Analytic cycles and vector bundles on non-compact algebraic varieties
Analytic cycles and vector bundles on non-compact algebraic varieties
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DOI：
10.1007/bf01389905
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发表时间：
1975-02
影响因子：
3.1
通讯作者：
M. Cornalba;P. Griffiths
中科院分区：
文献类型：
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作者：
M. Cornalba;P. Griffiths
附录 3：求解依赖于参数的 c5 方程....................................... 99 w 0. 引言 a) 在本文中，我们将使用具有生长条件的函数论来研究一些关于代数簇的问题，这些问题具有解析解，但不一定是代数解。主要的此类问题源于这样的观察：在光滑仿射簇 A 上，上同调 H2k (A, Q) 由解析子簇 1 的基本类生成。这些子簇一般不能是代数的，我们的主要发现之一是子簇的超越级可能与其基本类的 Hodge 类型有关。我们能够证明这个的下界（定理 VII），并证明下界对于除数来说是尖锐的（定理 V）。然而，我们无法找到更高余维循环的良好上限，因此必须满足于陈述一些猜想，其解决方案将完成我们的程序（w167 26和27）。一旦我们从先验的角度开始研究循环，我们很快就得出这样的结论：仿射代数簇上的大量自然提出的几何问题承认解析解，但不一定有代数解。正确理解这些问题的本质解决方案，似乎有必要在这样的变量 z 上证明具有生长条件的 Oka 原理 Oka 原理的主要例子是 Grauert 定理 [4]，该定理指出在 Stein 流形 M 上的自然映射。
Appendix 3: Solving the c5-Equation with Dependence on Parameters................... 99 w 0. Introduction a) In this pap6r we shall use function theory with growth conditions to study some questions on algebraic varieties which have analytic, but not necessarily algebraic, solutions. The main such question arises from the observation that, on a smooth affine variety A, the cohomology H2k (A, Q) is generated by fundamental classes of analytic subvarieties 1. These subvarieties cannot in general be algebraic, and one of our main discoveries is that the transcendental level of the subvariety may be related to the Hodge type of its fundamental class. We are able to prove a lower bound for this (Theorem VII), and to show that the lower bound is sharp for divisors (Theorem V). However, we could not find good upper bounds for higher codimension cycles, and so must be content to state some conjectures whose solution would complete our program (w167 26 and 27).Once we began the study of cycles from a" transcendental point of view, we were soon led to the conclusion that an extremely large number of naturally posed geometric problems on affine algebraic varieties admit analytic but not necessarily algebraic solutions. To properly understand the nature of these solutions, it seems necessary to prove the Oka principle with growth conditions on such a variety z. The primary example of the Oka principle is the theorem of Grauert [4], which states that on a Stein manifold M the natural mapping