Fano Varieties Research Articles

On the minimal model program for projective varieties with pseudo-effective tangent sheaf

In this paper, we develop a theory of pseudo-effective sheaves on normal projective varieties. As an application, by running the minimal model program, we show that projective klt varieties with pseudo-effective tangent sheaf can be decomposed into Fano varieties and Q-abelian varieties.

Apéry extensions

AbstractThe Apéry numbers of Fano varieties are asymptotic invariants of their quantum differential equations. In this paper, we initiate a program to exhibit these invariants as (mirror to) limiting extension classes of higher cycles on the associated Landau–Ginzburg (LG) models — and thus, in particular, as periods. We also construct an Apéry motive, whose mixed Hodge structure is shown, as an application of the decomposition theorem, to contain the limiting extension classes in question. Using a new technical result on the inhomogeneous Picard–Fuchs equations satisfied by higher normal functions, we illustrate this proposal with detailed calculations for LG‐models mirror to several Fano threefolds. By describing the “elementary” Apéry numbers in terms of regulators of higher cycles (i.e., algebraic ‐theory/motivic cohomology classes), we obtain a satisfying explanation of their arithmetic properties. Indeed, in each case, the LG‐models are modular families of surfaces, and the distinction between multiples of and (or ) translates ultimately into one between algebraic and of the family.

Arithmetic and geometric deformations of threefolds

AbstractWe show that mixed‐characteristic and equi‐characteristic small deformations of 3‐dimensional canonical (resp., terminal) singularities with perfect residue field of characteristic are canonical (resp., terminal). We discuss applications to arithmetic and geometric families of 3‐dimensional Fano varieties and minimal models with canonical singularities. Our results are contingent upon the existence of log resolutions for fourfolds.

Klt varieties of general type with small volume

By Hacon-McKernan-Xu, there is a positive lower bound in each dimension for the volume of all klt varieties with ample canonical class. We show that these bounds must go to zero extremely fast as the dimension increases, by constructing a klt $n$-fold with ample canonical class whose volume is less than $1/2^{2^n}$. These examples should be close to optimal. We also construct a klt Fano variety of each dimension $n$ such that $H^0(X,-mK_X)=0$ for all $1\leq m < b$ with $b$ roughly $2^{2^n}$. Here again there is some bound in each dimension, by Birkar's theorem on boundedness of complements, and we are showing that the bound must increase extremely fast with the dimension.

Positivity of Higher Exterior Powers of the Tangent Bundle

Abstract We prove that a smooth projective variety $X$ of dimension $n$ with strictly nef third, fourth, or $(n-1)$-th exterior power of the tangent bundle is a Fano variety. Moreover, in the first two cases, we provide a classification for $X$ under the assumption that $\rho (X)\ne 1$.

Asymptotic Chow Semistability Implies Ding Polystability for Gorenstein Toric Fano Varieties

In this paper, we prove that if a Gorenstein toric Fano variety (X,−KX) is asymptotically Chow semistable, then it is Ding polystable with respect to toric test configurations (Theorem ). This extends the known result obtained by others (Theorem ) to the case where X admits Gorenstein singularity. We also show the additivity of the Mabuchi constant for the product toric Fano varieties in Proposition based on the author’s recent work (Ono, Sano and Yotsutani in arxiv:2305.05924). Applying this formula to certain toric Fano varieties, we construct infinitely many examples that clarify the difference between relative K-stability and relative Ding stability in a systematic way (Proposition ). Finally, we verify the relative Chow stability for Gorenstein toric del Pezzo surfaces using the combinatorial criterion developed in (Yotsutani and Zhou in Tohoku Math. J. 71 (2019), 495–524.) and specifying the symmetry of the associated polytopes as well.

Machine learning the dimension of a Fano variety

Fano varieties are basic building blocks in geometry – they are ‘atomic pieces’ of mathematical shapes. Recent progress in the classification of Fano varieties involves analysing an invariant called the quantum period. This is a sequence of integers which gives a numerical fingerprint for a Fano variety. It is conjectured that a Fano variety is uniquely determined by its quantum period. If this is true, one should be able to recover geometric properties of a Fano variety directly from its quantum period. We apply machine learning to the question: does the quantum period of X know the dimension of X? Note that there is as yet no theoretical understanding of this. We show that a simple feed-forward neural network can determine the dimension of X with 98% accuracy. Building on this, we establish rigorous asymptotics for the quantum periods of a class of Fano varieties. These asymptotics determine the dimension of X from its quantum period. Our results demonstrate that machine learning can pick out structure from complex mathematical data in situations where we lack theoretical understanding. They also give positive evidence for the conjecture that the quantum period of a Fano variety determines that variety.

Making Waves

We study linear PDE constraints for vector-valued functions and distributions. Our focus lies on wave solutions, which give rise to distributions with low-dimensional support. Special waves from vector potentials are represented by syzygies. We parametrize all waves by projective varieties derived from the support of the PDE. These include determinantal varieties and Fano varieties, and they generalize wave cones in analysis.

On the Hodge conjecture for complete intersections

We study the Fano variety of k-planes Ω X ( k ) of a complete intersection X and show the surjectivity of the cylinder homomorphism in this setting, proving along the way the (classical) Hodge conjecture in cases where a numerical condition is satisfied. In particular, a new case is discovered in dimension 6. This work is a generalization of the second author’s paper Motives of some Fano varieties.

Projective varieties with nef tangent bundle in positive characteristic

Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties with nef tangent bundle and $T_M$ is numerically flat. We also prove that extremal contractions exist as smooth morphisms. As an application, we prove that, if the Frobenius morphism can be lifted modulo $p^2$, then $X$ admits, up to a finite étale Galois cover, a smooth morphism onto an ordinary abelian variety whose fibers are products of projective spaces.

Special issue on Fano varieties

Special issue on Fano varieties

On some invariants of cubic fourfolds

For a general cubic fourfold Xsubset mathbb {P}^5 with Fano variety F, we compute the Hodge numbers of the locus Ssubset F of lines of second type and the class of the locus Vsubset F of triple lines, using the description of the latter in terms of flag varieties. We also give an upper bound of 6 for the degree of irrationality of the Fano scheme of lines of any smooth cubic hypersurface.

Infinitesimal categorical Torelli theorems for Fano threefolds

Let X be a smooth Fano variety and Ku(X) its Kuznetsov component. A Torelli theorem for Ku(X) states that Ku(X) is uniquely determined by a certain polarized abelian variety associated to it. An infinitesimal Torelli theorem for X states that the differential of the period map is injective. A categorical variant of the infinitesimal Torelli theorem for X states that the morphism η:H1(X,TX)→HH2(Ku(X)) is injective. In the present article, we use the machinery of Hochschild (co)homology to relate the aforementioned three Torelli-type theorems for smooth Fano varieties via a commutative diagram. As an application, we prove infinitesimal categorical Torelli theorems for a class of prime Fano threefolds. We then prove, infinitesimally, a restatement of the Debarre–Iliev–Manivel conjecture regarding the general fibre of the period map for ordinary Gushel–Mukai threefolds.

Klt Varieties With Conjecturally Minimal Volume

Abstract We construct klt projective varieties with ample canonical class and the smallest known volume. We also find exceptional klt Fano varieties with the smallest known anti-canonical volume. We conjecture that our examples have the smallest volume in every dimension, and we give low-dimensional evidence for that. In order to improve on earlier examples, we are forced to consider weighted hypersurfaces that are not quasi-smooth. We show that our Fano varieties are exceptional by computing their global log canonical threshold (or $\alpha $-invariant) exactly; it is extremely large, roughly $2^{2^n}$ in dimension $n$. These examples give improved lower bounds in Birkar’s theorem on boundedness of complements for Fano varieties.

On the boundedness of globally F-split varieties

This paper proposes the use of F-split and globally F-regular conditions in the pursuit of BAB type results in positive characteristic. The main technical work comes in the form of a detailed study of threefold Mori fibre spaces over positive dimensional bases. As a consequence we prove the main theorem, which reduces birational boundedness for a large class of varieties to the study of prime Fano varieties.

Fano fourfolds having a prime divisor of Picard number 1

Abstract We prove a classification result for smooth complex Fano fourfolds of Picard number 3 having a prime divisor of Picard number 1, after a characterisation result in arbitrary dimension by Casagrande and Druel [5]. These varieties are obtained by blowing-up a ℙ1-bundle over a smooth Fano variety of Picard number 1 along a codimension 2 subvariety. We study in detail the case of dimension 4, and show that they form 28 families. We compute the main numerical invariants, determine the base locus of the anticanonical system, and study their deformations to give an upper bound to the dimension of the base of the Kuranishi family of a general member.

Varieties of general type with doubly exponential asymptotics

We construct smooth projective varieties of general type with the smallest known volume and others with the most known vanishing plurigenera in high dimensions. The optimal volume bound is expected to decay doubly exponentially with dimension, and our examples achieve this decay rate. We also consider the analogous questions for other types of varieties. For example, in every dimension we conjecture the terminal Fano variety of minimal volume, and the canonical Calabi-Yau variety of minimal volume. In each case, our examples exhibit doubly exponential behavior.

Geometry of polarised varieties

In this paper, we investigate the geometry of projective varieties polarised by ample and more generally nef and big Weil divisors. First we study birational boundedness of linear systems. We show that if X is a projective variety of dimension d with epsilon -lc singularities for epsilon >0, and if N is a nef and big Weil divisor on X such that N-K_{X} is pseudo-effective, then the linear system |mN| defines a birational map for some natural number m depending only on d,epsilon . This is key to proving various other results. For example, it implies that if N is a big Weil divisor (not necessarily nef) on a klt Calabi-Yau variety of dimension d, then the linear system |mN| defines a birational map for some natural number m depending only on d. It also gives new proofs of some known results, for example, if X is an epsilon -lc Fano variety of dimension d then taking N=-K_{X} we recover birationality of |-mK_{X}| for bounded m.We prove similar birational boundedness results for nef and big Weil divisors N on projective klt varieties X when both K_{X} and N-K_{X} are pseudo-effective (here X is not assumed epsilon -lc).Using the above, we show boundedness of polarised varieties under some natural conditions. We extend these to boundedness of semi-log canonical Calabi-Yau pairs polarised by effective ample Weil divisors not containing lc centres. We will briefly discuss applications to existence of projective coarse moduli spaces of such polarised Calabi-Yau pairs.

Bounds on the Picard rank of toric Fano varieties with minimal curve constraints

We study the Picard rank of smooth toric Fano varieties constrained to possess families of minimal rational curves of given degree. We discuss variants of a conjecture of Chen–Fu–Hwang and prove a version of their statement that recovers the original conjecture in sufficiently high dimension. We also prove new cases of the original conjecture for high degrees in all dimensions. Our main tools come from toric Mori theory and the combinatorics of Fano polytopes.

Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two

Abstract In this paper, we give a complete classification of full exceptional collections, up to cyclic permutations, normalizations and mutations, on smooth toric Fano threefolds and fourfolds with Picard rank two. For such varieties, we find all the exceptional collections of maximal length and show that they are in fact full. This gives a partial answer to a conjecture in [29] and [32]. Moreover, such full exceptional collections essentially arise from Orlov’s theorem on projective bundles.