A twistor approach to the Kontsevich complexes

Abstract

We study the V -filtration of the mixed twistor D-modules associated to algebraic meromorphic functions. We prove that their relative de Rham complexes are quasi-isomorphic to the family of Kontsevich complexes. It reveals a generalized Hodge theoretic meaning of Kontsevich complexes. On the basis of the quasiisomorphism, we revisit the results on the Kontsevich complexes due to H. Esnault, M. Kontsevich, C. Sabbah, M. Saito and J.-D. Yu from a viewpoint of mixed twistor D-modules.

The Künneth Formula for the Twisted de Rham and Higgs Cohomologies

“…Together with the isomorphism of (20), a decreasing induction on the index λ in (19) then gives the desired statements. Finally the isomorphism of (20) can be obtained by directly truncating the involved complexes and inductively using the Künneth formula for coherent sheaves [ [4]).…”

Hodge–Tate Conditions for Landau–Ginzburg Models

“…The counter-example suggests that we need to impose some conditions on the degeneration property of Y b as b → ∞. (Ω • f , λd + τ df ∧)) is known to be independent of the choice of (λ, τ ) ( [18], [30]). Let C λ , C τ be complex planes with coordinate λ and τ respectively.…”

“…It has been studied from the viewpoint of generalized Hodge theories. (See twistor D-modules [30], [29], irregular Hodge structures [10], [18], [36], [37], non-commutative Hodge structures [25], [26], TERP-structures [23], and so on. )…”

“…Let (λ, τ ) be a pair of complex numbers. The dimension of the hypercohomology H • (X; (Ω • f , λd + τ df ∧)) is known to be independent of the choice of (λ, τ ) ( [18], [30]). Let C λ , C τ be complex planes with coordinate λ and τ respectively.…”

“…[30, Proposition 2.3]). U • M is a V-filtration on M along τ indexed by integers with the standard order (up to shift of degree by 1)…”