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.

Duality for toric Landau-Ginzburg models

Abstract: We introduce a duality construction for toric Landau-Ginzburg models, applicable to complete intersections in toric varieties via the sigma model / Landau-Ginzburg model correspondence. This construction is shown to reconstruct those of Batyrev-Borisov, Berglund-H"ubsch, Givental, and Hori-Vafa. It can be done in more general situations, and provides partial resolutions when the above constructions give a singular mirror. An extended example is given: the Landau-Ginzburg models dual to elliptic curves in (P^1)^2 .

Homology and Cohomology of Toric Varieties

Abstract: The theory of toric varieties plays an important role as a bridge between algebraic geometry, combinatorial convex geometry, and commutative algebra. The interpretation and application of many notions and methods of these three theories in the case of toric varieties leads to fruitful examples and results. The relations between various algebraic, topological, and geometric properties of toric varieties ll extensive dictionaries. From the point of view of combinatorial convexity, all the structure of a toric variety can be encoded in a fan, i.e., a nite collection of strictly convex cones spanned by nitely many vectors with integral coordinates and satisfying very natural incidence relations. Now the obvious question is how to extract algebro-geometric or topological invariants of the associated toric variety from these data. A most striking example is given by the famous Theorem of Jurkiewicz-Danilov Da, Theorem 10.8 and Remark 10.9]: the integral cohomology ring of a smooth compact toric variety can be explicitly computed in terms of the associated fan. Allowing mild singularities, the analogous result holds with rational coeecients. For toric varieties with arbitrary singularities, but still in the compact case, there is a spectral sequence relating data of the fan to integral cohomology that has been investigated by Stephan Fischli in his dissertation Fi]. It admits the explicit computation of some integral cohomology groups in low and in high degrees; in particular, it yields complete results up to dimension three. In the work presented here, we also use spectral sequences to determine (co-) homological data of a toric variety in terms of the associated fan, but in a much more general setting: we investigate the homology with closed supports and the cohomology with compact supports and with arbitrary (constant) coeecients for not necessarily compact toric varieties with arbitrary singularities. We do not consider toric varieties over arbitrary elds but restrict ourselves to the complex case: The toric variety X associated to a fan in a vector space R n consists of (C) n-orbits O corresponding to the cones 2 ; in particular, the full-vi Preface dimensional cones 2 n correspond to xed points x. It turns out that the natural ltration of the toric variety X induced by its orbit structure provides convergent (co-)homology spectral sequences. An explicit calculation of the associated E 2-or E 2-terms yields formull | in low and in high degrees`| for the homology groups H cld`(X ; G), the cohomology groups …

Geometry of the Kimura 3-parameter model

Abstract: The Kimura 3-parameter model on a tree of n leaves is one of the most used in phylogenetics. The affine algebraic variety W associated to it is a toric variety. We study its geometry and we prove that it is isomorphic to a geometric quotient of the affine space by a finite group acting on it. As a consequence, we are able to study the singularities of W and prove that the biologically meaningful points are smooth points. Then we give an algorithm for constructing a set of minimal generators of the localized ideal at these points, for an arbitrary number of leaves n. This leads to a major improvement of phylogenetic reconstruction methods based on algebraic geometry.

Some Applications of the Mirror Theorem for Toric Stacks

Abstract: We use the mirror theorem for toric Deligne-Mumford stacks, proved recently by the authors and by Cheong-Ciocan-Fontanine-Kim, to compute genus-zero Gromov-Witten invariants of a number of toric orbifolds and gerbes. We prove a mirror theorem for a class of complete intersections in toric Deligne-Mumford stacks, and use this to compute genus-zero Gromov-Witten invariants of an orbifold hypersurface.

Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties

Abstract: We define a class of noncompact Fano toric manifolds which we call admissible toricmanifolds, for which Floer theory and quantum cohomology are defined. The class includes Fano toric negative line bundles, and it allows blow-ups along fixed point sets.We prove closed-string mirror symmetry for this class of manifolds: the Jacobian ring of the superpotential is the symplectic cohomology (not the quantum cohomology). Moreover, SH∗(M) is obtained from QH∗(M) by localizing at the toric divisors. We give explicit presentations of SH∗(M) and QH∗(M), using ideas of Batyrev, McDuff and Tolman.Assuming that the superpotential is Morse (or a milder semisimplicity assumption), we prove that the wrapped Fukaya category for this class of manifolds satisfies the toric generation criterion, ie is split-generated by the natural Lagrangian torus fibers of the moment map taken with suitable holonomies. In particular, the wrapped category is compactly generated and cohomologically finite.We prove a generic generation theorem: a generic deformation of the monotone toric symplectic form defines a local system for which the twisted wrapped Fukaya category satisfies the toric generation criterion. This theorem, together with a limiting argument about continuity of eigenspaces, are used to prove the untwisted generation results.We prove that for any closed Fano toric manifold, and a generic local system, the twisted Fukaya category satisfies the toric generation criterion. If the superpotential is Morse (or assuming semisimplicity), also the untwisted Fukaya category satisfies the criterion.The key ingredients are nonvanishing results for the open-closed string map, using tools from the paper by Ritter and Smith; we also prove a conjecture from that paper that any monotone toric negative line bundle contains a nondisplaceable monotone Lagrangian torus. The above presentation results require foundational work: we extend the class of Hamiltonians for which the maximum principle holds for symplectic manifolds conical at infinity, thus extending the class of Hamiltonian circle actions for which invertible elements can be constructed in SH∗(M). Computing SH∗(M) is notoriously hard and there are very few known examples beyond the cases of cotangent bundles and subcritical Stein manifolds. So this computation is significant in itself, as well as being the key ingredient in proving the above results in homological mirror symmetry.