Study on toric varieties, vector bundles on them and their subvarieties

已知，对于维数为n的射影环变上的一个充足的线束，其被自身扭转n-1次的整体截面定义了环变在射影空间中的嵌入。1999年，我们证明了当线束被扭转n次时，由二次曲面产生了定义嵌入环变的理想。摘要利用计算机研究了3维凸多边形及其上的点阵点，证明了由n-1次扭曲的充裕线束嵌入的环面变体是有限阿贝尔群在射影空间中的商时，其理想是由二次曲面生成的。在2000年，我们证明了定义由n-1倍扭曲的充足线束嵌入的n维的环向变异的理想一般是由二次曲面生成的。我们研究了Futaki不变量，它是对复形法诺流形切束上爱因斯坦卡勒度规存在性的一个抽象。在1999年和2000年，我们将其推广为Bando-Calabi-Futaki特征，这是对具有常标量曲率的Kaehler度量存在性的一种抽象，并且我们证明了Bando-Calabi-Futaki特征也是对几何不变理论中代数变量的半稳定性的一种抽象。我们还研究了阿贝尔曲面在4维非奇异环面变形中的嵌入问题。我们证明了只有椭圆曲线的乘积可以嵌入到维数为2的环向曲线的乘积中。

It is known that for an ample line bundle on a projective toric variety of dimension n the global sections of the line bundle twisted by itself over n-1 times defines an embedding of the toric variety into a projective space. In 1999 we proved that the ideal defining the embedded toric variety is generated by quadrics when the line bundle is twisted over n times. Ample line bundles on toric varieties are heavily related with convex polytopes, we investigated the convex polytopes of dimension 3 and lattice points in them by using computors, and we showed that the ideals of the toric varieties embedded by n-1 times twisted ample line bundles are generated by quadrics when the varieties are quotients of projective space by finite abelian groups.In 2000 we proved that the ideals defining the toric varieties of dimension n embedded by n-1 times twisted ample line bundles are generated by quadrics in general.We have studied the Futaki invariant which is an obstraction to the existence of an Einstein Kaehler metric on the tangent bundle over toric Fano manifold. In 1999 and 2000 we generalized it as the Bando-Calabi-Futaki character which is an obstraction to the existance of a Kaehler metric with constant sscalar curvature, and we showed that the Bando-Calabi-Futaki character is also an obstraction to semistability of algebraic varieties in the geometric invariant theory.We also studied the embedding problem of abelian surfaces into nonsingular toric varieties of dimension 4. We showed that only the product of elliptic curves can be embedded into the product of toric varieties of dimension 2.