Ginzburg model for a smooth Fano variety $X$ equals the dimension of the anticanonical system of $X$. We verify this conjecture for log Calabi--Yau compactifications of toric Landau--Ginzburg models for smooth Fano threefolds, complete intersections, and some toric varieties.
This is a review of the theory of toric Landau-Ginzburg models - the effective approach to mirror symmetry for Fano varieties. We mainly focus on the cases of dimensions 2 and 3, as well as on the case of complete intersections in weighted projective spaces and Grassmannians. Conjectures that relate invariants of Fano varieties and their Landau--Ginzburg models, such as Katzarkov-Kontsevich-Pantev conjectures, are also studied.
We study the Hodge numbers of Landau-Ginzburg models as defined by Katzarkov, Kontsevich and Pantev. First we show that these numbers can be computed using ordinary mixed Hodge theory, then we give a concrete recipe for computing these numbers for the Landau-Ginzburg mirrors of Fano threefolds. We finish by proving that for a crepant resolution of a Gorenstein toric Fano threefold $X$ there is a natural LG mirror $(Y,\mathsf{w})$ so that $h^{p,q}(X) = f^{3-q,p}(Y,\mathsf{w})$.
Given a Fano complete intersection defined by sections of a collection nef line bundles $L_1,\ldots, L_c$ on a Fano toric manifold $Y$, a construction of Givental/Hori-Vafa provides a mirror-dual Landau-Ginzburg model. This construction depends on a choice of suitable nef partition; that is, a partition of the rays of the fan determined by $Y$. We show that toric Landau-Ginzburg models constructed from different nef partitions representing the same complete intersection are related by a volume preserving birational map. In particular, various Laurent polynomial mirrors which may be obtained fr…
This paper is an introduction to D-branes in Landau-Ginzburg models and Homological Mirror Symmetry. The paper is based on a series of lectures which were given on Second Latin Congress on Symmetries in Geometry and Physics that took place at the University of Curitiba, Brazil in December 2010.
We summarize the description of the open-closed TFT datum for B-type Landau-Ginzburg models with Stein manifold targets and discuss various constructions which lead to large classes of examples of such models.
The Landau-Ginzburg/Calabi-Yau correspondence claims that the Gromov-Witten invariant of the quintic Calabi-Yau 3-fold should be related to the Fan-Jarvis-Ruan-Witten invariant of the associated Landau-Ginzburg model via wall crossings. In this paper, we consider the stack of quasi-maps with a cosection and introduce sequences of stability conditions which enable us to interpolate between the moduli stack for Gromov-Witten invariants and the moduli stack for Fan-Jarvis-Ruan-Witten invariants.
We prove the Landau-Ginzburg mirror symmetry conjecture between invertible quasi-homogeneous polynomial singularities at all genera. That is, we show that the FJRW theory (LG A-model) of such a polynomial is equivalent to the Saito-Givental theory (LG B-model) of the mirror polynomial.
Let V and F be holomorphic bundles over a complex manifold M, and s be a holomorphic section of V. We study the cohomology associated to the Koszul complex induced by s, and prove a generalized Serre duality theorem for them.
We interpret the Landau-Ginzburg potentials associated to Gross-Hacking-Keel-Kontsevich's partial compactifications of cluster varieties as F-polynomials of projective representations of Jacobian algebras. Along the way, we show that both the projective and the injective representations of Jacobi-finite quivers with potential are well-behaved under Derksen-Weyman-Zelevinsky's mutations of representations.
