Morse homology, tropical geometry, and homological mirror symmetry for toric varieties

Given a smooth projective toric variety X, we construct an A_∞ category of Lagrangians with boundary on a level set of the Landau–Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X.      

Projectivized Rank Two Toric Vector Bundles are Mori Dream Spaces

We prove that the Cox ring of the projectivization P(?) of a rank two toric vector bundle ?, over a toric variety X, is a finitely generated k-algebra. As a consequence, P(?) is a Mori dream space if the toric variety X is projective and simplicial.     

Wild automorphisms of varieties with Kodaira dimension 0

An automorphism σ of a projective variety X is said to be wild if σ(Y) ≠ Y for every non-empty subvariety Y \subsetneq X{Y \subsetneq X} . In [1] Z. Reichstein, D. Rogalski, and J.J. Zhang conjectured that if X is an irreducible projective variety admitting a wild automorphism then X is an abelian variety, and proved this conjecture for dim(X) ≤ 2. As a step toward answering this conjecture in higher dimensions we prove a structure theorem for projective varieties of Kodaira dimension 0 admitting wild automorphisms. This essentially reduces the Kodaira dimension 0 case to a study of Calabi-Yau varieties, which we also investigate. In support of this conjecture, we show that there are no wild automorphisms of certain Calabi-Yau varieties.     

Algebras with small homological dimensions

We consider the algebras Λ which satisfy the property that for each indecomposable module X, either its projective dimension pd_Λ X is at most one or its injective dimension id_Λ X is at most one. This clearly generalizes the so-called quasitilted algebras introduced by Happel–Reiten–Smal?. We show that some of the niciest features for this latter class of algebras can be generalized to the case we are considering, in particular the existence of a trisection in its module category. Received: 26 August 1998     

On Euler–Jaczewski sequence and Remmert–Van de Ven problem for toric varieties

Let X be a complete toric variety and Y a smooth projective variety with . We prove that, if is a surjective morphism then . Received: 15 May 2001; in final form: 22 October 2001/ Published online: 4 April 2002     

Minitwistor spaces,Severi varieties,and Einstein–Weyl structure

In this article, we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein–Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein–Weyl structure on the space of smooth rational curves on a complex surface, given by Hitchin. As geometric objects naturally associated to Einstein–Weyl structure, we investigate null surfaces and geodesics on the Severi varieties. Also, we see that if the projective surface has an appropriate real structure, then the real locus of the Severi variety becomes a positive definite Einstein–Weyl manifold. Moreover, we construct various explicit examples of rational surfaces having 3-dimensional Severi varieties of rational curves.     

Picard Groups of compact toric Varieties and combinatorial Classes of Fans

We consider the question what can be said about the rank of the Picard group Pic X_σ of a compact toric variety X_σ if we know only the combinatorial type of the associated fan σ. We establish upper and lower bounds for the rank of Pic X_σ and give conditions for Pic X_σ to be determined by the combinatorial type of σ. Furthermore, we show that for simple fans Pic X_σ is necessary isomorphic to {0} or Z and give an example for a compact toric variety having a trivial Picard group. Moreover in the projective case we study the relation between addition of T-invariant Cartier divisors on X_σ, taking tensor product of elements of Pic X_σ and piecewise linear functions on σ with Minkowski-addition of polytopes, where the latter operation is extended to a group operation. Finally, we explain the relation to strong cohomology in the projective case.     

On the unipotence of autoequivalences of toric complete intersection Calabi–Yau categories

Consider the derived category of coherent sheaves, D ^b (X), on a compact Calabi–Yau complete intersection X in a toric variety. The scope of this work is to establish the (quasi-)unipotence of a class of elements in the group of autoequivalences, Aut(D ^b (X)). This is achieved by associating singularity categories, modelled by matrix factorizations, to the toric data. Each of these triangulated categories is equivalent to the derived category of coherent sheaves on X. The idea is then that, although the singularity categories share the group of autoequivalences, on each category there are elements in Aut(D ^b (X)), whose (quasi-)unipotence relations are easier to see than on the other categories.     

On generators of ideals defining projective toric varieties

 For any ample line bundle L on a projective toric variety of dimension n, it is proved that the line bundle L ^⊗i is normally generated if i is greater than or equal to n−1, and examples showing that this estimate is best possible are given. Moreover we prove an estimate for the degree of the generators of the ideals defining projective toric varieties. In particular, when L is normally generated, the defining ideal of the variety embedded by the global sections of L has generators of degree at most n+1. When the variety is embedded by the global sections of L ^⊗(n−1) , then the defining ideal has generators of degree at most three. Received: 11 July 2001 / Revised version: 17 December 2001     

Gale duality and homogeneous toric varieties

A non-degenerate toric variety X is called S-homogeneous if the subgroup of the automorphism group Aut(X) generated by root subgroups acts on X transitively. We prove that maximal S-homogeneous toric varieties are in bijection with pairs (P,𝒜), where P is an abelian group and 𝒜 is a finite collection of elements in P such that 𝒜 generates the group P and for every a∈𝒜 the element a is contained in the semigroup generated by 𝒜?{a}. We show that any non-degenerate homogeneous toric variety is a big open toric subset of a maximal S-homogeneous toric variety. In particular, every homogeneous toric variety is quasiprojective. We conjecture that any non-degenerate homogeneous toric variety is S-homogeneous.     

Isolated rational curves on K3-fibered Calabi–Yau threefolds

In this paper we study 16 complete intersection K3-fibered Calabi--Yau variety types in biprojective space ℙ ^{n 1}}×ℙ¹. These are all the CICY-types that are K3 fibered by the projection on the second factor. We prove existence of isolated rational curves of bidegree (d,0) for every positive integer d on a general Calabi–Yau variety of these types. The proof depends heavily on existence theorems for curves on K3-surfaces proved by S. Mori and K. Oguiso. Some of these varieties are related to Calabi–Yau varieties in projective space by a determinantal contraction, and we use this to prove existence of rational curves of every degree for a general Calabi–Yau variety in projective space. Received: 14 October 1997 / Revised version: 18 January 1998     

Curve counting via stable pairs in the derived category

For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where C⊂X is an embedded curve and D⊂C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of X. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described.     

K-Theory of Algebraic Tori and Toric Varieties

It is proved that under certain conditions the group K _n (X) of a smooth projective variety X over a field F is a natural direct summand of K _n (A) for some separable F-algebra A. As an application we study the K-groups of toric models and toric varieties. A presentation in terms of generators and relations of the groupK ₀(T) for an algebraic torus T is given.     

Higher K-Theory of Toric Varieties

A natural higher K-theoretic analogue of the triviality of vector bundles on affine toric varieties is the conjecture on nilpotence of the multiplicative action of the natural numbers on the K-theory of these varieties. This includes both Quillen's fundamental result on K-homotopy invariance of regular rings and the stable version of the triviality of vector bundles on affine toric varieties. Moreover, it yields a similar behavior of not necessarily affine toric varieties and, further, of their equivariant closed subsets. The conjecture is equivalent to the claim that the relevant admissible morphisms of the category of vector bundles on an affine toric variety can be supported by monomials not in a nondegenerate corner subcone of the underlying polyhedral cone. We prove that one can in fact eliminate all lattice points in such a subcone, except maybe one point. The elimination of the last point is also possible in 0 characteristic if the action of the big Witt vectors satisfies a very natural condition. A partial result of this in the arithmetic case provides first nonsimplicial examples, actually an explicit infinite series of combinatorially different affine toric varieties, simultaneously verifying the conjecture for all higher groups.Supported by the Deutsche Forschungsgemeinschaft, INTAS grant 99-00817 and TMR grant ERB FMRX CT-97-0107     

Equivalence of Mirror Families Constructed from Toric Degenerations of Flag Varieties

Batyrev et al. constructed a family of Calabi–Yau varieties using small toric degenerations of the full flag variety G/B. They conjecture this family to be mirror to generic anticanonical hypersurfaces in G/B. Recently, Alexeev and Brion, as a part of their work on toric degenerations of spherical varieties, have constructed many degenerations of G/B. For any such degeneration we construct a family of varieties, which we prove coincides with Batyrev’s in the small case. We prove that any two such families are birational, thus proving that mirror families are independent of the choice of degeneration. The birational maps involved are closely related to Berenstein and Zelevinsky’s geometric lifting of tropical maps to maps between totally positive varieties.     

On the zeta function of divisors for projective varieties with higher rank divisor class group

Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of Cp. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case. In particular, we prove a p-adic meromorphic continuation theorem which applies to a large class of varieties. Examples of such varieties are projective nonsingular surfaces defined over a finite field (whose effective monoid is finitely generated) and all projective toric varieties (smooth or singular).     

Subvarieties of generic hypersurfaces in a nonsingular projective toric variety

We investigate the subvarieties contained in generic hypersurfaces of projective toric varieties and prove two main theorems. The first generalizes Clemens’ famous theorem on the genus of curves in hypersurfaces of projective spaces to curves in hypersurfaces of toric varieties and the second improves the bound in the special case of toric varieties in a theorem of Ein on the positivity of subvarieties contained in sufficiently ample generic hypersurfaces of projective varieties. Both depend on a hypothesis which deals with the surjectivity of multiplication maps of sections of line bundles on the toric variety. We also obtain an infinitesimal Torelli theorem for hypersurfaces of toric varieties.     

Is Every Toric Variety an M-Variety?

A complex algebraic variety X defined over the real numbers is called an M-variety if the sum of its Betti numbers (for homology with closed supports and coefficients in ) coincides with the corresponding sum for the real part of X. It has been known for a long time that any nonsingular complete toric variety is an M-variety. In this paper we consider whether this remains true for toric varieties that are singular or not complete, and we give a positive answer when the dimension of X is less than or equal to 3 or when X is complete with isolated singularities.An erratum to this article can be found at     

Gorenstein categories and Tate cohomology on projective schemes

We study Gorenstein categories. We show that such a category has Tate cohomological functors and Avramov–Martsinkovsky exact sequences connecting the Gorenstein relative, the absolute and the Tate cohomological functors. We show that such a category has what Hovey calls an injective model structure and also a projective model structure in case the category has enough projectives. As examples we show that if X is a locally Gorenstein projective scheme then the category ??????(X) of quasi‐coherent sheaves on X is such a category and so has these features. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)