Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties
Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties
复制标题
紧凑 PEL 型 Shimura 品种上扭转自守滑轮的消失定理
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
Junecue Suh
中科院分区：
文献类型：
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作者：
Kai;Junecue Suh
（1）在定义1.1中，括号内的说明“（如果\(L\neq\{0\}\)，那么\(r\)的值由\(g\)唯一确定。）”是不正确的，应该是“（这是一种符号的滥用，因为\(r\)并不总是由\(g\)确定。）”
（2）在引理1.20中，“\(W\in Rep_R(M_1)\)中的一个对象”应该是“一个对象\(W\in Rep_R(M_1)\)”。
（3）在2.1节的第二段中，“单的”应该是“不可分解的”，并且“每个投射\(O_1\)-模”应该更精确地表述为“每个有限生成投射\(O_1\)-模”。
（4）在2.4节的第三段中，“\(Spec(R_1)\)上的约化群概型\(G\)”应该是“\(Spec(R_1)\)上的约化群概型\(G_1\)”。
（5）在定义2.26和2.27之前的段落中，分布代数的作用是多余的。
（6）在定义2.29之后的两段中，我们应该只在\(\mathbb{Z}_{(p)}\)上定义分裂对象，而不是在\(\mathbb{Z}\)上，以避免说分裂（偶）正交群在\(2\)处是约化的。而且，再次说明，分布代数的作用是多余的。此外，“在可允许格中是极小的”应该更精确地表述为“在包含相同最高权向量的可允许格中是极小的”。
（7）在4.1节中，当定义和研究\(H_{dR}(A/M_{H,1})\)上的霍奇滤层时，所有“\(\Omega^{\bullet}_{A/S_1}\)”的实例都应该是“\(\Omega_{A_n/M_{H,1}}\)”。
（8）命题7.10中声称“如果\(\nu\)的系数\((k_{\tau})_{\tau\in\Upsilon}\)满足\(k_{\tau}+k_{\tau^{\circ}c}=0\)，那么\(W^{\nu}\)作为\(M_{H,1}\)上的线丛有平凡的张量平方”这一说法太强了。正确的说法是“如果\(\nu\)的系数\((k_{\tau})_{\tau\in\Upsilon}\)满足\(k_{\tau}+k_{\tau^{\circ}c}=0\)的条件，那么\(W^{\nu}\)在\(M_{H,1}\)的皮卡群中定义一个挠元”。最简单的证明是使用\(M_H\)的复纤维，为了方便读者，我们详细说明如下：
(1) In Def. 1.1, the parenthetical remark “(If L 6= {0}, then the value of r is uniquely determined by g.)” is incorrect and should be “(This is an abuse of notation, because r is not always determined by g.)” (2) In Lem. 1.20, “an object in W ∈ RepR(M1)” should be “an object W ∈ RepR(M1)”. (3) In the second paragraph of Section 2.1, “simple” should be “indecomposable”, and “every projective O1-module” should be more precisely “every finitely generated projective O1-module”. (4) In the third paragraph in Section 2.4, “reductive group scheme G over Spec(R1)” should be “reductive group scheme G1 over Spec(R1)”. (5) In the paragraphs preceding Def. 2.26 and 2.27, the action of the distribution algebras are redundant. (6) In the two paragraphes after Def. 2.29, we should only define the split objects over Z(p), but not Z, to avoid saying that split (even) orthogonal groups are reductive at 2. And, again, the action of distribution algebras are redundant. Moreover, “minimal among admissible lattices” should be more precisely only “minimal among the admissible lattices containing the same highest weight vector”. (7) In Section 4.1, when defining and studying the Hodge filtration on HdR(A /MH,1), all instances of “Ω • A/S1 ” should be “ΩAn/MH,1”. (8) The assertion in Prop. 7.10 that “W ν has trivial tensor square as a line bundle over MH,1 if its coefficients (kτ )τ∈Υ of ν satisfy kτ + kτ◦c = 0” is too strong. The correct assertion is that “W ν defines a torsion element in the Picard group of MH,1 if its coefficients (kτ )τ∈Υ of ν satisfy the condition that kτ + kτ◦c = 0”. The simplest proof is to use the complex fiber of MH, which we spell out as follows, for the convenience of the reader: