A determinant formula for period integrals
type: Item
instance of: scholarly article
title: A determinant formula for period integrals
zbMATH Open document ID: 0824.19003
author: Takeshi Saito
author: Tomohide Terasoma
published in: Proceedings of the Japan Academy. Series A
publication date: 1995-04-27T00:00:00Z
review text: Let \(k\) and \(F\) be subfields of \({\mathbb{C}}\) and \(U\) be a smooth algebraic variety. We define the category \(M_ k(U, F)\) by the triple \({\mathcal M}= (({\mathcal E}, \nabla), V, \rho)\) consisting of (1) A locally free \({\mathcal O}_ U\)-module \(\mathcal E\) of finite rank with an integrable connection \(\nabla: {\mathcal E}\to {\mathcal E}\otimes \Omega^ 1_ U\) which is regular singular along the boundary. (2) A local system \(V\) of \(F\)-vector spaces on the complex manifold \(U^{an}\). (3) An isomorphism \(\rho: V\otimes_ F {\mathbb{C}}\simeq \text{Ker }\nabla^{an}\) of local systems on the complex manifold \(U^{an}\). For an object \(\mathcal M\) of \(M_ k(U, F)\), by the comparison theorem of Deligne, one can define the object \(H^*(U, {\mathcal M})\). We describe the determinant object \(\text{det }H^*(U, {\mathcal M})\) of \(H^*(U, {\mathcal M})\) in terms of the pairing of the relative Chow group and the motivic Picard group \(\text{MPic}_ k(U, F)\). We give an outline of the construction of the pairing by using \(K\)-theory. This is an announcement of the full paper.
Mathematics Subject Classification ID: 19E08
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 19C20
zbMATH DE Number: 559496
zbMATH Keywords: determinant of period integrals
zbMATH Keywords: motive of rank 1
zbMATH Keywords: tame symbol
zbMATH Keywords: determinant object
zbMATH Keywords: relative Chow group
zbMATH Keywords: motivic Picard group
MaRDI profile type: Publication
full work available at URL: pjaa.69.131
OpenAlex ID: W2013535367
cites work: Q3895546
cites work: \(\varepsilon\)-factor of a tamely ramified sheaf on a variety
cites work: A product formula for period integrals
DOI: 10.3792/PJAA.69.131
description: scientific article; zbMATH DE number 559496