相关论文: Hirzebruch Surfaces, Tyurin Degenerations and Tori…

We extend the construction of Calabi-Yau manifolds to hypersurfaces in non-Fano toric varieties, requiring the use of certain Laurent defining polynomials, and explore the phases of the corresponding gauged linear sigma models. The…

Complex Ricci-flat (i.e., Calabi-Yau) hypersurfaces in spaces admitting a maximal (toric) $U(1)^n$ gauge symmetry of general type (encoded by certain non-convex and multi-layered multitopes) may degenerate, but can be smoothed by rational…

This is an outline of work in progress concerning an algebro-geometric form of the Strominger-Yau-Zaslow conjecture. We introduce a limited type of degeneration of Calabi-Yau manifolds, which we call toric degenerations. For these, the…

We discuss various topics on degenerations and special Lagrangian torus fibrations of Calabi-Yau manifolds in the context of mirror symmetry. A particular emphasis is on Tyurin degenerations and the Doran-Harder-Thompson conjecture, which…

Multiply-connected Calabi-Yau threefolds are of particular interest for both string theorists and mathematicians. Recently it was pointed out that one of the generic degenerations of these spaces (occurring at codimension one in moduli…

We investigate a potential relationship between mirror symmetry for Calabi-Yau manifolds and the mirror duality between quasi-Fano varieties and Landau-Ginzburg models. More precisely, we show that if a Calabi-Yau admits a so-called Tyurin…

We consider regular Calabi-Yau hypersurfaces in $N$-dimensional smooth toric varieties. On such a hypersurface in the neighborhood of the large complex structure limit point we construct a fibration over a sphere $S^{N-1}$ whose generic…

Using the algebraic geometric approach of Berenstein et {\it al} (hep-th/005087 and hep-th/009209) and methods of toric geometry, we study non commutative (NC) orbifolds of Calabi-Yau hypersurfaces in toric varieties with discrete torsion.…

For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the notion of a small toric degeneration. Using small toric degenerations of Fano n-folds $X$, we propose a general method for constructing mirrors of Calabi-Yau complete…

In this paper, we make progress on understanding the collapsing behavior of Calabi-Yau metrics on a degenerating family of polarized Calabi-Yau manifolds. In the case of a family of smooth Calabi-Yau hypersurfaces in projective space…

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…

We use toric geometry to study open string mirror symmetry on compact Calabi-Yau manifolds. For a mirror pair of toric branes on a mirror pair of toric hypersurfaces we derive a canonical hypergeometric system of differential equations,…

As a continuation of \lianyaufour, we study modular properties of the periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau varieties. In Part A of this paper, motivated by the recent work of Kachru-Vafa, we degenerate a…

We identify the twisted sectors of a compact simplicial toric variety. We do the same for a generic nondegenerate Calabi-Yau hypersurface of an $n$-dimensional simplicial Fano toric variety and then explicitly compute $h^{1,1}_{orb}$ and…

We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. The mirror cohomology correspondence in the context of mirror symmetry…

While Calabi-Yau hypersurfaces in toric ambient spaces provide a huge number of examples, theoretical considerations as well as applications to string phenomenology often suggest a broader perspective. With even the question of finiteness…

The most impressively prolific exploration of superstring models (aiming for our physical reality) has been focused on worldsheet-supersymmetric gauged linear sigma models and the closely associated complex-algebraic toric geometry. Mirror…

In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic…

We provide a sufficient condition for a general hypersurface in a $\mathbb Q$-Fano toric variety to be a Calabi-Yau variety in terms of its Newton polytope. Moreover, we define a generalization of the Berglund-H\"ubsch-Krawitz construction…

In this paper, we extend our geometrical derivation of expansion coefficients of mirror maps by localization computation to the case of toric manifolds with two K\"ahler forms. Especially, we take Hirzebruch surfaces F_{0}, F_{3} and…