Toric variety

About: Toric variety is a research topic. Over the lifetime, 2630 publications have been published within this topic receiving 65604 citations. The topic is also known as: torus embedding.

Remarks on the combinatorial intersection cohomology of fans

Abstract: We review the theory of combinatorial intersection cohomology of fans developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal properties but makes sense even for non-rational fans, which do not define a toric variety. As a result, a number of interesting results on the toric $g$ and $h$ polynomials have been extended from rational polytopes to general polytopes. We present explicit complexes computing the combinatorial IH in degrees one and two; the degree two complex gives the rigidity complex previously used by Kalai to study $g_2$. We present several new results which follow from these methods, as well as previously unpublished proofs of Kalai that $g_k(P) = 0$ implies $g_k(P^*) = 0$ and $g_{k+1}(P) = 0$.

Frobenius splittings of toric varieties

Abstract: We discuss a characteristic free version of Frobenius splittings for toric varieties and give a polyhedral criterion for a toric variety to be diagonally split. We apply this criterion to show that section rings of nef line bundles on diagonally split toric varieties are normally presented and Koszul, and that Schubert varieties are not diagonally split in general.

Residual kernels with singularities on coordinate planes

Abstract: A finite collection of planes {E v } in ℂd is called an atomic family if the top de Rham cohomology group of its complement is generated by a single element. A closed differential form generating this group is called a residual kernel for the atomic family. We construct new residual kernels in the case when E v are coordinate planes such that the complement ℂd/∪ E v admits a toric action with the orbit space being homeomorphic to a compact projective toric variety. They generalize the well-known Bochner-Martinelli and Sorani differential forms. The kernels obtained are used to establish a new formula of integral representations for functions holomorphic in Reinhardt polyhedra.

Examples of Kähler–Einstein toric Fano manifolds associated to non-symmetric reflexive polytopes

Abstract: In this note we report on examples of 7- and 8-dimensional toric Fano manifolds whose associated reflexive polytopes are not symmetric, but they still admit a Kahler–Einstein metric. This answers a question first posed by Batyrev and Selivanova. The examples were found in the classification of ≤8-dimensional toric Fano manifolds obtained by Obro. We also discuss related open questions and conjectures. In particular, we notice that the alpha-invariants of these examples do not satisfy the assumptions of Tian’s theorem.

Open mirror symmetry for Pfaffian Calabi-Yau 3-folds

Abstract: We investigate the open mirror symmetry of certain non-complete intersection Calabi- Yau 3-folds, so called pfaffian Calabi-Yau. We perform the prediction of the number of disk invariants of several examples by using the direct integration method proposed recently and the open mirror symmetry. We treat several pfaffian Calabi-Yau 3-folds in $\mathbb{P}^6$ and branes with two discrete vacua. Some models have the two special points in its moduli space, around both of which we can consider different A-model mirror partners. We compute disc invariants for both cases. This study is the first application of the open mirror symmetry to the compact non-complete intersections in toric variety.