Variations of mixed Hodge structure, Higgs fields, and quantum cohomology

Abstract

Following C. Simpson, we show that every variation of graded-polarized mixed Hodge structure defined over Q carries a natural Higgs bundle structure∂ + θ which is invariant under the C * action studied in [20]. We then specialize our construction to the context of [6], and show that the resulting Higgs field θ determines (and is determined by) the Gromov-Witten potential of the underlying family of Calabi-Yau threefolds.

Singularities of variations of mixed Hodge structure

“…Related results were presented in [6] [14] [17] and [18]. Some of the material from [8] and [14] is being published here for the first time.…”

“…As shown in [14] and [17], .Mis a homogeneous complex manifold which fits into an ascending sequence of homogeneous spaces -MR is the C^-submanifold of M consisting of the filtration F e M for which the associated mixed Hodge structure (F, W) is split over E. The corresponding sequence of Lie groups is…”

Degenerations of mixed Hodge structure

“…As discussed in [11], the data of such a variation V → S may be effectively encoded into its monodromy representation ρ : π 1 (S, s 0 ) → GL(V s 0 ), Image(ρ) = , (2.2) and its period map…”

“…In §2, we review the basic properties of the classifying spaces of graded-polarized mixed Hodge structures M = M W, S , {h p,q } constructed in [11] and recall how the isomorphism class of a variation of gradedpolarized mixed Hodge structure V → S may be recovered from the knowledge of its monodromy representation ρ : π 1 (S, s 0 ) → GL(V s 0 ), Image(ρ) = , and its period map ϕ : S → M / .…”

Naive boundary strata and nilpotent orbits

“…• -orbit of a nilpotent element (20) ) in g R . The image of (5.4) is obviously a boundary stratum, which we shall denote by B(N ).…”