Heights on a subvariety of an abelian variety

Extending Ullmo-Zhang's result on the Bogomolov conjecture, we give conditions that a closed subvariety of an abelian variety A defined over a number field is isomorphic to an abelian variety in terms of the value distribution of a Neron-Tate height function on the subvariety. As a corollary of the result, we prove the Bogomolov conjecture which claims that if an irreducible curve X in A is not isomorphic to an elliptic curve, then for the pseudodistance defined by the Neron-Tate height, the distribution of algebraic points on X is uniformly discrete. These results can be extended in the case where base fields are finitely generated over via Moriwaki's height theory.     

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties.In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group G^Mis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ? 2. A Galois cover f :X ? Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pal]. In this paper we prove two results about abelian covers:first, that if the building data are sufficiently ample, then the natural deformations surject on the Kuranishi family of X; second, that if the building data are sufficiently ample and generic, then Aut(X)= G.     

Generalized Greatest Common Divisors, Divisibility Sequences, and Vojta’s Conjecture for Blowups

We apply Vojta’s conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta’s conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.     

Generalized Greatest Common Divisors,Divisibility Sequences,and Vojta’s Conjecture for Blowups

We apply Vojta’s conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta’s conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.     

Wild automorphisms of varieties with Kodaira dimension 0

An automorphism σ of a projective variety X is said to be wild if σ(Y) ≠ Y for every non-empty subvariety Y \subsetneq X{Y \subsetneq X} . In [1] Z. Reichstein, D. Rogalski, and J.J. Zhang conjectured that if X is an irreducible projective variety admitting a wild automorphism then X is an abelian variety, and proved this conjecture for dim(X) ≤ 2. As a step toward answering this conjecture in higher dimensions we prove a structure theorem for projective varieties of Kodaira dimension 0 admitting wild automorphisms. This essentially reduces the Kodaira dimension 0 case to a study of Calabi-Yau varieties, which we also investigate. In support of this conjecture, we show that there are no wild automorphisms of certain Calabi-Yau varieties.     

A spherical Bernstein theorem for minimal submanifolds of higher codimension

Combining the tools of geometric analysis with properties of Jordan angles and angle space distributions, we derive a spherical and a Euclidean Bernstein theorem for minimal submanifolds of arbitrary dimension and codimension, under the condition that the Gauss image is contained in some geometrically defined closed region of a Grassmannian manifold. The proof depends on the subharmoncity of an auxiliary function, the Codazzi equations and geometric measure theory.     

Petits points et conjecture de Bogomolov

This paper is devoted to the statement known as the Bogomolov conjecture on small points. We present the outline of Zhang's proof of the generalized version of the conjecture. An explicit bound for the height of a non-torsion variety of an abelian variety is obtained in the frame of Arakelov theory. Some further developments are mentioned.     

On some examples of obstructed irregular surfaces

We determine the base space of the Kuranishi family of some complete intersections in the product of an abelian variety and a projective space.As a consequence,we obtain new examples of obstructed irregular surfaces with ample canonical bundle and maximal Albanese dimension.     

On Hitchin's connection

The aim of the paper is to give an explicit expression for Hitchin's connection in the case of stable rank 2 bundles on genus 2 curves. Some general theory (in the algebraic geometric setting) concerning heat operators is developed. In particular the notion of compatibility of a heat operator with respect to a closed subvariety is introduced. This is used to compare the heat operator in the nonabelian rank 2 genus 2 case to the abelian heat operator (on theta functions) for abelian surfaces. This relation allows one to perform the computation; the resulting differential equations are similar to the Knizhnik-Zalmolodshikov equations.

     

Mumford's example and a general construction

We construct a line bundle on a complex projective manifold (a general ruled variety over a curve) which is not ample, but whose restriction to every proper subvariety is ample. This example is of interest in connection with ampleness questions of vector bundles on varieties of dimension greater than one. The method of construction shows that a stable bundle of positive degree on a curve is ample. The example can be used to show that there is no restriction theorem for Bogomolov stability.     

Equidistribution over function fields

We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For number fields, these results were proved by Yuan and we transfer here his methods to function fields. If X is a closed subvariety of an abelian variety, then we can describe the equidistribution measure explicitly in terms of convex geometry.     

The Bogomolov conjecture for totally degenerate abelian varieties

We prove the Bogomolov conjecture for a totally degenerate abelian variety A over a function field. We adapt Zhang’s proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place v. As a corollary, we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v.     

Minoration de la hauteur de Néron-Tate sur les surfaces abéliennes

This paper contains results concerning a conjecture made by Lang and Silverman, predicting a lower bound for the canonical height on abelian varieties of dimension 2 over number fields. The method used here is a local height decomposition. We derive as corollaries uniform bounds on the number of torsion points on families of abelian surfaces and on the number of rational points on families of genus 2 curves.     

等参函数及相关问题研究

著名的Yau 猜想断言单位球面中的紧致嵌入极小超曲面的Laplace 算子的第一特征值等于其维数. 近年来有许多几何学家致力于对Yau 猜想的研究, 但是到目前为止, 已有的结论只是一些关于第一特征值估计的不等式. 作为本文的一个主要结果, 本文证明了对于单位球面中的等参极小超曲面,Yau 猜想是正确的. 进一步地, 对于等参超曲面的焦流形(实际上是球面的极小子流形), 本文还证明了在一定维数条件下, 它的第一特征值也是其维数.
作为本文的第二个主要结果, 以著名的Schoen-Yau-Gromov-Lawson 的关于数量曲率的手术理论为出发点, 本文在一个Riemann 流形的嵌入超曲面处作手术, 构造了一个新的具有丰富几何性质的流形, 称为double 流形. 特别地, 本文在单位球面的极小等参超曲面处实行了这一手术, 发现得到的double 流形不仅有很复杂的拓扑(但其示性类有精确描述), 还存在数量曲率为正的度量, 更重要的是保持了等参叶状结构.
比Willmore 曲面更广泛的定义是Willmore 子流形, 即Willmore 泛函在球面中的的极值子流形.单位球面中的Willmore 子流形的例子在已有文献中是非常罕见的. 作为本文的另外两个主要结果, 通过深入挖掘单位球面上的OT-FKM- 型等参函数的焦流形的性质, 本文发现其极大值对应的焦流形是单位球面的一系列Willmore 子流形; 之后, 本文用几何办法统一证明了单位球面中具有4 个不同主曲率的等参超曲面的焦流形都是单位球面的Willmore 子流形. 这些新的Willmore 子流形是极小的,但一般不是Einstein 的.     

Minimal faithful representations of abelian Jordan algebras and Lie superalgebras

Over an algebraically closed field of characteristic zero, all the abelian subalgebras of the maximal dimension are classified for any special Jordan algebra. As a consequence, the minimal dimension of the faithful representations of any finite-dimensional abelian Jordan algebra is determined and the minimal faithful representations are classified for the so-called nice abelian Jordan algebras. The same is done for the purely odd Lie superalgebras.     

Visibility of Shafarevich-Tate Groups of Abelian Varieties

We investigate Mazur's notion of visibility of elements of Shafarevich-Tate groups of abelian varieties. We give a proof that every cohomology class is visible in a suitable abelian variety, discuss the visibility dimension, and describe a construction of visible elements of certain Shafarevich-Tate groups. This construction can be used to give some of the first evidence for the Birch and Swinnerton-Dyer conjecture for abelian varieties of large dimension. We then give examples of visible and invisible Shafarevich-Tate groups.     

On the B-canonical splittings of flag varieties

Let G be a semisimple algebraic group over an algebraically closed field of positive characteristic. In this note, we show that an irreducible closed subvariety of the flag variety of G is compatibly split by the unique canonical Frobenius splitting if and only if it is a Richardson variety, i.e. an intersection of a Schubert and an opposite Schubert variety.     

On height functions on Jacobian surfaces

The canonical height on an abelian variety is useful and important for the study of the Mordell-Weil group. But it is difficult to calculate the canonical height in general. We give an effective method to calculate the canonical height on a Jacobian surface. As an application, we verify the Birch-Swinnerton-Dyer conjecture for the Jacobian surface of a twisted modular curve. Received: 15 July 1996 / Revised version: 19 January 1997     

Minimal sets and varieties

The aim of this paper is twofold. First some machinery is established to reveal the structure of abelian congruences. Then we describe all minimal, locally finite, locally solvable varieties. For locally solvable varieties, this solves problems 9 and 10 of Hobby and McKenzie. We generalize part of this result by proving that all locally finite varieties generated by nilpotent algebras that have a trivial locally strongly solvable subvariety are congruence permutable.