对称张量与$\Bbb P^N$的子簇几何
Symmetric tensors and the geometry of subvarieties of $\Bbb P^N$
摘要：
本文用几何方法证明了扭对称微分空间$H^0(X,S^mΩ^1_X\otimes \Cal O_X(k))$与$k\le m$在子簇$X\subset \Bbb P^N$上的消失和非消失的新结果，并改进了旧结果. $k=m$的情况是特殊的，非零结果与包含$X$的二次曲面空间有关，并导致与$X$相关的有趣几何对象，例如$X$的所有相切三割线的变化。同样的技术给结果的对称微分的子品种的阿贝尔品种。本文最后给出了关于对称复属的投射簇$X_t$，$Q_m(X)= \dim H^0(X,S^mΩ^1_X)$，或$α$ -扭曲对称复属的投射簇$Q_{α,m}(X)= \dim H^0(X,S^m(Ω^1_X\otimes αK_X))$沿光滑簇跳跃的新结果和例子.
英文摘要：
This paper following a geometric approach proves new, and reproves old, vanishing and nonvanishing results on the space of twisted symmetric differentials, $H^0(X,S^mΩ^1_X\otimes \Cal O_X(k))$ with $k\le m$, on subvarieties $X\subset \Bbb P^N$. The case of $k=m$ is special and the nonvanishing results are related to the space of quadrics containing $X$ and lead to interesting geometrical objects associated to $X$, as for example the variety of all tangent trisecant lines of $X$. The same techniques give results on the symmetric differentials of subvarieties of abelian varieties. The paper ends with new results and examples about the jump along smooth families of projective varieties $X_t$ of the symmetric plurigenera, $Q_m(X)= \dim H^0(X,S^mΩ^1_X)$, or of the $α$-twisted symmetric plurigenera, $Q_{α,m}(X)= \dim H^0(X,S^m(Ω^1_X\otimes αK_X))$.
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