Fano Complete Intersections Research Articles

Canonical and log canonical thresholds of Fano complete intersections

It is proved that the global log canonical threshold of a Zariski general Fano complete intersection of index 1 and codimension k in {mathbb P}^{M+k} is equal to one, if Mgeqslant 2k+3 and the maximum of the degrees of defining equations is at least 8. This is an essential improvement of the previous results about log canonical thresholds of Fano complete intersections. As a corollary we obtain the existence of Kähler–Einstein metrics on generic Fano complete intersections described above.

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 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.

Mori dream spaces and birational rigidity of Fano 3-folds

We highlight a relation between the existence of Sarkisov links and the finite generation of (certain) Cox rings. We introduce explicit methods to use this relation in order to prove birational rigidity statements. To illustrate, we complete the birational rigidity results of Okada for Fano complete intersection 3-folds in singular weighted projective spaces.

Laurent phenomenon for Landau-Ginzburg models of complete intersections in Grassmannians

In 1997 Batyrev, Ciocan-Fontanine, Kim, and van Straten suggested a construction of Landau–Ginzburg models for Fano complete intersections in Grassmannians similar to Givental’s construction for complete intersections in smooth toric varieties. We show that for a Fano complete intersection in a Grassmannian the result of the above construction is birational to a complex torus. In other words, the complete intersections under consideration have very weak Landau–Ginzburg models.

On the global log canonical threshold of Fano complete intersections

We show that the global log canonical threshold of generic Fano complete intersections of index 1 and codimension k in \({\mathbb {P}}^{M+k}\) is equal to 1 if \(M\geqslant 3k+4\) and the highest degree of defining equations is at least 8. This improves the earlier result where the inequality \(M\geqslant 4k+1\) was required, so the class of Fano complete intersections covered by our theorem is considerably larger. The result implies, in particular, that the Fano complete intersections satisfying our assumptions admit a Kahler–Einstein metric. We also show the existence of Kahler–Einstein metrics for a new finite set of families of Fano complete intersections.

On Hodge Numbers of Complete Intersections and Landau–Ginzburg Models

We prove that the Hodge number $h^{1,N-1}(X)$ of an $N$-dimensional ($N\geqslant 3$) Fano complete intersection $X$ is less by one then the number of irreducible components of the central fiber of (any) Calabi--Yau compactification of Givental's Landau--Ginzburg model for $X$.

Very free curves on Fano complete intersections

In this paper, we show that general Fano complete intersections over an algebraically closed field of arbitrary characteristics are separably rationally connected. Our proof also implies that general log Fano complete intersections with smooth tame boundary divisors admit very free $A^1$-curves.

On the Space of Conics on Complete Intersections

We get sharp degree bound for generic smoothness and connectedness of the space of lines and conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on hypersurfaces. As a consequence, we prove that for a Fano complete intersection $$X$$ with index $$\ge 2$$ , the $$1$$ -Griffiths group generated by algebraic $$1$$ -cycles homologous to $$0$$ modulo algebraic equivalence is trivial, which is a conjecture for general rationally connected varieties.

One-cycles on rationally connected varieties

AbstractWe prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. Applying the same technique, we also show that the Chow group of 1-cycles on a separably rationally connected Fano complete intersection of index at least 2 is generated by lines. As a consequence, we give a positive answer to a question of Professor Totaro about integral Hodge classes on rationally connected 3-folds. And by a result of Professor Voisin, the general case is a consequence of the Tate conjecture for surfaces over finite fields.

Birational rigidity of Fano complete intersections

Birational rigidity of Fano complete intersections

Birationally rigid complete intersections of quadrics and cubics

We prove the birational superrigidity of generic Fano complete intersections of type in the projective space provided that and , and of certain families of Fano complete intersections of dimensions 10 and 11.

Birationally rigid Fano complete intersections. II

Abstract. We prove that a generic (in the sense of Zariski topology) Fano complete intersection V of the type ( d 1 , ⋯ , d k ) $(d_1,\dots ,d_k)$ in ℙ M + k ${\mathbb {P}}^{M+k}$ , where d 1 + ⋯ + d k = M + k $d_1+\dots +d_k=M+k$ , is birationally superrigid if M ≥ 7 $M\ge 7$ , M ≥ k + 3 $M\ge k+3$ and max { d i } ≥ 4 $\operatorname{max} \lbrace d_i\rbrace \ge 4$ . In particular, on the variety V there is exactly one structure of a Mori fibre space (or a rationally connected fibre space), the groups of birational and biregular self-maps coincide, Bir V = Aut V $\operatorname{Bir} V= \operatorname{Aut} V$ , and the variety V is non-rational. This fact covers a considerably larger range of complete intersections than our result of [J. reine angew. Math 541 (2001), 55–79], which required the condition M ≥ 2 k + 1 $M\ge 2k+1$ .

K-trivial structures on Fano complete intersections

It is proved that any fiber space structure into varieties of Kodaira dimension zero on a generic Fano complete intersection of index 1 and of dimension M in ℙ M+k is a pencil of hyperplane sections provided that M ≥ 2k + 1. The K-trivial structures on the varieties with a pencil of Fano complete intersections are described.

Hori-Vafa mirror models for complete intersections in weighted projective spaces and weak Landau-Ginzburg models

Abstract We prove that Hori-Vafa mirror models for smooth Fano complete intersections in weighted projective spaces admit an interpretation as Laurent polynomials.

Existence of the Kähler-Einstein metric on certain Fano complete intersections

For certain families of Fano complete intersections we prove an estimate from below for the global (log) canonical threshold, which implies the existence of Kahler-Einstein metric on generic varieties in these families.

Birational geometry of algebraic varieties with a pencil of Fano complete intersections

We prove birational superrigidity of generic Fano fiber spaces \({V/\mathbb {P}^{1}}\) , the fibers of which are Fano complete intersections of index 1 and dimension M in \({\mathbb {P}^{M+k}}\) , provided that M ≥ 2k + 1. The proof combines the traditional quadratic techniques of the method of maximal singularities with the linear techniques based on the connectedness principle of Shokurov and Kollár. Certain related results are also considered.

Generalized Tsen's theorem and rationally connected Fano fibrations

The existence is proved and an explicit algebraic description is given for sections of a fibration over a curve whose general fibre is a Fano complete intersection in a product of weighted projective spaces. It is proved also that a fibration whose general fibre is a smooth Fano threefold is rationally connected.

Birationally rigid Fano complete intersections

Birationally rigid Fano complete intersections

On the semi-simplicity of the quantum cohomology algebras of complete intersections

In this short note, we study the semi-simplicity of the quantum cohomology algebra H(V, C) of a Fano complete intersection V, and show that the subspace of H(V, C) generated by the hyperplane sections is semi-simple in the sense of Dubrovin when the degree of V satisfies certain conditions.