Fano Varieties Research Articles

Birational geometry of Fano double spaces of index two

We study birational geometry of Fano varieties realized as double covers , , branched over generic smooth hypersurfaces of degree . We prove that the only structures of a rationally connected fibre space on are pencil-subsystems of the free linear system . The groups of birational and biregular self-maps of coincide: .

Configurations of points and strings

Let X be a smooth projective variety of dimension n ≥ 2 . It is shown that a finite configuration of points on X subject to certain geometric conditions possesses rich inner structure. On the mathematical level this inner structure is a variation of Hodge-like structure. As a consequence one can attach to such point configurations: (i) Lie algebras and their representations; (ii) a Fano toric variety whose hyperplane sections are Calabi–Yau varieties. These features imply that the points cease to be zero-dimensional objects and acquire dynamics of linear operators “propagating” along the paths of a particular trivalent graph. Furthermore, following particular linear operators along the “shortest” paths of the graph, one creates, for every point of the configuration, a distinguished hyperplane section of the Fano variety in (ii), i.e. the points “open up” to become Calabi–Yau varieties. Thus one is led to a picture which is very suggestive of quantum gravity according to string theory.

THE PSEUDO-INDEX OF HOROSPHERICAL FANO VARIETIES

We prove a conjecture of Bonavero et al. on the pseudo-index of smooth Fano varieties, in the special case of horospherical varieties.

The variety of reductions for a reductive symmetric pair

We define and study the variety of reductions for a complex reductive symmetric pair (G, θ), which is the natural compactification of the set of its Cartan subspaces. These varieties generalize the varieties of reductions for the Severi varieties studied by Iliev and Manivel, which are Fano varieties.We develop a theoretical basis to the study of these varieties of reductions, and relate their geometry to some problems in representation theory. A very useful result is the rigidity of semisimple elements in deformations of algebraic subalgebras of Lie algebras. We use it to show that the closure of a decomposition class is a union of decomposition classes.We apply this theory to the study of other varieties of reductions in a companion paper, which yields two new Fano varieties.

Pseudotoric Lagrangian fibrations of toric and nontoric fano varieties

We introduce the notion of a pseudotoric structure on a symplectic manifold, generalizing the notion of a toric structure. We show that such a pseudotoric structure can exist on toric and nontoric symplectic manifolds. For the toric manifolds, it describes deformations of the standard toric Lagrangian fibrations; for the nontoric ones, it gives Lagrangian fibrations with singularities that are very close to the toric fibrations. We present examples of toric manifolds with different pseudotoric structures and prove that certain nontoric manifolds (smooth complex quadrics) have such structures. In the future, introducing this new structure can be useful for generalizing the geometric quantization and mirror symmetry methods that work well in the toric case to a broader class of Fano varieties.

Helices on del Pezzo surfaces and tilting Calabi–Yau algebras

We study tilting for a class of Calabi–Yau algebras associated to helices on Fano varieties. We do this by relating the tilting operation to mutations of exceptional collections. For helices on del Pezzo surfaces the algebras are of dimension three, and using an argument of Herzog, together with results of Kuleshov and Orlov, we obtain a complete description of the tilting process in terms of quiver mutations.

Nontoric foliations by Lagrangian tori of toric Fano varieties

A construction of a foliation of a toric Fano variety by Lagrangian tori is presented; it is based on linear subsystems of divisor systems of various degrees invariant under the Hamiltonian action of distinguished function-symbols. It is shown that known examples of foliations (such as the Clifford foliation and D. Auroux’s example) are special cases of this construction. As an application, nontoric Lagrangian foliations by tori of two-dimensional quadrics and projective space are constructed.

Globally F-regular and log Fano varieties

We prove that every globally F-regular variety is log Fano. In other words, if a prime characteristic variety X is globally F-regular, then it admits an effective Q -divisor Δ such that − K X − Δ is ample and ( X , Δ ) has controlled (Kawamata log terminal, in fact globally F-regular) singularities. A weak form of this result can be viewed as a prime characteristic analog of de Fernex and Hacon's new point of view on Kawamata log terminal singularities in the non- Q -Gorenstein case. We also prove a converse statement in characteristic zero: every log Fano variety has globally F-regular type. Our techniques apply also to F-split varieties, which we show to satisfy a “log Calabi–Yau” condition. We also prove a Kawamata–Viehweg vanishing theorem for globally F-regular pairs.

Tangent bundle of a complete intersection

Let X be a Fano variety of Picard number one defined over an algebraically closed field. We give conditions under which the tangent bundle of a complete intersection on X is stable or strongly stable.

A note on restriction theorems for semistable sheaves

We prove a new restriction theorem for semistable sheaves on varieties in all characteristics strengthening previous results. We also prove restriction theorem for strong semistability for varieties with some non-negativity constrains on the cotangent bundle (e.g., most of Fano and Calabi-Yau varieties).

$\boldsymbol{Q}$-factorial Gorenstein toric Fano varieties with large Picard number

In dimension $d$, ${\boldsymbol Q}$-factorial Gorenstein toric Fano varieties with Picard number $\rho_X$ correspond to simplicial reflexive polytopes with $\rho_X + d$ vertices. Casagrande showed that any $d$-dimensional simplicial reflexive polytope has at most $3 d$ and $3d-1$ vertices if $d$ is even and odd, respectively. Moreover, for $d$ even there is up to unimodular equivalence only one such polytope with $3 d$ vertices, corresponding to the product of $d/2$ copies of a del Pezzo surface of degree six. In this paper we completely classify all $d$-dimensional simplicial reflexive polytopes having $3d-1$ vertices, corresponding to $d$-dimensional ${\boldsymbol Q}$-factorial Gorenstein toric Fano varieties with Picard number $2d-1$. For $d$ even, there exist three such varieties, with two being singular, while for $d > 1$ odd there exist precisely two, both being nonsingular toric fiber bundles over the projective line. This generalizes recent work of the second author.

Classification of Gorenstein Toric Del Pezzo Varieties in Arbitrary Dimension

An n-dimensional Gorenstein toric Fano variety X is called Del Pezzo variety if the anticanonical class KX is an (n 1)-multiple of a Cartier divisor. Our purpose is to give a complete biregular class- fication of Gorenstein toric Del Pezzo varieties in arbitrary dimension n > 2. We show that up to isomorphism there exist exactly 37 Goren- stein toric Del Pezzo varieties of dimension n which are not cones over (n 1)-dimensional Gorenstein toric Del Pezzo varieties. Our results are closely related to the classification of all Minkowski sum decompositions of reflexive polygons due to Emiris and Tsigaridas and to the classifi- cation up to deformation of n-dimensional almost Del Pezzo manifolds obtained by Jahnke and Peternell.

Geometry of the genus 9 Fano 4-folds

We study the geometry of a general Fano variety of dimension four, genus nine, and Picard number one. We compute its Chow ring and give an explicit description of its variety of lines. We apply these results to study the geometry of non quadratically normal varieties of dimension three in a five dimensional projective space.

On Fano manifolds with a birational contraction sending a divisor to a curve

Let X be a smooth Fano variety of dimension at least 4. We show that if X has an elementary birational contraction sending a divisor to a curve, then the Picard number of X is smaller or equal to 5.

On Fano threefolds with canonical Gorenstein singularities

We classify three-dimensional Fano varieties with canonical Gorenstein singularities whose degree is greater than 64.Bibliography: 32 titles.

On the Fano variety of a class of real four-dimensional cubics

The topological type of the real part of the Fano variety parametrizing the set of lines on a nonsingular real hypersurface of degree three in a five-dimensional projective space is evaluated provided that the hypersurface belongs to a special rigid projective class. In the paper by Finashin and Kharlamov on the rigid projective classification of real four-dimensional cubics, this class is said to be irregular. The results of the author of the present paper from the article devoted to the equivariant topological classification of the Fano varieties of real cubic fourfolds are also used.

Seshadri constants and the generation of jets

In this paper we explore the connection between Seshadri constants and the generation of jets. It is well known that one way to view Seshadri constants is to consider them as measuring the rate of growth of the number of jets that multiples of a line bundle generate. Here we ask, conversely, what we can say about the number of jets once the Seshadri constant is known. As an application of our results, we prove a characterization of projective space among all Fano varieties in terms of Seshadri constants.

Birational maps and special Lagrangian fibrations

Birational maps give the main research tool for the theory of Fano varieties, as we know from the fundamental works of V.A. Iskovskikh and his school. Nowadays one can exploit them in the new approach of D. Auroux to Mirror Symmetry of Fano varieties, which is based on a certain generalization of the notion of special Lagrangian fibration suitable for Fano varieties. We present a very simple example of how a special Lagrangian fibration can be transferred by a birational map.

An update on semisimple quantum cohomology and F-manifolds

To the memory of V.A. Iskovskikh Abstract—In the first section of this note, we show that Theorem 1.8.1 of Bayer-Manin can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold V is generically semisimple, then V has no odd cohomology and is of Hodge-Tate type. In particular, this answers a question discussed by G. Ciolli. In the second section, we prove that an analytic (or formal ) supermanifold M with a given supercommutative associative OM -bilinear multiplication on its tangent sheaf TM is an F -manifold in the sense of Hertling- Manin if and only if its spectral cover, as an analytic subspace of the cotangent bundle T ∗ M , is coisotropic of maximal dimension. This answers a question of V. Ginzburg. Finally, we discuss these results in the context of mirror symmetry and Landau-Ginzburg models for Fano varieties.

Birational rigidity of Fano varieties and field extensions

This note studies the behavior of birational rigidity and universal birational rigidity in extensions of algebraically closed fields.