Classes de Chern des variétés uniréglées

非实性的一类单直纹簇

Non-solidity of uniruled varieties

本文研究维数至少为2的单直纹簇的非实性条件，为Abban-Okada关于Fano三维簇实性的猜想提供新证据，并构造了加权射影空间中高余维Fano三维簇的显式双有理链。

On varieties which are uniruled by lines

Using the $\sharp$-minimal model program of uniruled varieties we show that for any pair $(X, \H)$ consisting of a reduced and irreducible variety $X$ of dimension $k \geq 3$ and a globally generated big line bundle $\H$ on $X$ with $d:= \H^k$ and $n:= h^0(X, \H)-1$ such that $d<2(n-k)-4$, then $X$ is uniruled of $\H$-degree one, except if $(k,d,n)=(3,27,19)$ and a ${\sharp}$-minimal model of $(X, \H)$ is $(\PP^3,Ø_{\PP^3}(3))$. We also show that the bound is optimal for threefolds.

Finite groups of bimeromorphic selfmaps of non-uniruled Kähler threefolds

We prove the Jordan property for groups of bimeromorphic selfmaps of three-dimensional compact Kähler varieties of non-negative Kodaira dimension and positive irregularity.

On Nonvanishing for uniruled log canonical pairs

We prove the Nonvanishing conjecture for uniruled projective log canonical pairs of dimension $n$, assuming the Nonvanishing conjecture for smooth projective varieties in dimension $n-1$. We also show that the existence of good minimal models for non-uniruled projective klt pairs in dimension $n$ implies the existence of good minimal models for projective log canonical pairs in dimension $n$.

Uniruled varieties with split tangent bundle

Beauville asked if a compact Kähler manifold with split tangent bundle has a universal covering that is a product of manifolds. We use Mori theory and elementary results about holomorphic foliations to study this problem for projective uniruled varieties. In particular we obtain an affirmative answer for rationally connected varieties in any dimension and uniruled varieties in dimension 4.

Polarized parallel transport and uniruled divisors on deformations of generalized Kummer varieties

In this note we characterize polarized parallel transport operators on irreducible holomorphic symplectic varieties which are deformations of generalized Kummer varieties. We then apply such characterization to show the existence of ample uniruled divisors on these varieties and derive some interesting consequences on their Chow group of 0-cycles.

相对典范除子的伪有效性与正特征中的无理性覆盖

Pseudo-effectivity of the relative canonical divisor and uniruledness in positive characteristic

本文证明在正特征代数闭域上，若纤维化映射的通用纤维几何整且非无理性覆盖，则相对典范除子伪有效；证明基于有理曲线覆盖和上同调无理性条件。

Symplectic Criteria on Stratified Uniruledness of Affine Varieties and Applications

We develop criteria for affine varieties to admit uniruled subvarieties of certain dimensions. The measurements are from long exact sequences of versions of symplectic cohomology, which is a Hamiltonian Floer theory for some open symplectic manifolds including affine varieties. Symplectic cohomology is hard to compute, in general. However, certain vanishing and invariance properties of symplectic cohomology can be used to prove that our criteria for finding uniruled subvarieties hold in some cases. We provide applications of the criteria in birational geometry of log pairs in the direction of …

Polarized endomorphisms of uniruled varieties (with Appendix by Y. Fujimoto and N. Nakayama)

We show that polarized endomorphisms of rationally connected threefolds with at worst terminal singularities are equivariantly built up from those on Q-Fano threefolds, Gorenstein log del Pezzo surfaces and P^1. Similar results are obtained for polarized endomorphisms of uniruled threefolds and fourfolds. As a consequence, we show that every smooth Fano threefold with a polarized endomorphism of degree > 1, is rational.

三维实代数簇的拓扑

Topologie des variétés algébriques réelles de dimension 3

本文综述了实单有理三维簇分类的最新进展。

Birational geometry of defective varieties

Here we investigate the birational geometry of projective varieties of arbitrary dimension having defective higher secant varieties. We apply the classical tool of tangential projections and we determine natural conditions for uniruledness, rational connectivity, and rationality.

Comment on: Pseudo-effectivity of the relative canonical divisor ... by Zsolt Patakfalvi

We discuss an elementary approach to the existence of non-uniruled ramified covers of varieties.

The intermediate type of certain moduli spaces of curves

A well-known principle of Mumford asserts that all moduli spaces of curves are varieties of general type, except a finite number of cases that occur for relatively small genus, when these varieties tend to be uniruled. In all known cases, the transition from uniruledness to being of general type is quite sudden. For the first time, we show that naturally defined moduli spaces of curves can have intermediate birational type. We show that the moduli space of even spin curves of genus 8 is of Calabi-Yau type. Then we prove that the 41-dimensional moduli space M_{11, 11} of 11-pointed curves of ge…

Jordan groups, conic bundles and abelian varieties

A group $G$ is called Jordan if there is a positive integer $J=J_G$ such that every finite subgroup $\mathcal{B}$ of $G$ contains a commutative subgroup $\mathcal{A}\subset \mathcal{B}$ such that $\mathcal{A}$ is normal in $\mathcal{B}$ and the index $[\mathcal{B}:\mathcal{A}] \le J$ (V.L. Popov). In this paper we deal with Jordaness properties of the groups $Bir(X)$ of birational automorphisms of irreducible smooth projective varieties $X$ over an algebraically closed field of characteristic zero. It is known (Yu. Prokhorov - C. Shramov) that $Bir(X)$ is Jordan if $X$ is non-uniruled. On the …

具有伪有效典范除子的簇及其小平维数

Varieties with pseudoeffective canonical divisor and their Kodaira dimension

本文研究复数域上光滑射影簇，围绕Shigefumi Mori提出的“簇为单直纹的当且仅当其小平维数为负”的猜想展开探讨，属于代数几何领域的理论研究。

Algebraic geometry and projective differential geometry, Seoul National University concentrated lecture series, 1997

This is an expanded and updated version of a lecture series I gave at Seoul National University in September 1997. It is in some sense an update of the 1979 Griffiths and Harris paper with a similar title. I discuss: Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with degenerate Gauss images, When can a uniruled variety be smooth?, Dual varieties, Linear systems of bounded and constant rank, Secant and tangential varieties, Systems of quadrics with tangential defects, Recognizing uniruled varieties, Recognizing intersections of quadrics, Recogniz…

Rational curves on primitive symplectic varieties of OG6 singular type

We prove that any ample class on a primitive symplectic variety that is locally trivial deformation of O'Grady's singular 6 dimensional example is proportional to the first Chern class of a uniruled divisor. This result answers a question of Lehn, Mongardi and Pacienza, extending their result for primitive symplectic varieties of this deformation type.

光滑四次和五次曲面上的有理点计数

Counting rational points on non-uniruled surfaces

本文研究了光滑四次和五次曲面上有理点的计数问题，通过切割和法尔廷斯定理改进了已有界。

Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity

The prolongation g^{(k)} of a linear Lie algebra g \subset gl(V) plays an important role in the study of symmetries of G-structures. Cartan and Kobayashi-Nagano have given a complete classification of irreducible linear Lie algebras g \subset gl(V) with non-zero prolongations. If g is the Lie algebra aut(\hat{S}) of infinitesimal linear automorphisms of a projective variety S \subset \BP V, its prolongation g^{(k)} is related to the symmetries of cone structures, an important example of which is the variety of minimal rational tangents in the study of uniruled projective manifolds. From this p…

Families of rational curves on holomorphic symplectic varieties and applications to zero-cycles

We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of $K3^{[n]}$-type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed $n$, we show that there are only finitely many polarization types of holomorphic symplectic variety of $K3^{[n]}$-type that do not contain such a uniruled divisor. As an application we provide a generalization of a result due to Beauville-Voisin on the Chow group of 0-…

极小有理曲线邻域的几何

Geometry of Neighborhoods of Minimal Rational Curves

本文综述了单有理射影流形上极小有理曲线的芽等价问题，重点研究了当极小有理切线的相伴簇构成射影簇的等平凡族时，Zariski开子集上自然出现的G-结构及其与几何结构的微分几何的相互作用，并讨论了极小有理曲线芽等价的形式原理。

Calabi-Yau Threefolds and Moduli of Abelian Surfaces I

We describe birational models and decide the rationality/unirationality of moduli spaces AA_{d} (and AA^{lev}_{d}) of (1,d)-polarized abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi-Yau threefolds with Euler characteristic zero.

光滑射影簇上对称张量的代数

The algebra of symmetric tensors on smooth projective varieties

本文讨论光滑复射影簇上对称张量代数H^0(X, Sym*TX)，通过实例计算并给出其Krull维数精确界，同时提出对具伪有效切丛的非不可直纹射影流形的猜想刻画。

Quotients of Calabi-Yau varieties

Let $X$ be a complex Calabi-Yau variety, that is, a complex projective variety with canonical singularities whose canonical class is numerically trivial. Let $G$ be a finite group acting on $X$ and consider the quotient variety $X/G$. The aim of this paper is to determine the place of $X/G$ in the birational classification of varieties. That is, we determine the Kodaira dimension of $X/G$ and decide when it is uniruled or rationally connected. If $G$ acts without fixed points, then $κ(X/G)=κ(X)=0$; thus the interesting case is when $G$ has fixed points. We answer the above questions in terms o…

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.

具有良好极小模型的簇上的无处消失1-形式

Nowhere vanishing 1-forms on varieties admitting a good minimal model

本研究在良好极小模型存在的假设下，证明了非消失1-形式与阿贝尔簇上光滑态射相关的若干猜想，给出Calabi-Yau簇族的分解结果，并得到单直纹情形下相关簇的结构定理。

On the projective geometry of homogeneous spaces

We study the projective geometry of homogeneous varieties $X= G/P\subset P(V)$, where $G$ is a complex simple Lie group, $P$ is a maximal parabolic subgroup and $V$ is the minimal $G$-module associated to $P$. Our study began with the observation that Freudenthal's magic chart could be derived from Zak's theorem on Severi varieties and standard geometric constructions. Our attempt to understand this observation led us to discover further connections between projective geometry and representation theory. Among other things, we calculate the variety of tangent directions to lines on $X$ through …

Rational curves of minimal degree and characterizations of ${\mathbb P}^n$

In this paper we investigate complex uniruled varieties $X$ whose rational curves of minimal degree satisfy a special property. Namely, we assume that the tangent directions to such curves at a general point $x\in X$ form a linear subspace of $T_xX$. As an application of our main result, we give a unified geometric proof of Mori's, Wahl's, Campana-Peternell's and Andreatta-Wiśniewski's characterizations of ${\mathbb P}^n$.

On the uniformity of the Iitaka fibration

We study pluricanonical systems on smooth projective varieties of positive Kodaira dimension, following the approach of Hacon-McKernan, Takayama and Tsuji succesfully used in the case of varieties of general type. We prove a uniformity result for the Iitaka fibration of smooth projective varieties of positive Kodaira dimension, provided that the base of the Iitaka fibration is not uniruled, the variation of the fibration is maximal, and the generic fiber has a good minimal model.

具有大量微分形式的代数簇的丰度

Abundance for varieties with many differential forms

假设低维极小模型纲领成立，证明具有温和奇点且存在大量带 pluricanonical 丛系数自反微分形式的代数簇$X$满足丰度猜想，推出埃尔米特半正典范除子几乎半丰富等结论，且$K_X$数值维数为1时结果无条件成立，还处理 Calabi-Yau 簇上 nef 线丛半丰富性问题。

关于环面流形 Gromov 宽度的一个注记

A note on the Gromov width of toric manifolds

本文利用单直纹射影 Kähler 流形中极小曲线辛面积控制 Gromov 宽度的结果，将其应用于环面簇，得到用环面组合不变量表示的上界。

On smooth projective D-affine varieties

We show various properties of smooth projective D-affine varieties. In particular, any smooth projective D-affine variety is algebraically simply connected and its image under a fibration is D-affine. In characteristic zero such D-affine varieties are also uniruled. We also show that (apart from a few small characteristics) a smooth projective surface is D-affine if and only if it is isomorphic to either ${\mathbb P}^2$ or ${\mathbb P}^1\times {\mathbb P}^1$. In positive characteristic, a basic tool in the proof is a new generalization of Miyaoka's generic semipositivity theorem.

Foliations with vanishing Chern classes

In this paper we aim at the description of foliations having tangent sheaf $T\mathcal F$ with $c_1(T\mathcal F)=c_2(T\mathcal F)=0$ on non-uniruled projective manifolds. We prove that the universal covering of the ambient manifold splits as a product, and that the Zariski closure of a general leaf of $\mathcal F$ is an Abelian variety. It turns out that the analytic type of the Zariski closures of leaves may vary from leaf to leaf. We discuss how this variation is related to arithmetic properties of the tangent sheaf of the foliation.

Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents

We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety $Z$, a family of minimal rational curves with $Z$-isotrivial varieties of minimal rational tangents is locally equivalent to the flat model. We show that this is the case when $Z$ satisfies certa…

Picard数为1的光滑射影horospherical簇的刻画

Characterizations of smooth projective horospherical varieties of Picard number one

本文通过极小有理切向量簇的射影等价性，刻画了Picard数为1的光滑射影horospherical簇，并利用$W$-正规完全步延拓方法与李代数上同调证明了局部平坦性与全纯截面消失。

Canonical heights on hyper-Kähler varieties and the Kawaguchi-Silverman conjecture

The Kawaguchi--Silverman conjecture predicts that if $f\colon X \dashrightarrow X$ is a dominant rational-self map of a projective variety over $\overline{\mathbb{Q}}$, and $P$ is a $\overline{\mathbb{Q}}$-point of $X$ with Zariski-dense orbit, then the dynamical and arithmetic degrees of $f$ coincide: $λ_1(f) = α_f(P)$. We prove this conjecture in several higher-dimensional settings, including all endomorphisms of non-uniruled smooth projective threefolds with degree larger than $1$, and all endomorphisms of hyper-Kähler varieties in any dimension. In the latter case, we construct a canonical…

n-Dimensional Projective Varieties with the Action of an Abelian Group of Rank n-1

Let X be a normal projective variety of dimension n > 2 admitting the action of the group G := Z^{n-1} such that every non-trivial element of G is of positive entropy. We show: `X is not rationally connected' ==> `X is G-equivariant birational to the quotient of a complex torus' <==> `K_X + D is pseudo-effective for some G-periodic effective fractional divisor D.' See Main Theorem 2.5. To apply, one uses the above and fact: `the Kodaira dimension of X is at least 0' ==> `X is not uniruled' ==> `X is not rationally connected.' We may generalize the result to the case of solvable G as in Remark …

Irreducible components of module varieties: projective equations and rationality

We expand the existing arsenal of methods for exploring the irreducible components of the varieties $Rep(A,\bold d)$ which parametrize the representations with dimension vector $\bold d$ of a finite dimensional algebra $A$. To do so, we move back and forth between $Rep(A,\bold d)$ and a projective variety, $GRASS(A,\bold d)$, parametrizing the same set of isomorphism classes of modules. In particular, we show the irreducible components to be accessible in a highly compressed format within the projective setting. Our results include necessary and sufficient conditions for unirationality, smooth…

Zariski density of points with maximal arithmetic degree

Given a dominant rational self-map on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical degree. In this article, we show that there are densely many $\overline{\mathbb Q}$-rational points with maximal arithmetic degree (i.e. whose arithmetic degree is equal to the first dynamical degree) for self-morphisms on projective varieties. For unirational varieties and abelian varieties, we show that there are densely many rational points with maximal arithme…

Potential density of projective varieties having an int-amplified endomorphism

We consider the potential density of rational points on an algebraic variety defined over a number field $K$, i.e., the property that the set of rational points of $X$ becomes Zariski dense after a finite field extension of $K$. For a non-uniruled projective variety with an int-amplified endomorphism, we show that it always satisfies potential density. When a rationally connected variety admits an int-amplified endomorphism, we prove that there exists some rational curve with a Zariski dense forward orbit, assuming the Zariski dense orbit conjecture in lower dimensions. As an application, we p…