Towards a generalization of Mumford’s Theorem
Towards a generalization of Mumford’s Theorem
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芒福德定理的推广
DOI：
10.1215/kjm/1250520265
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发表时间：
1989
影响因子：
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通讯作者：
James D. Lewis
中科院分区：
文献类型：
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作者：
James D. Lewis
在本文中，我们研究了芒福德定理，基于一个技术的塞维里，在确定的作用，经常2-形式的“大小”的周群的0-圈光滑，复杂，投影代数曲面（cf [6]）。设X B是n维光滑复射影代数簇，CH k（X）= CH（X）= k维代数圈的Chow群（codimn-k）模有理等价，且n d Ak（X）= Iv E CH k（X）1）代数等价于0}.对于（X的）有理上同调，有一个算术滤子，其结果与[2 ]中的相同。. .体现了[该方案] X“的深层算术性质。我们的目的是应用芒福德定理来展示过滤的分级片段与A5（X）的“大小”（k > O）之间的联系。根据下面的主要定理的陈述，当k = 0时，我们得到了n = 2时Mumford定理的结论，以及Roitman对n > 2的推广（见[8 ]），即对p > 2的h ' °（X）0 0蕴涵A0（X）“无穷维”.我们开始这篇论文的背景，回顾两个过滤（也见[2]）。Fak（i）= Hi 2 4（Y，Q）在Hi（X，Q）中的Gysin像，其中Y = T的去奇异化，其中Tc X是余维数q > k的子簇;（i）={Fi-1 i（X，C）} f116 X，Q）中的最大子Hodge结构.有著名的包含Fa（i）FL'（i），被证明是一个等式（[Grothendieck修正的] Hodge猜想），和相应的分次态射Tk，i：Grak（i）= Fak（i）/Fak+1（i）n（i）/Fr（i）= Grti（i），如果假设Hodge猜想为B e真，则可将其转化为同构。现在设V B e是一个smooth射影簇，j：W c 4 V是一个smooth超平面截面，并回忆弱Lefschetz定理，即j*：H i（W）内射（相应地.同构），对于i = dim W（分别i < dim W）。设（j*）1 B e是j * 的左逆（如[4]中所介绍的）。我们的主要假设是Lefschetz型的标准猜想：A（*）：（j*）是代数的，即由一个具有有理系数的n-代数圈导出。
In this paper, we examine a theorem of Mumford, based on a technique of Severi, on the role of regular 2-forms in determining the "size" of the Chow group of 0-cycles on a smooth, complex, projective algebraic surface (cf [6]). L et X b e a smooth, complex projective algebraic variety of dim ension n, CH k (X) = C H (X ) = Chow group of algebraic cycles of dimension k (codim n — k) modulo rational equivalence, a n d Ak ( X ) = Iv E CHk (X)11) is algebraically equivalent to zero}. O n rational cohomology (of X) there is equipped an arithmetic filtration, fo r w h ich to quo te i n [2 ], " . . . embodies deep arithmetic properties of [the scheme] X " . O ur purpose is to apply Mumford's theorem to exhibit a connection between graded pieces o f th is filtration and the "size" of A 5 (X ) for k > O. According to the statement of the main theorem below, when k = 0, we arrive at the conclusion of Mumford's theorem for the case n = 2, and Roitman's generalization (see [8 ]) fo r n > 2, namely h ' ° (X) 0 0 fo r som e p > 2 implies A 0 (X) "infinite dimensional". We begin with the setting of this paper by recalling two filtrations (see also [2]). Fak (i)=Gysin images of Hi 2 4 (Y, Q) in H i (X, Q), where Y = desingularization o f T a n d w here T c X i s a subvariety o f codimension q > k; ( i ) = largest subHodge structure in {F i-I i (X, C)} fl 16X , Q ). There is the well known inclusion Fa (i) FL' (i), conjectured to be an equality ([Grothendieck amended] Hodge conjecture), and corresponding graded morphisms Tk ,i : Grak ( i ) = F ak(i)/Fak+1(i) n (i) /Fr(i) = G rti (i), w hich aga in translate to isom orphism s if one assum es th e Hodge conjecture to b e t r u e . N ow le t V b e a sm ooth projective variety, j : W c4 V a sm ooth hyperplane section, and recall the weak Lefschetz theorem, namely j*: H i ( W ) injective (resp. isom orphism ) for i = dim W (resp. i < dim W ). Let( j*) 1 b e the left inverse to j * (as introduced in [4]). Our main assumption is a standard conjecture of Lefschetz type: A (* ): ( j* ) is algebraic, i.e. induced by a n algebraic cycle with rational coeffi-