Mixed Hodge structures on log smooth degenerations

Polarizations on limiting mixed Hodge structures

“…In this section, we first recall results in [6] and adjust them to the case of a log deformation. The definition and the notation are slightly changed from that in [6].…”

“…In this section, we first recall results in [6] and adjust them to the case of a log deformation. The definition and the notation are slightly changed from that in [6]. In addition to this change, the method in Section 2 is used instead of the simplicial method in [6] because it can work without fixing the total order on the index set.…”

“…The definition and the notation are slightly changed from that in [6]. In addition to this change, the method in Section 2 is used instead of the simplicial method in [6] because it can work without fixing the total order on the index set. After recalling the results in Steenbrink [22], Fujisawa-Nakayama [8] briefly, we construct a morphism from Steenbrink's cohomological mixed Hodge complex to the complex in [6].…”

An ideal proof for Fujisawa's result and its generalization

“…(cf. [F,Corollary 5.7]). ) Here note that the filtration F on H X/s C /C is not finite; we need a formalism of derived categories of filtered complexes in e. g., [NS], which is a special case of the formalism of derived categories of filtered complexes in [B].…”

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].…”