A hypersurface defect relation for a class of meromorphic maps
A hypersurface defect relation for a class of meromorphic maps
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一类亚纯贴图的超曲面缺陷关系
DOI：
10.1090/s0002-9947-1982-0642329-1
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发表时间：
1982
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影响因子：
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通讯作者：
A. Biancofiore
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文献类型：
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作者：
A. Biancofiore
让D . Dq是P中的p次超曲面，具有法向交叉。本文证明了对一类亚纯映射f：Cm P，，有一个亏关系3f（DI）+ . ?-+ 8 f（Dq）<（n +1）/p由Ph. Griffiths和B证明。希夫曼导论.设f：CtmPn是一个亚纯映射.令D，…，Dq是Pn中的p次超曲面，使得f（Cm）D，j = 1，.，q.定义了f对D的Nevanlinna亏Sf（DJ），其中j = 1，. . .，q.当亏损关系q（1）E8（DJ）？（n + l）/p j=1保持？我们将对这个问题提供一个部分的答案。如果p = 1，这是经典的亏损关系（Nevanlinna [8]，H. [12][13][14][15][16][17][18][19][这里我们关注的是第二个案例。Carlson-Griffiths-King理论[3，6]暗示（1）如果DI，. . . Dq有正规交叉，如果m2 n = rank f.希望这个秩条件可以被一个更自然的假设所取代，这个假设允许m < n。P. Griffiths [5]证明了（1）成立，如果f的象不包含在任何p次超曲面中，且如果DI . Dq有正常的交叉口。我们给出一个反例（？5）。如果f（Cm）不包含在任何超曲面中，则猜想（Shiffman [9]）仍然没有解决，即使m = 1，尽管进行了许多尝试。1979年B。Shiffman [10]研究了一类特殊的有限阶亚纯映射。他认为D1...，Dq的不同的p次超曲面使得Pn的任何点都不包含在n + 1个不同的Dj中.如果f E（和f（Cm）δ t DJ，对于j = 1. ., q他表明q：E Sf（Dj）< 2n。j=1在这些假设下，2n是最好的可能界限。我们对更一般的9 P亚纯映射类证明了（1），但自然地，我们对因子施加了更严格的正规交叉条件。我们的地图由编辑1980年5月12日收到。论文发表于784 th A.M.S. 1981年3月20日至21日，巴黎圣母院会议。1980年数学学科分类小学32A 22、32 H25、32 H30;中学30 D35。
Let D . Dq be hypersurfaces of degree p in P,, with normal crossings. We prove for a certain class of meromorphic maps f: Cm P,, a defect relation 3f (DI) + . ?-+8f (Dq) < (n + l)/p conjectured by Ph. Griffiths and B. Shiffman. Introduction. Let f: Ctm Pn be a meromorphic map. Let D,..., Dq be hypersurfaces of degree p in Pn such that f(Cm) D for j = 1,... , q. The Nevanlinna defect Sf (DJ) of f for D is defined forj = 1,. . . , q. When does the defect relation q (1) E 8(DJ) ? (n + l)/p j=1 hold? We shall provide a partial answer to this question. If p = 1, this is the classical defect relation (Nevanlinna [8], H. Cartan [4], Ahlfors [1], Weyl [13], Stoll [11], Vitter [12]). Here we are concerned with the casep 2 2. The Carlson-Griffiths-King theory [3, 6] implies (1) if DI,. . . Dq have normal crossings and if m 2 n = rank f. Hopefully this rank condition can be replaced by a more natural assumption which permits m < n. P. Griffiths [5] conjectured that (1) holds if the image of f is not contained in any hypersurface of degree p and if DI . Dq have normal crossings. We provide a counterexample (?5). If f(Cm) is not contained in any hypersurface, the conjecture (Shiffman [9]) remains unresolved, even if m = 1, despite many attempts. In 1979 B. Shiffman [10] investigated a particular class (5 of meromorphic maps of finite order. He considers D1,... , Dq distinct hypersurfaces of degree p such that no point of Pn is contained in n + 1 distinct Dj. If f E ( and f(Cm) 5t DJ for j = 1 ... ., q he shows q :E Sf(Dj) < 2n. j=1 Under these assumptions, 2n is the best possible bound. We prove (1) for a more general class 9P of meromorphic maps, but, naturally we impose the stricter condition of normal crossings on the divisors. The maps of our Received by the editors May 12, 1980. Paper presented at the 784th A.M.S. Meeting, Notre Dame, March 20-21, 1981. 1980 Mathematics Subject Classification. Primary 32A22, 32H25, 32H30; Secondary 30D35.