On the de rham cohomology of algebraic varieties
On the de rham cohomology of algebraic varieties
复制标题
DOI：
10.1007/bf02684298
复制
发表时间：
1975-12
期刊：
影响因子：
--
通讯作者：
R. Hartshorne
中科院分区：
文献类型：
--
作者：
R. Hartshorne
使用微分形式和积分来定义代数簇的数值不变量的想法可以追溯到Picard和Lefschetz。最近，Atiyah和Hodge [2]和Grothendieck [16]证明了代数微分形式可以用来计算C上光滑格式的奇异上同调。这个代数拓扑比较定理已由Lieberman和Herrera [29]和Deligne（未发表）推广到包括C上奇异格式的情况。Lieberman和Herrera还证明了一阶微分算子的对偶定理，其中我们的定理（II. 5.（1）是一个特殊的情况。另一个比较定理参见Lieberman [49]。在另一个上下文中，代数微分形式被证明在研究一族复簇的单值性时是有用的，使用高斯-马宁联络。[34]见KatzandOda [34]，Katz [32]和[33]，Deligne [10]和Brieskorn [8]。在纯解析的背景下，Reiffen [44]和Bloom和Herrera [5]研究了奇异解析空间上的全纯微分。最后，在研究特征p中的簇时，代数微分形式在Monsky的形式上同调[38]中很重要，并且是Grothendieck的结晶上同调的动机（[18]和[4]）。
The idea of using differential forms and their integrals to define numerical invariants of algebraic varieties goes back to Picard and Lefschetz. More recently, Atiyah and Hodge [2] and Grothendieck [16] showed that algebraic differential forms could be used to calculate the singular cohomology of a smooth scheme over C. This algebraictopological comparison theorem has been generalized by Lieberman and Herrera [29] and by Deligne (unpublished) to include the case of singular schemes over C. Lieberman and Herrera also proved a duality theorem for first order differential operators, of which our Theorem (II. 5. i) is a special case. See also Lieberman [49] for another comparison theorem.In another context, algebraic differential forms have proved to be useful in the study of the monodromy of a family of complex varieties, using the Gauss-Manin connection. SeeKatzandOda [34], Katz ([32] and [33]), Deligne [10], and Brieskorn [8]. In the purely analytic context, holomorphic differentials on a singular analytic space have been studied by Reiffen [44] and Bloom and Herrera [5]. Finally in the study of varieties in characteristic p, algebraic differential forms are important in Monsky's formal cohomology [38], and as motivation for Grothendieck's crystalline cohomology ([18] and [4]).