Igor Krylov
Varieties of Fano type are very well behaved with respect to MMP, and they are known to be rationally connected. We study the relation between classes of rationally connected varieties and varieties of Fano type. We give examples of rationally connected varieties of dimension $\geqslant 3$ which are not birationally equivalent to Fano type varieties, thereby answering the question of Cascini and Gongyo.
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We prove that a fibration X \to \Bbb P_1, the general fiber of which is a smooth Fano threefold, is rationally connected. The proof is based on a generalization of Tsen's classical theorem: a fibration X/C over a curve the general fiber of which is a Fano complete intersection in a product of weighted projective spaces has a section.
Rationally connected Galois covers of Fano-Mori fibre spaces
本文证明了在特定稳定性与扭转条件下，Fano-Mori 纤维空间上的有理连通 Galois 循环有理覆盖必从底拉回，并据此给出 $V$ 不含此类覆盖的判据。
Rationally connected rational double covers of primitive Fano varieties
本研究证明M≥5时P^{M+1}中M+1次Zariski一般超曲面不存在阿贝尔群的高次Galois有理连通覆盖，尤其不存在2次有理连通有理映射，该结论可推广至其他本原Fano簇并启发绝对刚性猜想。
We prove that a degeneration rationally connected varieties over a field of characteristic zero always contains a geometrically irreducible subvariety which is rationally connected.
Nonsingular projective varieties which are both convex and rationally connected are considered. We ask whether such varieties must be algebraic homogeneous spaces G/P. In case X is a complete intersection, an affirmative answer is obtained by an elementary argument.
We prove that a one-parameter family of rationally connected varieties (over an algebraically closed field of characteristic 0) always has a section.
We establish a conjecture of Mumford characterizing rationally connected complex projective manifolds in several cases.
We study for rationally connected varieties $X$ the group of degree 2 integral homology classes on $X$ modulo those which are algebraic. We show that the Tate conjecture for divisor classes on surfaces defined over finite fields implies that this group is trivial for any rationally connected variety.
In this short note we prove that if X is a separably rationally connected variety over an algebraically closed field of positive characteristic, then H^1(X, O_X)=0.
We prove that every curve on a rationally connected variety is algebraically equivalent to a (non-effective) integral sum of rational curves.
A perfect PAC field containing an algebraically closed field is known to be $C_1$, i.e., every degeneration of a Fano complete intersection has a point. We prove that also every degeneration of a separably rationally connected variety has a point.
We prove a structure theorem for non-isomorphic endomorphisms of weak Q-Fano threefolds, or more generally for threefolds with big anti-canonical divisor. Also provided is a criterion for a fibred rationally connected threefold to be rational. As a consequence, we show (without using the classification) that every smooth Fano threefold having a non-isomorphic surjective endomorphism is rational.
This is an expository paper on rationally connected varieties. The aim is to provide an introduction to the subject, as well as to discuss a recent result by T. Graber, J. Harris and J. Starr. The paper is based on the talk I gave at the 2004 Summer Research Conference on Algebraic Geometry at Snowbird.
Let $k$ be an algebraically closed field. Chambert-Loir proved that the étale fundamental group of a normal rationally chain connected variety over $k$ is finite. We prove that the fundamental group scheme of a normal rationally chain connected variety over $k$ is finite and étale. In particular, the fundamental group scheme of a Fano variety is finite and étale.
In this short note, we show that a construction by Ottem provides an example of a rationally connected variety that is not birationally equivalent to a Mori dream space. This answers in the negative (at least in the category of terminal varieties) a question posed by Krylov.
This is a survey of weak approximation over complex function fields, touching on the Koll'ar-Miyaoka-Mori theorem, places of good and bad reduction, the special case of rational surfaces, rationally simply connected varieties, and applications to low-degree complete intersections.
Let U be an open subset of a unirational variety (or more generally of a separably rationally connected variety). We prove that there is rational curve C in U such that the fundamental group of C surjects onto the fundamental group of U. The proof improves the earlier proof (math.AG/0003138) and works in any characteristic. Much improved version, thanks to comments of Colliot-Thélène.
Zero-cycles in families of rationally connected varieties
该研究聚焦有理连通簇族中的零闭链，证明亨泽尔离散赋值环上光滑射影概形的特殊纤维可分有理连通时，相对零闭链到特殊纤维的限制诱导周群同构，还推广到高阶周群并提出非光滑情形猜想，推广了Kollár的相关结果。
We discuss a possible approach to the study of the vanishing of the Kobayashi pseudometric of a projective variety X, using chains of rational or elliptic curves contained in an arbitrarily small neighborhood of X in projective space for the Euclidean topology.
Let X be a geometrically rational (or more generally, separably rationally connected) variety over a finite field K. We prove that if K is large enough then X contains many rational curves defined over K. As a consequence we prove that R-equivalence is trivial on X if K is large enough. These imply that if Y is defined over a local field and it has good, separably rationally connected reduction then the Chow group of zero cycles is trivial for any residue field. R-equivalence is also trivial if the residue field is large enough.
Holomorphic Lie algebroid connections over rationally connected varieties
研究有理连通光滑复射影簇上的全纯李代数胚联络，证明在特定条件下向量丛存在联络当且仅当其平凡，且所有联络平坦。
A result of Graber, Harris, and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a finite field or the function field of a curve over any algebraically closed field will have a rational point. Here we show that rationally connected varieties over the maximally unramified extension of the p-adics usually, in a precise sense, have rational points. This result is in the spirit of Ax and Kochen's result saying that the p-adics are usually $C_{2}$ …
All curves on a separably rationally connected variety are rationally equivalent to a (non-effective) integral sum of rational curves, hence the first Chow group is generated by rational curves. Applying the same techniques, we also proved that the first Chow group of all separably rationally connected Fano complete intersections with index at least 2 is generated by lines. As a consequence, a question of Professor Burt Totaro about integral Hodge classess on rationally connected 3-folds is solved, and positive answer to the question for general n-fold due to Professor János Kollár will follow…
In this short note we prove that in many cases the failure of a variety to be separably rationally connected is caused by the instability of the tangent sheaf (if there are no other obvious reasons). A simple application of the results proves that a smooth Fano complete intersection is separably rationally connected if and only if it is separably uniruled. In particular, a general such Fano complete intersection is separably rationally connected.
Let X be a smooth, projective variety defined over a local field K. Following Manin, two K-points of X are called R-equivalent if they can be joined by a rational curve defined over K. The main result of this note shows that if there are only finitely many R-equivalence classes over the algebraic closure of K then the same holds over K. This also yields the unirationality of several classes of varieties over K.
Let U be an open subset of a unirational variety. We prove that there is rational curve C in U such that the fundamental group of C surjects onto the fundamental group of U. As a consequence we obtain new proofs of the theorems of Harbater and Colliot-Thélène on Galois covers and torsors over the p-adic line. We also obtain examples of pencils of curves of genus <14 whose monodromy group is the full Teichmüller group.
Assume that R is a local regular ring containing an infinite perfect field, or that R is the local ring of a point on a smooth scheme over an infinite field. Let K be the field of fractions of R and the characteristic of K is not 2. Let X be an exceptional projective homogeneous scheme over R. We prove that in most cases the condition that X has a K-point implies that X has an R-point.
Under suitable hypotheses, we prove that a form of a projective homogeneous variety $G/P$ defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of simple connectedness replacing the unit interval by the projective line. As a consequence, we complete the proof of Serre's Conjecture II in Galois cohomology for function fields over an algebraically closed field.
We study smooth complex projective polarized varieties $(X,H)$ of dimension $ n \ge 2$ which admit a dominating family $V$ of rational curves of $H$-degree $3$, such that two general points of $X$ may be joined by a curve parametrized by $V$ and which do not admit a covering family of lines (i.e. rational curves of $H$-degree one). We prove that such manifolds are obtained from RCC manifolds of Picard number one by blow-ups along smooth centers. If we further assume that $X$ is a Fano manifold, we obtain a stronger result, classifying all Fano RCC manifolds of Picard number $ρ_X \ge 3$
We study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine Fano varieties.
We prove a more general and precise version of the Noether-Fano inequalities for birational maps between Mori fiber spaces. This is applied to give descriptions of global canonical thresholds on Fano varieties of Picard number one.
On analogues of the Kato conjectures and proper base change for $1$-cycles on rationally connected varieties
本文针对代数闭域上的光滑射影有理连通簇及其在 henselian 离散赋值环上的族，发展了关于 $1$-闭链的 Kato 猜想类比框架，并基于近期结果证明了部分特殊情形。
In this paper, we give a simple proof of a triviality criterion due to I.Biswas and J.Pedro and P.Dos Santos. We also prove a vector bundle on a homogenous space is trivial if and only if the restrictions of the vector bundle to Schubert lines are trivial. Using this result and Chern classes of vector bundles, we give a general criterion of a uniform vector bundle on a homogenous space to be splitting. As an application, we prove a uniform vector bundle on classical Grassmannians and quadrics of low rank is splitting.
We consider rationally connected complex projective manifolds M and show that their loop spaces--infinite dimensional complex manifolds--have properties similar to those of M. Furthermore, we give a finite dimensional application concerning holomorphic vector bundles over rationally connected complex projective manifolds.
This paper is a survey about cylinders in Fano varieties and related problems.
Fano varieties with conjecturally largest Fano index
本文构造了具有典范和末端奇点的 Fano 簇中大 Fano 指标的示例，并基于低维证据猜想这些示例在所有维度上达到最大 Fano 指标。
In this paper, we give an affirmative answer to a conjecture in the Minimal Model Program. We prove that log $Q$-Fano varieties of dim $n$ are rationally connected. We also study the behavior of the canonical bundles under projective morphisms.
We overview some recent results on Fano varieties giving evidence of their rigid nature under small deformations.
We give sufficient conditions for the semisimplicity of quantum cohomology of Fano varieties of Picard rank 1. We apply these techniques to prove new semisimplicity results for some Fano varieties of Picard rank 1 and large index. We also give examples of Fano varieties having a non semisimple small quantum cohomology but a semisimple big quantum cohomology.
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.
We prove a conjecture of L.Bonavero, C. Casagrande, O. Debarre and S. Druel, on the pseudo-index of smooth Fano varieties, in the special case of horospherical varieties.
The symmetric projective varieties of rank one are all smooth and Fano by a classic result of Akhiezer. We classify the locally factorial (respectively smooth) projective symmetric $G$-varieties of rank 2 which are Fano. When $G$ is semisimple we classify also the locally factorial (respectively smooth) projective symmetric $G$-varieties of rank 2 which are only quasi-Fano. Moreover, we classify the Fano symmetric $G$-varieties of rank 3 obtainable from a wonderful variety by a sequence of blow-ups along $G$-stable varieties. Finally, we classify the Fano symmetric varieties of arbitrary rank …
We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach.
Using deformation theory of rational curves, we prove a conjecture of Sommese on the extendability of morphisms from ample subvarieties when the morphism is a smooth (or mildly singular) fibration with rationally connected fibers. We apply this result in the context of Fano fibrations and prove a classification theorem for projective bundle and quadric fibration structures on ample subvarieties.
A characterization of Fano type varieties
本文通过证明Fano型变种的特征化，揭示了其几何结构的核心属性。
We give examples of Fano varieties $X$ with Picard number 1, which have terminal singularities and admit endomorphisms with degree larger than 1.
We prove the birational boundedness of Q-Fano varieties with Picard number one and dimension $n$ for every $n$. We do not use Mori theory.
We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described.
We prove divisorial canonicity of Fano hypersurfaces and double spaces of general position with elementary singularities.
We report on progress in the qualitative study of rational points on rationally connected varieties over number fields, also examining integral points, zero-cycles, and non-rationally connected varieties. One of the main objectives is to highlight and explain the many recent interactions with analytic number theory.
In this paper, we obtain a complete classification of smooth toric Fano varieties equipped with extremal contractions which contract divisors to curves for any dimension. As an application, we obtain a complete classification of smooth projective toric varieties which can be equivariantly blown-up to Fano along curves.
In his work on birational classification of foliations on projective surfaces, Brunella showed that every regular foliation on a rational surface is algebraically integrable with rational leaves. This led Touzet to conjecture that every regular foliation on a rationally connected manifold is algebraically integrable with rationally connected leaves. Druel proved this conjecture for the case of weak Fano manifolds. In this paper, we extend this result showing that Touzet's conjecture is true for codimension one foliations on threefolds with nef anti-canonical bundle.
We give a simple necessary and sufficient condition for uniform K-stability of $\mathbb{Q}$-Fano varieties.
In this article we give the definitions of log Fano varieties and log Calabi-Yau varieties in the framework of theory of log schemes of Fontain-Illusie-Kato and give congruences of the cardinalities of rational points of them over the log points of finite fields.
We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci.
Generalizing the notions of reflexive polytopes and nef-partitions of Batyrev and Borisov, we propose a mirror symmetry construction for Calabi-Yau complete intersections in Fano toric varieties.
Fano varieties are 'atomic pieces' of algebraic varieties, the shapes that can be defined by polynomial equations. We describe the role of computation and database methods in the construction and classification of Fano varieties, with an emphasis on three-dimensional Fano varieties with mild singularities called Q-Fano threefolds. The classification of Q-Fano threefolds has been open for several decades, but there has been significant recent progress. These advances combine computational algebraic geometry and large-scale data analysis with new ideas that originated in theoretical physics.
In this paper we classify mildly singular Fano varieties with maximal Picard number whose effective divisors are numerically eventually free.
We observe a method for finding weak Landau-Ginzburg models for Fano varieties and find them for smooth Fano threefolds of genera 9, 10, and 12.
We study zero cycles on rationally connected varieties defined over characteristic zero Laurent fields with algebraically closed residue fields. We show that the degree map induces an isomorphism for rationally connected threefolds defined over such fields. In general, the degree map is an isomorphism if rationally connected varieties defined over algebraically closed fields of characteristic zero satisfy the integral Hodge/Tate conjecture for one cycles, or if the Tate conjecture is true for divisor classes on surfaces defined over finite fields. To prove these results, we introduce technique…
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