The classification of smooth toric weakened Fano 3-folds

 We obtain a complete classification of toric weakened Fano 3-folds, that is, smooth toric weak Fano 3-folds which are not Fano but are deformed to smooth Fano 3-folds. There exist exactly 15 toric weakened Fano 3-folds up to isomorphisms. Received: 30 January 2002 / Revised version: 6 May 2002     

Classification of Toric 2-Fano 4-folds

In this notes we classify toric Fano 4-folds having positive second Chern character.     

On the classification of toric Fano 4-folds

The biregular classification of smoothd-dimensional toric Fano varieties is equivalent to the classification of special simplicial polyhedraP in ℝ ^d , the so-called Fano polyhedra, up to an isomorphism of the standard lattice . In this paper, we explain the complete biregular classification of all 4-dimensional smooth toric Fano varieties. The main result states that there exist exactly 123 different types of toric Fano 4-folds somorphism. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory Vol. 56. Algebraic Geometry-9, 1998.     

Local Euler obstructions of toric varieties

We use Matsui and Takeuchi's formula for toric A-discriminants to give algorithms for computing local Euler obstructions and dual degrees of toric surfaces and 3-folds. In particular, we consider weighted projective spaces. As an application we give counterexamples to a conjecture by Matsui and Takeuchi. As another application we recover the well-known fact that the only defective normal toric surfaces are cones.     

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.     

Abelian surfaces in projective toric 4-folds

We show fundamental properties on embedding of abelian surfaces into projective toric 4-folds, and study the case of the toric Del Pezzo 4-fold from the viewpoint of the moduli space of abelian surfaces with polarization of type (1, 5). Received: 31 January 2005; revised: 15 April 2005     

Toric degenerations of spherical varieties

We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.     

A tropical approach to secant dimensions

Tropical geometry yields good lower bounds, in terms of certain combinatorial-polyhedral optimisation problems, on the dimensions of secant varieties. The approach is especially successful for toric varieties such as Segre-Veronese embeddings. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for the quadratic one and the quartic one are non-defective; and indeed, no Segre-Veronese embeddings are known where the tropical lower bound does not give the correct dimension. Short self-contained introductions to secant varieties and the required tropical geometry are included.     

On the quantum homology algebra of toric Fano manifolds

We study certain algebraic properties of the small quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semisimplicity of this algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily verifiable sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non-semisimple quantum homology, and others in which the Calabi quasi-morphism is not unique.      

Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds

We prove the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the Gromov-Witten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting.     

An integrable system of K3-Fano flags

Given a K3 surface S, we show that the relative intermediate Jacobian of the universal family of Fano 3-folds V containing S as an anticanonical divisor is a Lagrangian fibration.     

Compactifications of contractible affine 3-folds into smooth Fano 3-folds with <Emphasis Type="Italic">B</Emphasis><Subscript>2</Subscript>=2

After the contributions of Furushima, Nakayama, Peternell and Schneider, in 1993, Furushima [Fur93] finally succeeded in the classification of the compactifications of the affine 3-space into smooth Fano 3-folds with B₂=1. In this paper, we consider the compactifications of the contractible affine 3-folds X (not necessarily X=) into smooth Fano 3-folds V with B₂=2. Consequently, we classify all such compactifications X↪(V,D₁∪D₂) in the case where K_V+D₁+D₂ is not nef. Furthermore, we see that infinitely many mutually non-isomorphic exotic 's can be compactified into Fano 3-folds with B₂=2. This phenomenon never occurs when B₂=1. During this research the author was supported as a Twenty-First Century COE Kyoto Mathematics Fellow.     

Complete toric varieties with reductive automorphism group

We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism group of a complete toric variety is proven by studying the set of Demazure roots.     

Moduli stacks of stable toric quasimaps

We construct new “virtually smooth” modular compactifications of spaces of maps from nonsingular curves to smooth projective toric varieties. They generalize Givental's compactifications, when the complex structure of the curve is allowed to vary and markings are included, and are the toric counterpart of the moduli spaces of stable quotients introduced by Marian, Oprea, and Pandharipande to compactify spaces of maps to Grassmannians. A brief discussion of the resulting invariants and their (conjectural) relation with Gromov-Witten theory is also included.     

Classification of Fano 3-folds with B2≥2

This article contains the classification of Fano 3-folds with B₂2.There exist exactly 87 types of such 3-folds up to deformations; a Fano 3-fold is isomorphic to a product of P^l and a del Pezzo surface if its second Betti number is not less than 6. In particular, the second Betti number of a Fano 3-fold is not greater than 10.Firstly we classify Fano 3-folds which are either primitive or have B₂=2 by the tools developed in [2]; then we study Fano 3-folds obtained from them by successive curve-blow-ups by using their conic bundle structures or the existence of lines on them.Partially supported by Educational Projects for Japanese Mathematical Scientists and NSF Grants MCS 77-15524 and MCS 77-18723 (A04)Partially supported by Educational Projects for Japanese Mathematical Scientists     

Three-Dimensional Toric Morphisms with Anti-Nef Canonical Divisors

In this article, we classify projective toric birational morphisms from Gorenstein toric 3-folds onto the 3-dimensional affine space with relatively ample anti-canonical divisors.     

Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels

V.G. Berkovich’s non-Archimedean analytic geometry provides a natural framework to understand the combinatorial aspects in the theory of toric varieties and toroidal embeddings. This point of view leads to a conceptual and elementary proof of the following results: if X is an algebraic scheme over a perfect field and if D is the exceptional normal crossing divisor of a resolution of the singularities of X, the homotopy type of the incidence complex of D is an invariant of X. This is a generalization of a theorem due to D. Stepanov.