A Theorem Concerning the Differential Equations Satisfied by Normal Functions Associated to Algebraic Cycles
A Theorem Concerning the Differential Equations Satisfied by Normal Functions Associated to Algebraic Cycles
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DOI：
10.2307/2373941
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发表时间：
1979-02
影响因子：
1.7
通讯作者：
P. Griffiths
中科院分区：
文献类型：
--
作者：
P. Griffiths
本文对正规函数作了一般性的讨论，最终得到一个结果，它把由代数圈产生的正规函数表示为一类常微分方程的解，这类方程是由通过给定圈或与之同系的大次超曲面所参数化的。在第一节中，我们将非正式地讨论曲面上曲线的情形，其中技术机器是必要的。沿着这条路，我们给出了正规函数理论中主要经典结果的证明，并发现了关于由Lefschetz曲线束产生的Hodge丛的一些新信息。在第二节中，我们转向更高的维度。在讨论了正规函数的定义和基本性质之后，我们分析了产生于Lefschetz束中的中间Jacobian的余切空间的Hodge丛（c.f.（2.13 c）和（2.14 c）），然后应谴责的结果（c.f. (2.9)（对于语句）通过它们的Hodge型来表征正规函数的基本类。最后，在对Picard-Fuchs方程作了一些一般性的观察之后，我们给出并证明了我们的主要结果定理2.2，然后用关于构造代数圈问题的一些观察来结束本文。除非另有说明，同源性将与Z-系数和上同调与C-系数。希望其他符号和术语是标准的。
In this paper we shall give a general discussion of normal functions culminating in a result characterizing those normal functions arising from algebraic cycles as being solutions to a family of ordinary differential equations parametrized by the hypersurfaces of large degree passing through the given cycle or through one homologous to it. In Section 1 the theorem will be informally discussed for the case of curves on a surface where a minimum of technical machinery is necessary. Along the way we give proofs of the main classical results in the theory of normal functions and find some new information on the Hodge bundles arising from a Lefschetz pencil of curves. Then in Section 2 we turn to higher dimensions. Following a discussion of the definition and basic properties of normal functions we analyze the Hodge bundles arising from the cotangent spaces to the intermediate Jacobians in a Lefschetz pencil (c.f. (2.13c) and (2.14c)), and then shall reprove the result (c.f. (2.9) for the statement) characterizing the fundamental classes of normal functions by their Hodge type. Finally, after some general observations on Picard-Fuchs equations we formulate and prove our main result Theorem 2.2, and then conclude the paper with some observations concerning the problem of constructing algebraic cycles. Unless otherwise specified, homology will be with Z-coefflcients and cohomology with C-coefficients. Hopefully the other notations and terminology are standard.