Archimedean Cohomology Revisited

C. Deninger produced a unified description of the local factors at arithmetic infinity and at the finite places where the local Frobenius acts semi-simply, in the form of a Ray–Singer determinant of a “logarithm of Frobenius” Φ on an infinite dimensional vector space (the archimedean cohomology H · ar(X) at the archimedean places, cf. [11]). The first author gave a cohomological interpretation of the space H · ar(X), in terms of a double complex K ·,· of real differential forms on a smooth projective algebraic variety X (over C or R), with Tate-twists and suitable cutoffs, together with an endomorphism N , which represents a “logarithm of the local monodromy at arithmetic infinity”. Moreover, in this theory the cohomology of the complex Cone(N)· computes real Deligne cohomology of X (cf. [9]). The construction of [9] is motivated by a dictionary of analogies between the geometry of the tubular neighborhoods of the “fibers at arithmetic infinity” of an arithmetic variety X and the geometric theory of the limiting mixed Hodge structure of a degeneration over a disk. Thus, the formulation and notation used in [9] for the double complex and archimedean cohomology mimics the definition, in the geometric case, of a resolution of the complex of nearby cycles and its cohomology(cf. [28]). In Section 2 and 3 we give an equivalent description of Consani’s double complex, which allows us to investigate further the structure induced on the complex and the archimedean cohomology by the operators N , Φ, and the Lefschetz operator L. In Section 4 we illustrate the analogies between the complex and archimedean cohomology and a resolution of the complex of nearby cycles in the classical geometry of an analytic degeneration with normal crossings over a disk. In Section 5 we show that, using the Connes–Kreimer formalism of renormalization, we can identify the endomorphism N with the residue of a Fuchsian connection, in analogy to the log of the monodromy in the geometric case. In Section 6 we recall Deninger’s approach to the archimedean cohomology through an interpretation as global sections of a real analytic Rees sheaf over R. In Section 7 we show how the action of the endomorphisms N and L and the Frobenius operator Φ define a noncommutative manifold (a spectral triple in the sense of Connes), where the algebra is related to the SL(2,R) representation associated to the Lefschetz L, the Hilbert space is obtained by considering Kernel and Cokernel of powers of N , and the log of Frobenius Φ gives the Dirac operator. The archimedean part of the Hasse-Weil L-function is obtained from a zeta function of the spectral triple. In Section 8 we outline some formal analogies between the complex and cohomology at arithmetic infinity and the equivariant Floer cohomology of loop spaces considered in Givental’s homological geometry of mirror symmetry.