Families of rationally connected varieties

Curve Classes on Rationally Connected Varieties

We prove that every curve on a rationally connected variety is algebraically equivalent to a (non-effective) integral sum of rational curves.

Degenerations of rationally connected varieties and PAC fields

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.

Rationally connected varieties

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.

On the fundamental group scheme of rationally chain connected varieties

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.

A remark on rationally connected varieties and Mori dream spaces

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.

Weak approximation and rationally connected varieties over function fields of curves

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.

Rationally connected varieties and fundamental groups

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.

有理连通簇族中的零闭链

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。

Approximately rationally or elliptically connected varieties

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.

Rationally connected varieties over finite fields

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.

本原Fano簇的有理连通有理二重覆盖

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。

有理连通簇上的全纯李代数胚联络

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。

Rationally connected varieties over the maximally unramified extension of p-adic fields

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 $C_{2}$ …

One Cycles on Rationally Connected Varieties

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…

Rationally connected varieties over local fields

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.

Fundamental groups of rationally connected varieties

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.

Rationally isotropic exceptional projective homogeneous varieties are locally isotropic

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.

Families of rationally simply connected varieties over surfaces and torsors for semisimple groups

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.

Rationally Connected Varieties and Loop Spaces

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.

关于有理连通簇上 $1$-闭链的 Kato 猜想类比与正常基变换

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。

Triviality and Split of Vector Bundles on Rationally Connected Varieties

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.

Rationally elliptic toric varieties

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.

Rational points and zero-cycles on rationally connected varieties over number fields

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.

Zero cycles on rationally connected varieties over Laurent fields

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…

On rationally connected varieties over $C_1$ 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 $0$.

Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism

We prove Kawaguchi-Silverman conjecture for all surjective endomorphisms on every smooth rationally connected variety admitting an int-amplified endomorphism.

Rational points of rationally simply connected varieties over global function fields

A complex projective manifold is rationally connected, resp. rationally simply connected, if finite subsets are connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally connected. We prove that a projective scheme over a global function field with vanishing "elementary obstruction" has a rational point if it deforms to a rationally simply connected variety in characteristic 0. This gives new, uniform proofs over these fields of the Period-Index Theorem, the quasi-split case of Serre's "Conjecture II", and Lang's $C_2$ property.

有限交换群作用于有理连通三维流形 II：K3 类型群

Finite abelian groups acting on rationally connected threefolds II: groups of K3 type

本文研究有限交换群作用于三维有理连通流形的性质，重点分析K3类型群的分类及其与K3表面的关系。

具有理性椭圆空间性质的奇异代数簇族

Families of singular algebraic varieties that are rationally elliptic spaces

本文研究了具有有限理性同调秩和同调群秩之和的奇异代数簇族，证明了此类无限光滑理性椭圆 3 折叠族不存在。

有理单纯连通品种的紧近似

Tight approximation for rationally simply connected varieties

研究证明了从复平滑射影曲线开子集到有理单纯连通品种的全纯映射可被代数映射在紧开拓扑下近似，并推导出相关结论。

Rationally generated arcs

One distinguishing feature of rational curves is that they have algebraic parameterizations. Arc spaces are a way of describing approximations to parameterizations of all curves in some fixed space. Playing on these descriptions, this paper introduces rationally generated arcs, and applies them towards the description and enumeration of rational curves in algebraic varieties.

Rationally smooth Schubert varieties and inversion hyperplane arrangements

We show that an element $w$ of a finite Weyl group $W$ is rationally smooth if and only if the hyperplane arrangement $I$ associated to the inversion set of $w$ is inductively free, and the product $(d_1+1) \cdots (d_l+1)$ of the coexponents $d_1,\ldots,d_l$ is equal to the size of the Bruhat interval $[e,w]$, where $e$ is the identity in $W$. As part of the proof, we describe exactly when a rationally smooth element in a finite Weyl group has a chain Billey-Postnikov decomposition. For finite Coxeter groups, we show that chain Billey-Postnikov decompositions are connected with certain modular…

Pattern characterization of rationally smooth affine Schubert varieties of type A

Schubert varieties in finite dimensional flag manifolds G/P are a well-studied family of projective varieties indexed by elements of the corresponding Weyl group W. In particular, there are many tests for smoothness and rational smoothness of these varieties. One key result due to Lakshmibai-Sandhya is that in type A the smooth Schubert varieties are precisely those that are indexed by permutations that avoid the patterns 4231 and 3412. Recently, there has been a flurry of research related to the infinite dimensional analogs of flag manifolds corresponding with G being a Kac-Moody group and W …

Smooth rationally connected threefolds contain all smooth curves

We show that if X is a smooth rationally connected threefold and C is a smooth projective curve then C can be embedded in X. Furthermore, a version of this property characterises rationally connected varieties of dimension at least 3. We give some details about the toric case.

Free Curves on Varieties

We study various generalisations of rationally connected varieties, allowing the connecting curves to be of higher genus. The main focus will be on free curves $f:C\to X$ with large unobstructed deformation space as originally defined by Kollár, but we also give definitions and basic properties of varieties $X$ covered by a family of curves of a fixed genus $g$ so that through any two general points of $X$ there passes the image of a curve in the family. We prove that the existence of a free curve of genus $g \geq 1$ implies the variety is rationally connected in characteristic zero and initia…

Rationally simply connected varieties and pseudo algebraically closed fields

The cohomological dimension of a field is the largest degree with non-vanishing Galois cohomology. Serre's "Conjecture II" predicts that for every perfect field of cohomological dimension $2$, every torsor over the field for a semisimple, simply connected algebraic group is trivial. A field is perfect and "pseudo algebraically closed" (PAC) if every geometrically irreducible curve over the field has a rational point. These have cohomological dimension $1$. Every transcendence degree $1$ extension of such a field has cohomological degree $2$. We prove Serre's "Conjecture II" for such fields of …

Billey-Postnikov decompositions and the fibre bundle structure of Schubert varieties

A theorem of Ryan and Wolper states that a type A Schubert variety is smooth if and only if it is an iterated fibre bundle of Grassmannians. We extend this theorem to arbitrary finite type, showing that a Schubert variety in a generalized flag variety is rationally smooth if and only if it is an iterated fibre bundle of rationally smooth Grassmannian Schubert varieties. The proof depends on deep combinatorial results of Billey-Postnikov on Weyl groups. We determine all smooth and rationally smooth Grassmannian Schubert varieties, and give a new proof of Peterson's theorem that all simply-laced…

关于全局+正则簇的有理链连通性

On rational chain connectedness of globally +-regular varieties

本文研究了全局$+$-正则簇的有理链连通性，证明了其在特定条件下的连通性，并引入了强全局$+$-正则的概念。

Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras

For any $n\geq 3$, we prove that there exist equivalences between these apparently unrelated objects: irreducible $n$-dimensional non degenerate projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal scrolls and 3-covered by twisted cubic curves, up to projective equivalence; quadro-quadric Cremona transformations of $ \mathbb P^{n-1}$, up to linear equivalence; $n$-dimensional complex Jordan algebras of rank three, up to isotopy. We also provide some applications to the classification of particular classes of varieties in the class defined above and of quadro-quadric …

有理连通光滑射影簇的动力视角刚性

Rigidity of rationally connected smooth projective varieties from dynamical viewpoints

本文从动力系统视角研究了有理连通光滑射影簇的刚性，证明了其成为环面簇或$(\mathbb{P}^1)^{\times n}$的充要条件分别取决于其自同态的分支除子不变性与特征值性质。

有理椭圆空间与代数簇的同伦群和上同调群的秩

Ranks of homotopy and cohomology groups for rationally elliptic spaces and algebraic varieties

本文研究了有理椭圆空间与代数簇同伦群和上同调群的秩，证明了自乘积 Poincaré 多项式及混合 Hodge 结构生成函数值间的不等式，并引入稳定化阈值不变量探讨其与 Hilali 猜想的联系。

Fundamental group-scheme of some rationally connected fibrations

In this paper we describe the fundamental group-scheme of a proper variety fibered over an abelian variety with rationally connected fibers over an algebraically closed field. We use old and recent results for the Nori fundamental group-scheme, and of finite group-schemes in general to prove that the kernel of such a fibration is finite, and that the homotopy exact sequence holds in this case. As an application, we describe the fundamental group-schemes of certain proper varieties that are connected by curves or are related to them.

Rational connectedness of log $Q$-Fano varieties

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.

Hilali猜想与复代数簇

Hilali conjecture and complex algebraic varieties

本文围绕有理椭圆空间上的Hilali猜想，回顾基本性质并给出满足猜想的复代数簇例子，进而提出修正公式与新猜想，并讨论与混合Hodge多项式相关的模乘积不等式。

Billey-Postnikov 偏序集、有理光滑 Schubert 簇与 Poincaré 对偶

Billey-Postnikov posets, rationally smooth Schubert varieties, and Poincaré duality

本文发展了 Billey-Postnikov 分解理论，证明有限型中可由模式条件识别，并应用于有理光滑 Schubert 簇的结构常数三角性和 Poincaré 对偶，同时分类了有限型中允许广义 Lehmer 码的有理光滑 Bruhat 区间，并否定了无限型中的相关猜想。

允许极化伽罗瓦自同态的簇

Varieties Admitting Polarized Galois Endomorphisms

本文证明了允许内放大伽罗瓦自同态的光滑复射影簇的极大有理连通纤维化为环面纤维化，从而推出有理连通时该簇为环面簇，皮卡德数一时为射影空间，并验证了对数 Calabi-Yau 型猜想。

Rational smoothness of varieties of representations for quivers of Dynkin type

With the help of Lusztig's canonical basis, we study local intersection cohomology of the Zariski closures of orbits of representations of a quiver of type A, D or E. In particular, we characterize the rationally smooth orbits and prove that orbit closures are smooth if and only if they are rationally smooth. This provides an analogue of theorems of V. Deodhar, and J. Carrell and D. Peterson on Schubert varieties.

具有固定指数的有理连通对数 Calabi–Yau 对的双有理有界性

Birational boundedness of rationally connected log Calabi-Yau pairs with fixed index

本文研究固定维数的有理连通 klt 对数 Calabi–Yau 对，证明在反对数典范类的固定 Cartier 指数与 nef 条件下，相应射影簇的集合模 flops 有界。

Rationally cubic connected manifolds I: manifolds covered by lines

In this paper 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 such that there is a covering family of rational curves of H-degree one. Our main result is that the Picard number of these manifolds is at most three, and that, if equality holds, (X,H) has an adjuction theoretic scroll structure over a smooth variety.

具有 nef 反典范除子的射影簇的自同构群与齐次纤维化

Automorphism groups and Homogeneous fibrations on projective varieties with nef anticanonical divisors

本文研究具有 nef 反典范除子的代数簇的 Albanese 态射与极大有理链连通纤维化的齐次结构，构造自然群同态诱导 Aut^0 的 Chevalley 分解（相差等源），并类比 Nishi-Matsumura 定理，用现代语言重证之。

Weak Approximation for Isotrivial Families

We prove weak approximation for isotrivial families of rationally connected varieties defined over the function field of a smooth projective complex curve.

A local-global principle for weak approximation on varieties over function fields

We present a new perspective on the weak approximation conjecture of Hassett and Tschinkel: formal sections of a rationally connected fibration over a curve can be approximated to arbitrary order by regular sections. The new approach involves the study of how ideal sheaves pullback to Cartier divisors.

某些簇的局部-整体原则与整Tate猜想

Local-global principle and integral Tate conjecture for certain varieties

本文提出几何判据验证余维数一可分有理连通簇的整Tate猜想及零循环局部-整体原则的唯一Brauer-Manin阻碍，并证明了几何有理曲面与四次del Pezzo曲面的相关结论。

Rationally connected manifolds and semipositivity of the Ricci curvature

This work establishes a structure theorem for compact Kähler manifolds with semipositive anticanonical bundle. Up to finite étale cover, it is proved that such manifolds split holomorphically and isometrically as a product of Ricci flat varieties and of rationally connected manifolds. The proof is based on a characterization of rationally connected manifolds through the non existence of certain twisted contravariant tensor products of the tangent bundle, along with a generalized holonomy principle for pseudoeffective line bundles. A crucial ingredient for this is the characterization of unirul…