论文标题
来自线性部分的任意部分的hodge数量
Hodge Numbers of Arbitrary Sections from Linear Sections
论文作者
论文摘要
令$ y $为一个非常宽敞的线条束$π的倒数的投影子手：l^{ - 1} \ rightarrow b $，上面是投影歧管$ b $。 $ l^{ - 1} \ rightArrow b $的任何部分都是同构至$ b $，任何合理的平滑多个的hodge数字取决于该多节的度量$ d $，就像任何平滑的完整的多组的hodge数字是$ \ weft（d_ {1} $} $的任何平滑完整的多个多部分的完整完整交叉点。在本文中，递归公式是根据整数$ \ weft \ {d_ {1}，\ ldots，d_ {r} \ right \} $和线性段的hodge编号给出的hodge编号。递归是通过归纳对维度和程度进行的。它的证明依赖于渐近混合霍奇结构的理论。一个有趣的推论是，在这种情况下，Lefschetz超平面特性削弱了一个程度。也就是说，相对消失不会达到超平面段的中间程度，而仅比中度小的程度少于一度。作为一个应用程序，在附录中，我们计算了所有平滑的完整交叉点的封闭公式，用于$ \ dim b = 3 $。
Let $Y$ be a projective submanifold of the total space of the inverse of a very ample line bundle $π:L^{-1}\rightarrow B$ over a projective manifold $B$. Any section of $L^{-1}\rightarrow B$ is isomorphic to $B$ and the Hodge numbers of any proper smooth multisection are determined by the degree $d$ of that multi-section as are the Hodge numbers of any smooth complete intersection of multi-sections of degrees $\left(d_{1},\ldots,d_{r}\right)$. In this paper recursive formulae are given for those Hodge numbers in terms of the integers $\left\{ d_{1},\ldots,d_{r}\right\} $ and the Hodge numbers of the linear sections. The recursion proceeds by induction on dimension and degree. Its proof relies on the theory of asymptotic mixed Hodge structures. An interesting corollary is that the Lefschetz hyperplane property is weakened by one degree in this setting. That is, relative vanishing does not reach the middle degree of the hyperplane section but only to degree one less than the middle degree. As an application, in an Appendix, we calculate closed formulae for all Hodge numbers of all smooth complete intersections for the case $\dim B=3$.
