Determinant of period integrals
Determinant of period integrals
复制标题
周期积分的行列式
DOI：
10.1090/s0894-0347-97-00243-9
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发表时间：
1997
影响因子：
3.9
通讯作者：
T. Terasoma
中科院分区：
文献类型：
--
作者：
Takeshi Saito;T. Terasoma
证明了周期积分的一个行列式公式。周期积分是由Betti上同调和de Rham上同调的比较而产生的。我们在第4节中的公式定理1将周期积分的行列式表示为在相对标准圈处求出的周期与Γ-函数的特殊值的乘积。该公式是[S ~ 2]中定理1的Hodge形式。同时给出了一个动机公式，即第5节定理2。它特别意味着与代数Hecke特征标相关的秩为1的动机范畴在取上同调行列式下是封闭的。因此，它支持了Deligne的猜想，猜想8.1 iii [D4];秩为1的动机与代数Hecke特征有关。我们公式的一个典型例子是费马超曲面的周期是Γ-函数的特殊值的乘积（参见：引理5.4）。X = P的主要定理是第二个署名作者[T]定理1.2的重新表述。定理证明减少到这种情况下，通过归纳的维数使用莱夫谢茨铅笔。我们描述了结果。设U是C的一个子域k上的光滑拟投射簇，F是C的另一子域。本文的主要内容是对三元组M =（（E，λ），V，ρ）定义了周期的行列式：（1）E是U上的局部自由层，其可积连接是沿边界沿着正则奇异的。它产生了一个本地系统
We prove a formula for the determinant of period integrals. Period integrals arise from comparison between Betti cohomologies and de Rham cohomologies. Our formula Theorem 1 in Section 4 expresses the determinant of period integrals as the product of the periods evaluated at the relative canonical cycles and special values of the Γ-function. The formula is a Hodge version of Theorem 1 of [S2] for `-adic cohomology. Together with this, it gives a motivic formula, Theorem 2 in Section 5. It particularly implies that the category of motives of rank 1 associated to an algebraic Hecke character is closed under taking the determinant of cohomology. Hence it gives a support to a conjecture of Deligne, Conjecture 8.1 iii [D4]; a motive of rank 1 is associated to an algebraic Hecke character. A typical example of our formula is that the period of a Fermat hypersurface is a product of special values of the Γ-function (cf. proof of Lemma 5.4). The main theorem for X = P is a reformulation of a theorem of the second named author [T], Theorem 1.2. The theorem is proved by reducing to this case by induction on dimension using a Lefschetz pencil. We describe the result. Let U be a smooth quasi-projective variety over a subfield k of C and F another subfield of C. The determinant of the periods, the main subject of the paper, is defined for a triple M = ((E ,∇), V, ρ) consisting of the following three data: (1) E is a locally free sheaf on U with an integrable connection ∇ regular singular along the boundary. It gives rise to a local system