Fano Varieties Research Articles

Stability of Equivariant Vector Bundles over Toric Varieties

We give a complete answer to the question of (semi)stability of tangent bundles on any nonsingular complex projective toric variety with Picard number 2 by using combinatorial criterion of (semi)stability of an equivariant sheaf. We also give a complete answer to the question of (semi)stability of tangent bundles on all toric Fano 4-folds with Picard number \leq 3 which are classified by Batyrev [1]. We construct a collection of equivariant indecomposable rank 2 vector bundles on Bott towers and pseudo-symmetric toric Fano varieties. Furthermore, in case of Bott towers, we show the existence of an equivariant stable rank 2 vector bundle with certain Chern classes with respect to a suitable polarization.

Higher codimensional alpha invariants and characterization of projective spaces

We generalize the definition of alpha invariant to arbitrary codimension. We also give a lower bound of these alpha invariants for K-semistable [Formula: see text]-Fano varieties and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. Our results demonstrate the relation between alpha invariants of any codimension and volumes of Fano manifolds in the characterization of projective spaces.

Duality for systems of conservation laws

For one-dimensional systems of conservation laws admitting two additional conservation laws we assign a ruled surface of codimension two in projective space. We call two such systems dual if the corresponding ruled surfaces are dual. We show that a Hamiltonian system is autodual, its ruled surface sits in some quadric, and the generators of this ruled surface form a Legendre submanifold for the contact structure on Fano variety of this quadric. We also give a complete geometric description of 3-component nondiagonalizable systems of Temple class: such systems admit two additional conservation laws, they are dual to systems with constant characteristic speeds, constructed via maximal rank 3-webs of curves in space.

The wall-crossing formula and Lagrangian mutations

We prove a general form of the wall-crossing formula which relates the disk potentials of monotone Lagrangian submanifolds with their Floer-theoretic behaviour away from a Donaldson divisor. We define geometric operations called mutations of Lagrangian tori in del Pezzo surfaces and in toric Fano varieties of higher dimension, and study the corresponding wall-crossing formulas that compute the disk potential of a mutated torus from that of the original one.In the case of del Pezzo surfaces, this justifies the connection between Vianna's tori and the theory of mutations of Landau-Ginzburg seeds. In higher dimension, this provides new Lagrangian tori in toric Fanos corresponding to different chambers of the mirror variety, including ones which are conjecturally separated by infinitely many walls from the chamber containing the standard toric fibre.

Embedding non-projective Mori dream space

This paper is devoted to extend some Hu-Keel results on Mori dream spaces (MDS) beyond the projective setup. Namely, $\Q$-factorial algebraic varieties with finitely generated class group and Cox ring, here called \emph{weak} Mori dream spaces (wMDS), are considered. Conditions guaranteeing the existence of a neat embedding of a (completion of a) wMDS into a complete toric variety are studied, showing that, on the one hand, those which are complete and admitting low Picard number are always projective, hence Mori dream spaces in the sense of Hu-Keel. On the other hand, an example of a wMDS does not admitting any neat embedded \emph{sharp} completion (i.e. Picard number preserving) into a complete toric variety is given, on the contrary of what Hu and Keel exhibited for a MDS. Moreover, termination of the Mori minimal model program (MMP) for every divisor and a classification of rational contractions for a complete wMDS are studied, obtaining analogous conclusions as for a MDS. Finally, we give a characterization of a wMDS arising from a small $\Q$-factorial modification of a projective weak $\Q$-Fano variety.

Big Vector Bundles on Surfaces and Fourfolds

The aim of this note is to exhibit explicit sufficient cohomological criteria ensuring bigness of globally generated, rank-r vector bundles, $$r {\geqslant }2$$, on smooth, projective varieties of even dimension $$d {\leqslant }4$$. We also discuss connections of our general criteria to some recent results of other authors, as well as applications to tangent bundles of Fano varieties, to suitable Lazarsfeld–Mukai bundles on fourfolds, etcetera.

On stringy de Sitter spacetimes

We reexamine a family of models with a 3+1-dimensional de Sitter spacetime obtained in the standard tree-level low-energy limit of string theory with a non-trivial anisotropic axion-dilaton background. While such limiting approximations are encouraging but incomplete, our analysis reveals a host of novel features, and shows these models to relate standard and well understood supersymmetric string theory solutions. Finally, we conjecture that this de Sitter spacetime naturally arises by including more of the stringy degrees of freedom, such as a recently advanced variant of the non-commutative phase-space formalism, as well as the analytic continuation of a complex two-dimensional Fano variety arising as a small resolution in a Calabi-Yau 5-fold.

Monotonic Lagrangian Tori of Standard and Nonstandard Types in Toric and Pseudotoric Fano Varieties

Monotonic Lagrangian Tori of Standard and Nonstandard Types in Toric and Pseudotoric Fano Varieties

The Generalized Franchetta Conjecture for Some Hyperkähler Fourfolds

We obtain a "generalized Franchetta conjecture" type of statement for the Hilbert squares of low genus K3 surfaces, and for the Fano varieties of lines on certain cubic fourfolds.

Rational curves in holomorphic symplectic varieties and Gromov–Witten invariants

We use Gromov–Witten theory to study rational curves in holomorphic symplectic varieties. We present a numerical criterion for the existence of uniruled divisors swept out by rational curves in the primitive curve class of a very general holomorphic symplectic variety of K3[n] type. We also classify all rational curves in the primitive curve class of the Fano variety of lines in a very general cubic 4-fold, and prove the irreducibility of the corresponding moduli space. Our proofs rely on Gromov–Witten calculations by the first author, and in the Fano case on a geometric construction of Voisin. In the Fano case a second proof via classical geometry is sketched.

Uniqueness of K-polystable degenerations of Fano varieties

We prove that K-polystable degenerations of $\mathbb{Q}$-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable $\mathbb{Q}$-Fano varieties is separated. Together with recently proven boundedness and openness statements, the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable $\mathbb{Q}$-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable $\mathbb{Q}$-Fano variety is finite.

Anti-pluricanonical systems on Fano varieties

In this paper, we study the linear systems $|-mK_X|$ on Fano varieties $X$ with klt singularities. In a given dimension $d$, we prove $|-mK_X|$ is non-empty and contains an element with good singularities for some natural number $m$ depending only on $d$; if in addition $X$ is $\epsilon$-lc for some $\epsilon>0$, then we show that we can choose $m$ depending only on $d$ and $\epsilon$ so that $|-mK_X|$ defines a birational map. Further, we prove Shokurov's conjecture on boundedness of complements, and show that certain classes of Fano varieties form bounded families.

K-stability of birationally superrigid Fano varieties

We prove that every birationally superrigid Fano variety whose alpha invariant is greater than (respectively no smaller than) $\frac{1}{2}$ is K-stable (respectively K-semistable). We also prove that the alpha invariant of a birationally superrigid Fano variety of dimension $n$ is at least $1/(n+1)$ (under mild assumptions) and that the moduli space (if it exists) of birationally superrigid Fano varieties is separated.

Stringy -functions of canonical toric Fano threefolds and their applications

Let be a -dimensional lattice polytope containing exactly one interior lattice point. We give a simple combinatorial formula for computing the stringy -function of the -dimensional canonical toric Fano variety associated with . Using the stringy Libgober– Wood identity and our formula, we generalize the well-known combinatorial identity which holds for -dimensional reflexive polytopes .

On Automorphisms of Moduli Spaces of Parabolic Vector Bundles

Abstract Fix $n\geq 5$ general points $p_1, \dots , p_n\in{\mathbb{P}}^1$ and a weight vector ${\mathcal{A}} = (a_{1}, \dots , a_{n})$ of real numbers $0 \leq a_{i} \leq 1$. Consider the moduli space $\mathcal{M}_{{\mathcal{A}}}$ parametrizing rank two parabolic vector bundles with trivial determinant on $\big ({\mathbb{P}}^1, p_1,\dots , p_n\big )$ that are semistable with respect to ${\mathcal{A}}$. Under some conditions on the weights, we determine and give a modular interpretation for the automorphism group of the moduli space $\mathcal{M}_{{\mathcal{A}}}$. It is isomorphic to $\left (\frac{\mathbb{Z}}{2\mathbb{Z}}\right )^{k}$ for some $k\in \{0,\dots , n-1\}$ and is generated by admissible elementary transformations of parabolic vector bundles. The largest of these automorphism groups, with $k=n-1$, occurs for the central weight ${\mathcal{A}}_{F}= \left (\frac{1}{2},\dots ,\frac{1}{2}\right )$. The corresponding moduli space ${\mathcal M}_{{\mathcal{A}}_F}$ is a Fano variety of dimension $n-3$, which is smooth if $n$ is odd, and has isolated singularities if $n$ is even.

The failure of Kodaira vanishing for Fano varieties, and terminal singularities that are not Cohen-Macaulay

We show that the Kodaira vanishing theorem can fail on smooth Fano varieties of any characteristic p > 0 p>0 . Taking cones over some of these varieties, we give the first examples of terminal singularities which are not Cohen-Macaulay. By a different method, we construct a terminal singularity of dimension 3 (the lowest possible) in characteristic 2 which is not Cohen-Macaulay.

Embedding derived categories of Enriques surfaces in derived categories of Fano varieties

We show that the bounded derived category of coherent sheaves on a general Enriques surface can be realized as a semi-orthogonal component in the derived category of a smooth Fano variety with diagonal Hodge diamond.

On the proper moduli spaces of smoothable Kähler–Einstein Fano varieties

In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano K\"ahler-Einstein manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski open condition and establish the uniqueness for the Gromov-Hausdorff limit for a punctured flat family of Fano K\"ahler-Einstein manifolds. Based on these, we construct a proper scheme parameterizing the S-equivalent classes of $\QQ$-Gorenstein smoothable, K-semistable Fano varieties, and verify various necessary properties to guarantee that it is a good moduli space.

Gorenstein Formats, Canonical and Calabi–Yau Threefolds

Gorenstein formats present the equations of regular canonical, Calabi–Yau and Fano varieties embedded by subcanonical divisors. We present a new algorithm for the enumeration of these formats based on orbifold Riemann–Roch and knapsack packing-type algorithms. We apply this to extend the known lists of threefolds of general type beyond the well-known classes of complete intersections and also to find classes of Calabi–Yau threefolds with canonical singularities.

Geometric Manin’s conjecture and rational curves

Let$X$be a smooth projective Fano variety over the complex numbers. We study the moduli space of rational curves on$X$using the perspective of Manin’s conjecture. In particular, we bound the dimension and number of components of spaces of rational curves on$X$. We propose a geometric Manin’s conjecture predicting the growth rate of a counting function associated to the irreducible components of these moduli spaces.