Fano Varieties Research Articles

Weakly special test configurations of log canonical Fano varieties

Let $X$ be a strictly log canonical Fano variety, we show that every lc place of complements is dreamy, and there exists a correspondence between weakly special test configurations of $(X,-K_X)$ and lc places of complements.

The Lefschetz defect of Fano varieties

Smooth, complex Fano varieties are a very natural class of projective varieties, and have been intensively studied both classically and in recent times, due to their significance in the framework of the Minimal Model Program. The Lefschetz defect is an invariant of smooth Fano varieties that has been recently introduced in [7]; it is related to the Picard number \(\rho _X\) of X, and to the Picard number of prime divisors in X. This paper is a survey on this new invariant and its properties: we explain the definition of the Lefschetz defect \(\delta _X\) and its origin, the known results, and several examples, especially in dimensions 3 and 4; we do not give new theoretical results. In particular, we explain how the results on the Lefschetz defect allow to recover the classification of Fano 3-folds with \(\rho _X\ge 5\), due to Mori and Mukai. We also review the known examples of Fano 4-folds with \(\rho _X\ge 6\), that are remarkably few: apart products and toric examples, there are only 6 known families, with \(\rho _X\in \{6,7,8,9\}\).

Optimal destabilization of K–unstable Fano varieties via stability thresholds

We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano variety degenerates to a uniquely determined twisted K-polystable Fano variety. We also show that the stability threshold can be approximated by divisorial valuations induced by special test configurations. As an application of the above results and the analytic work of Datar, Sz\'ekelyhidi, and Ross, we deduce that greatest Ricci lower bounds of Fano manifolds of fixed dimension form a finite set of rational numbers. As a key step in the proofs, we adapt the process of Li and Xu producing special test configurations to twisted K-stability in the sense of Dervan.

Curve classes on Calabi–Yau complete intersections in toric varieties

AbstractWe prove the integral Hodge conjecture for curve classes on smooth varieties of dimension at least three constructed as a complete intersection of ample hypersurfaces in a smooth projective toric variety, such that the anticanonical divisor is the restriction of a nef divisor. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric minimal model program, we prove that in each case, is generated by classes of rational curves.

On rationally connected varieties over C1 fields of characteristic 0

We use birational geometry to show that the existence of rational points on proper rationally connected varieties over fields of characteristic $0$ is a consequence of the existence of rational points on terminal Fano varieties. We discuss several consequences of this result, especially in relation to the $C_1$-conjecture. We also provide evidence that supports the conjecture in dimension $3$ for $C_1$ fields of characteristic $0$.

Equivariant $${\mathbb {R}}$$-Test Configurations and Semistable Limits of $${\mathbb {Q}}$$-Fano Group Compactifications

Let G be a connected, complex reductive group. In this paper, we classify $$G\times G$$ -equivariant normal $${\mathbb {R}}$$ -test configurations of a polarized G-compactification. Then, for $${\mathbb {Q}}$$ -Fano G-compactifications, we express the H-invariants of their equivariant normal $${\mathbb {R}}$$ -test configurations in terms of the combinatory data. Based on Han and Li “Algebraic uniqueness of Kähler–Ricci flow limits and optimal degenerations of Fano varieties”, we compute the semistable limit of a K-unstable Fano G-compactification. As an application, we show that for the two smooth K-unstable Fano SO $$_4({\mathbb {C}})$$ -compactifications, the corresponding semistable limits are indeed the limit spaces of the normalized Kähler–Ricci flow.

Hilbert squares of degeneracy loci

Let S be the first degeneracy locus of a morphism of vector bundles corresponding to a general matrix of linear forms in \({\mathbb {P}}^s\). We prove that, under certain positivity conditions, its Hilbert square \({{\mathrm{Hilb}}}^2(S)\) is isomorphic to the zero locus of a global section of an irreducible homogeneous vector bundle on a product of Grassmannians. Our construction involves a naturally associated Fano variety, and an explicit description of the isomorphism.

A mirror theorem for multi-root stacks and applications

Let X be a smooth projective variety with a simple normal crossing divisor D:=D_1+D_2+cdots +D_n, where D_isubset X are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks X_{D,{overrightarrow{r}}} by constructing an I-function lying in a slice of Givental’s Lagrangian cone for Gromov–Witten theory of multi-root stacks. We provide three applications: (1) We show that some genus zero invariants of X_{D,overrightarrow{r}} stabilize for sufficiently large overrightarrow{r}. (2) We state a generalized local-log-orbifold principle conjecture and prove a version of it. (3) We show that regularized quantum periods of Fano varieties coincide with classical periods of the mirror Landau–Ginzburg potentials using orbifold invariants of X_{D,overrightarrow{r}}.

Higher index Fano varieties with finitely many birational automorphisms

A famous problem in birational geometry is to determine when the birational automorphism group of a Fano variety is finite. The Noether–Fano method has been the main approach to this problem. The purpose of this paper is to give a new approach to the problem by showing that in every positive characteristic, there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphism group. To do this, we prove that these varieties admit ample and birationally equivariant line bundles. Our result applies the differential forms that Kollár produces on $p$-cyclic covers in characteristic $p > 0$.

K-stability of Gorenstein Fano group compactifications with rank two

We give a classification of Gorenstein Fano bi-equivariant compactifications of semi-simple complex Lie groups with rank two, and determine which of them are equivariant K-stable and admit (singular) Kähler–Einstein metrics. As a consequence, we obtain several explicit examples of K-stable Fano varieties admitting (singular) Kähler–Einstein metrics. We also compute the greatest Ricci lower bounds, equivalently the delta invariants for K-unstable varieties. This gives us three new examples on which each solution of the Kähler–Ricci flow is of type II.

Derived categories of flips and cubic hypersurfaces

A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by describing the complement. As an application, we can lift the "quadratic Fano correspondence" (due to Galkin-Shinder) in the Grothendieck ring of varieties between a smooth cubic hypersurface, its Fano variety of lines, and its Hilbert square, to a semiorthogonal decomposition. We also show that the Hilbert square of a cubic hypersurface of dimension at least 3 is again a Fano variety, so in particular the Fano variety of lines on a cubic hypersurface is a Fano visitor. The most interesting case is that of a cubic fourfold, where this exhibits the first higher-dimensional hyperk\"ahler variety as a Fano visitor.

Special lines on contact manifolds

In a series of two articles Kebekus studied deformation theory of minimal rational curves on contact Fano manifolds. Such curves are called contact lines. Kebekus proved that a contact line through a general point is necessarily smooth and has a fixed standard splitting type of the restricted tangent bundle. In this paper we study singular contact lines and those with special splitting type. We provide restrictions on the families of such lines, and on contact Fano varieties which have reducible varieties of minimal rational tangents. We also show that the results about singular lines naturally generalise to complex contact manifolds, which are not necessarily Fano, for instance, quasiprojective contact manifolds or contact manifolds of Fujiki class C. In particular, in many cases the dimension of a family of singular lines is at most 2 less than the dimension of the contact manifold.

Expectations, concave transforms, chow weights and Roth’s theorem for varieties

We present a new perspective, which is at the intersection of K-stability and Diophantine arithmetic geometry. It builds on work of Boucksom and Chen (Compos Math 147(4):1205–1229, 2011), Boucksom et al. (Math Ann 361(3–4):811–834, 2015), Boucksom et al. (Ann Inst Fourier (Grenoble) 67(2):743–841, 2017), Codogni and Patakfalvi (Invent Math 223(3):811–894, 2021), Fujita (Amer J Math 140(2):391–414, 2018), Fujita (J reine angew Math 751:309–338, 2019), Li (Duke Math J 166(16):3147–3218, 2017), Ferretti (Compos Math 121(3):247–262, 2000), Ferretti (Duke Math J 118(3): 493–522, 2003), McKinnon and Roth (Invent Math 200(2): 513–583, 2015), Ru and Vojta (Amer J Math 142(3):957–991, 2020), Ru and Wang (Algebra Number Theory 11(10): 2323–2337, 2017), Ru and Wang (Int J Number Theory 18(1):61–74, 2022), Heier and Levin (Amer J Math 143(1): 213–226, 2021) and our related works (including Grieve (Michigan Math J 67(2):371–404, 2018), Grieve (Houston J Math 44(4):1181–1202, 2018), Grieve (Asian J Math 24(6):995–1005, 2020), Grieve (Acta Arith 195(4):415–428, 2020), Grieve (Res Number Theory 7(1):1, 14, 2021) and Grieve (C R Math Acad Sci Soc R Can 44(1):16–32, 2022)). Our main results may be described via the expectations of the Duistermaat-Heckman measures. For example, we prove that the expected orders of vanishing for very ample linear series along subvarieties may be calculated via the theory of Chow weights and Okounkov bodies. This result provides a novel unified viewpoint to the existing works that surround these topics. It expands on work of Mumford (Enseignement Math (2) 23(1-2):39–110, 1977), Odaka (Ann of Math (2) 177(2):645–661, 2013) and Fujita (Amer J Math 140(2):391–414, 2018). Moreover, it has applications for Diophantine approximation and K-stability. As one result of this flavour, we define a concept of uniform arithmetic $$\mathrm {K}$$ -stability for Kawamata log terminal pairs. We then establish a form of K. F. Roth’s celebrated approximation theorem for certain collections of divisorial valuations which fail to be uniformly arithmetically $$\mathrm {K}$$ -stable. This result builds on the Roth type theorems of McKinnon and Roth (Invent Math 200(2):513–583, 2015). Moreover, it complements the concept of arithmetic canonical boundedness, in the sense of McKinnon and Satriano (Trans Amer Math Soc 374(5):3557–3577, 2021), in addition to our results that establish instances of Vojta’s Main Conjecture for $${\mathbb {Q}}$$ -Fano varieties Grieve (Asian J Math 24(6):995–1005, 2020).

Maximal log Fano manifolds are generalized Bott towers

We prove that maximal log Fano manifolds are generalized Bott towers. As an application, we prove that in each dimension, there is a unique maximal snc Fano variety satisfying Friedman's d-semistability condition.

Very free rational curves in Fano varieties

Let X be a projective variety and let C be a rational normal curve on X. We compute the normal bundle of C in a general complete intersection of hypersurfaces of sufficiently large degree in X. As a result, we establish the separable rational connectedness of a large class of varieties, including general Fano complete intersections of hypersurfaces of degree at least three in flag varieties, in arbitrary characteristic. In addition, we give a new way of computing the normal bundle of certain rational curves in products of varieties in terms of their restricted tangent bundles and normal bundles on each factor.

Fano varieties with large Seshadri constants in positive characteristic

We prove that a Fano variety (with arbitrary singularities) of dimension n in positive characteristic is isomorphic to ℙ n if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than n and classify Fano varieties whose anti-canonical divisors have Seshadri constants n. In characteristic p>5 and dimension 3, we also show that Fano varieties X with Seshadri constants ϵ(-K X ,x)>2+ϵ at some smooth point x∈X (for some fixed ϵ>0) have bounded anti-canonical degrees.

The Anticanonical Complex for Non-degenerate Toric Complete Intersections

The anticanonical complex generalizes the Fano polytope from toric geometry and has been used to study Fano varieties with torus action so far. We provide an explicit description of the anticanonical complex for complete intersections in toric varieties defined by non-degenerate systems of Laurent polynomials. As an application, we classify the terminal Fano threefolds that are embedded into a fake weighted projective space via a general system of Laurent polynomials.

Vertices of the Harder and Narasimhan polygons and the laws of large numbers

AbstractWe build on the recent techniques of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894), which were used to establish theorems about semi-positivity of the Chow Mumford line bundles for families of $\mathrm {K}$ -semistable Fano varieties. Here, we apply the Central Limit Theorem to ascertain the asymptotic probabilistic nature of the vertices of the Harder and Narasimhan polygons. As an application of our main result, we use it to establish a filtered vector space analogue of the main technical result of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894). In doing so, we expand upon the slope stability theory, for filtered vector spaces, that was initiated by Faltings and Wüstholz (1994, Inventiones Mathematicae 116, 109–138). One source of inspiration for our abstract study of Harder and Narasimhan data, which is a concept that we define here, is the lattice reduction methods of Grayson (1984, Commentarii Mathematici Helvetic 59, 600–634). Another is the work of Faltings and Wüstholz (1994, Inventiones Mathematicae 116, 109–138), and Evertse and Ferretti (2013, Annals of Mathematics 177, 513–590), which is within the context of Diophantine approximation for projective varieties.

On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties

Firstly, we see that the bases of the miniversal deformations of isolated Q \mathbb {Q} -Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension ≤ 2 \leq 2 which are the bases of the miniversal deformations of isolated Q \mathbb {Q} -Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric 3 3 -fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension ≥ 3 \geq 3 . As a consequence, we prove that the number of local branches of the K-moduli stack of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano varieties is unbounded in each dimension ≥ 3 \geq 3 .

BIRATIONAL RIGIDITY OF ORBIFOLD DEGREE 2 DEL PEZZO FIBRATIONS

AbstractVarieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees$1$and$2$over the projective line with smooth total space satisfying the so-called$K^2$-condition are birationally rigid: their Mori fiber space structure is unique. This implies that they are not birational to any Fano varieties, conic bundles, or other del Pezzo fibrations. In particular, they are irrational. The families of del Pezzo fibrations with smooth total space of degree$2$are rather special, as formostfamilies a general del Pezzo fibration has the simplest orbifold singularities. We prove that orbifold del Pezzo fibrations of degree$2$over the projective line satisfying explicit generality conditions as well as a generalized$K^2$-condition are birationally rigid.