Fano Varieties Research Articles

Toric Degenerations and Laurent Polynomials Related to Givental's Landau–Ginzburg Models

AbstractFor an appropriate class of Fano complete intersections in toric varieties, we prove that there is a concrete relationship between degenerations to speciûc toric subvarieties and expressions for Givental's Landau–Ginzburg models as Laurent polynomials. As a result, we show that Fano varieties presented as complete intersections in partial flag manifolds admit degenerations to Gorenstein toric weak Fano varieties, and their Givental Landau–Ginzburg models can be expressed as corresponding Laurent polynomials.We also use this to show that all of the Laurent polynomials obtained by Coates, Kasprzyk and Prince by the so–called Przyjalkowski method correspond to toric degenerations of the corresponding Fano variety. We discuss applications to geometric transitions of Calabi–Yau varieties.

On K-stability of finite covers

We show that certain Galois covers of K-semistable Fano varieties are K-stable. We use this to give some new examples of Fano manifolds admitting K\"ahler-Einstein metrics, including hypersurfaces, double solids and threefolds.

ON THE MODIFIED FUTAKI INVARIANT OF COMPLETE INTERSECTIONS IN PROJECTIVE SPACES

Let $M$ be a Fano manifold. We call a Kähler metric ${\it\omega}\in c_{1}(M)$ a Kähler–Ricci soliton if it satisfies the equation $\text{Ric}({\it\omega})-{\it\omega}=L_{V}{\it\omega}$ for some holomorphic vector field $V$ on $M$. It is known that a necessary condition for the existence of Kähler–Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian and Zhu. In a recent work of Berman and Nyström, it was generalized for (possibly singular) Fano varieties, and the notion of algebrogeometric stability of the pair $(M,V)$ was introduced. In this paper, we propose a method of computing the modified Futaki invariant for Fano complete intersections in projective spaces.

On singular Fano varieties with a divisor of Picard number one

In this paper we study the geometry of mildly singular Fano varieties on which there is an effective prime divisor of Picard number one. Afterwards, we address the case of toric varieties. Finally, we treat the lifting of extremal contractions to universal covering spaces in codimension 1.

The generalized Mukai conjecture for symmetric varieties

We associate to any complete spherical variety $X$ a certain nonnegative rational number $\wp ({X})$, which we conjecture to satisfy the inequality $\wp ({X}) \le \dim X - \mathrm {rank} X$ with equality holding if and only if $X$ is isomorphic to a toric variety. We show that, for spherical varieties, our conjecture implies the generalized Mukai conjecture on the pseudo-index of smooth Fano varieties due to Bonavero, Casagrande, Debarre, and Druel. We also deduce from our conjecture a smoothness criterion for spherical varieties. It follows from the work of Pasquier that our conjecture holds for horospherical varieties. We are able to prove our conjecture for symmetric varieties.

Homological projective duality for determinantal varieties

In this paper we prove Homological Projective Duality for categorical resolutions of several classes of linear determinantal varieties. By this we mean varieties that are cut out by the minors of a given rank of a m×n matrix of linear forms on a given projective space. As applications, we obtain pairs of derived-equivalent Calabi–Yau manifolds, and address a question by A. Bondal asking whether the derived category of any smooth projective variety can be fully faithfully embedded in the derived category of a smooth Fano variety. Moreover we discuss the relation between rationality and categorical representability in codimension two for determinantal varieties.

SMOOTH HOROSPHERICAL VARIETIES OF PICARD NUMBER ONE AS LINEAR SECTIONS OF RATIONAL HOMOGENEOUS VARIETIES

We construct projective embeddings of horospherical varieties of Picard number one by means of Fano varieties of cones over rational homogeneous varieties. Then we use them to give embeddings of smooth horospherical varieties of Picard number one as linear sections of rational homogeneous varieties.

A closer look at mirrors and quotients of Calabi-Yau threefolds

Let X be the toric variety (P1)4 associated with its four-dimensional polytope 1. Denote by X˜ the resolution of the singular Fano variety Xo associated with the dual polytope 1o. Generically, anticanonical sections Y of X and anticanonical sections Y˜ of X˜ are mirror partners in the sense of Batyrev. Our main result is the following: the Hodge-theoretic mirror of the quotient Z associated to a maximal admissible pair (Y, G) in X is not a quotient Z˜ associated to an admissible pair in X˜ . Nevertheless, it is possible to construct a mirror orbifold for Z by means of a quotient of a suitable Y˜. Its crepant resolution is a Calabi-Yau threefold with Hodge numbers (8, 4). Instead, if we start from a non-maximal admissible pair, in the same case, its mirror is the quotient associated to an admissible pair.

Proof of the gamma conjecture for Fano 3-folds of Picard rank 1

We verify the (first) gamma conjecture, which relates the gamma class of a Fano variety to the asymptotics at infinity of the Frobenius solutions of its associated quantum differential equation, for all 17 of the deformation classes of Fano 3-folds of rank 1. This involves computing the corresponding limits (`Frobenius limits') for the Picard–Fuchs differential equations of Apéry type associated by mirror symmetry with the Fano families, and is achieved using two methods, one combinatorial and the other using the modular properties of the differential equations. The gamma conjecture for Fano 3-folds always contains a rational multiple of the number . We present numerical evidence suggesting that higher Frobenius limits of Apéry-like differential equations may be related to multiple zeta values.

Balanced line bundles on Fano varieties

Abstract A conjecture of Batyrev and Manin relates arithmetic properties of varieties with ample anticanonical class to geometric invariants. We analyze the geometry underlying these invariants using the Minimal Model Program and then apply our results to primitive Fano threefolds.

On Fano varieties whose effective divisors are numerically eventually free

In this paper we classify mildly singular Fano varieties with maximal Picard number whose effective divisors are numerically eventually free.

O-minimality on twisted universal torsors and Manin's conjecture over number fields

Manin’s conjecture predicts the distribution of rational points on Fano varieties. Using explicit parameterizations of rational points by integral points on universal torsors and lattice-point-counting techniques, it was proved for several specific varieties over Q, in particular del Pezzo surfaces. We show how this method can be implemented over arbitrary number fields, by proving Manin’s conjecture for a singular quartic del Pezzo surface of type A3 + A1. The parameterization step is treated in high generality with the help of twisted integral models of universal torsors. To make the counting step feasible over arbitrary number fields, we deviate from the usual approach over Q by placing higher emphasis on the geometry of numbers in the framework of o-minimal structures.

SOME REMARKS ON CONES OF PARTIALLY AMPLE DIVISORS

We study the cones of q-ample divisors on smooth complex varieties. In favourable cases, we identify a part where the closure of this cone and the nef cone have the same boundary. This is especially interesting for Fano (or almost Fano) varieties.

Some remarks on cones of partially ample divisors

We study the cones of q-ample divisors on smooth complex varieties. In favourable cases, we identify a part where the closure of this cone and the nef cone have the same boundary. This is especially interesting for Fano (or almost Fano) varieties.

On log K-stability for asymptotically log Fano varieties

The notion of asymptotically log Fano varieties was given by Cheltsov and Rubinstein. We show that, if an asymptotically log Fano variety $(X, D)$ satisfies that $D$ is irreducible and $-K_X-D$ is big, then $X$ does not admit Kahler-Einstein edge metrics with angle $2\pi\beta$ along $D$ for any sufficiently small positive rational number $\beta$. This gives an affirmative answer to a conjecture of Cheltsov and Rubinstein.

On Berman–Gibbs stability and K-stability of -Fano varieties

The notion of Berman–Gibbs stability was originally introduced by Berman for$\mathbb{Q}$-Fano varieties$X$. We show that the pair$(X,-K_{X})$is K-stable (respectively K-semistable) provided that$X$is Berman–Gibbs stable (respectively semistable).

Variation of Log Canonical Thresholds in Linear Systems

We investigate the variation of log canonical thresholds in (graded) linear systems. For toric log Fano varieties, we give a sharp lower bound for log canonical thresholds of the anticanonical members in terms of the global minimal log discrepancy.

Asymptotically log Fano varieties

Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional assumption of log smoothness, and give a complete classification of two dimensional strongly asymptotically log smooth log Fano varieties. Based on this classification we formulate an asymptotic logarithmic version of Calabi's conjecture for del Pezzo surfaces for the existence of Kähler–Einstein edge metrics in this regime. We make some initial progress towards its proof by demonstrating some existence and non-existence results, among them a generalization of Matsushima's result on the reductivity of the automorphism group of the pair, and results on log canonical thresholds of pairs. One by-product of this study is a new conjectural picture for the small angle regime and limit which reveals a rich structure in the asymptotic regime, of which a folklore conjecture concerning the case of a Fano manifold with an anticanonical divisor is a special case.

Smoothings of Fano Varieties With Normal Crossing Singularities

AbstractThis paper obtains criteria for a Fano variety X defined over an algebraically closed field of characteristic zero with normal crossing singularities to be smoothable. In particular, we show that X is smoothable by a flat deformation X → Δ with smooth total space X if and only if where D is the singular locus of X.

CLASSIFICATION OF POLARIZED SYMPLECTIC AUTOMORPHISMS OF FANO VARIETIES OF CUBIC FOURFOLDS

AbstractWe classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci.