射影簇和向量丛的数字不变量对超二次曲面的刻画

Characterization of hyperquadrics by the numerical invariant of projective varieties and vector bundles

设X是光滑的n维射影簇,E是X上的丰富向量丛,E的秩r<n.如果E在X上的数字有效值为nr,且X的皮卡数1,则X是超二次曲面Qn,E是线丛OQn(1)的直和.

Let X be a smooth projective variety of dimension n and E an ample vector bundle with rank r 〈 n. Assume that numerically effective value of E over X is n and that the Picard number of X is r one, then X is a hyperquadric Q^n and E is a direct sum of line bundles OQ^n （ 1 ）.