Polarizations on limiting mixed Hodge structures

Abstract

We construct a polarization on the relative log de Rham cohomology groups of a projective log deformation. To this end, we study the behavior of weight and Hodge filtrations under the cup product and construct a trace morphism for a log deformation.Comment: 47 pages, corrections and improvement

Geometric polarized log Hodge structures with a base of log rank one

“…defined in 5.11 of [3]. Therefore the morphism (4.7.1) coincides with the composite As in Remark 5.10 of [3], we have…”

“…and by reducing to the main result in [3] that the right-hand-side carries a polarized log Hodge structure. Here the notation is as in [5].…”

“…The log Hodge structure given by Theorem 3.2 is polarized by [3] via the cohomologies of the Koszul complex. More precisely, we have proved Theorem 3.2 in [5] by constructing the comparison isomorphism of Q-structures…”

“…where Θ C,0 is the morphism defined in Definition 7.7 of [3], from the fact that C → W 0 K C factors through the subcomplex (δW ) 0 C(ω · X• ) and from the commutativity of the diagram [3]. Furthermore, we have the commutative diagram…”

Limits of Hodge structures via holonomic D-modules

“…Steenbrink later pointed out the limiting mixed Hodge structure he constructed only depends on the log structure associated to the semistable family f ∶ X → ∆ [Ste95]. Inspired by the idea in [Ste95], Fujisawa extended Steenbrink's results in [Ste76,Ste95] to semistable Kähler families over the polydisk and furthermore to the log geometry setting [Fuj99,Fuj08,Fuj14]. Recently, Nakkajima announced a simpler proof of Fujisawa's results [Nak21].…”