A determinant formula for period integrals
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.
scientific article; zbMATH DE number 559496