E-Factors for the Period Determinants of Curves

0.1. Let X be a smooth compact complex curve, M be a holonomic D-module on X (so outside a finite subset T ⊂ X , our M is a vector bundle with a connection ∇). Denote by dR(M) the algebraic de Rham complex of M placed in degrees [−1, 0]; this is a complex of sheaves on the Zariski topology XZar. Its analytic counterpart dR(M) is a complex of sheaves on the classical topology Xcl. Viewed as an object of the derived category of C-sheaves, this is a perverse sheaf, which we denote by B(M); outside T , it is the local system M X\T of ∇horizontal sections (placed in degree −1). Set H · dR(X,M) := H (XZar, dR(M)), H · B(X,M) := H (Xcl, B(M)); these are the de Rham and Betti cohomology. We have the period isomorphism ρ : H · dR(X,M) ∼ → H · B(X,M). The cohomology H · dR and H · B have, respectively, algebraic and topological nature that can be tasted as follows. Let k, k ⊂ C be subfields. Then: For (X,M) defined over k, we have de Rham k-structure H · dR(Xk,Mk) on H · dR(X,M); A k-structure on B(M), i.e., a perverse k-sheaf Bk′ on Xcl together with an isomorphism Bk′ ⊗ k′ C ∼ → B(M), yields Betti k-structure H (Xcl, Bk′) on H · B(X,M). If both (Xk,Mk) and Bk′ are at hand, then, computing det ρ with respect to rational bases, one gets a number whose class [det ρ] in C/kk does not depend on the choice of the bases. In his farewell seminar at Bures [Del], Deligne, guided