The $l$-adic bifiltered El Zein-Steenbrink-Zucker complex of a proper SNCL scheme with a relative SNCD

For a family of log points with constant log structure and for a proper SNCL scheme with an SNCD over the family, we construct a fundamental l-adic bifiltered complex as a geometric application of the theory of the derived category of (bi)filtered complexes in our papers. By using this bifiltered complex, we give the formulation of the log l-adic relative monodromy-weight conjecture with respect to the filtration arising from the SNCD. That is, we state that the relative l-adic monodromy filtration should exist for the Kummer log etale cohomological sheaf of the proper SNCL scheme with an SNCD and it should be equal to the l-adic weight filtration.

💡 Research Summary

The paper develops a new ℓ‑adic bifiltered complex, called the El Zein‑Steenbrink‑Zucker (ESZ) complex, for a proper SNCL (simple normal crossing log) scheme equipped with an SNCD (simple normal crossing divisor) over a family of log points with constant log structure. The construction rests on a substantial extension of the derived category of filtered complexes: the author defines a bounded bifiltered derived category DᵇF₂(ctf) of constructible ℤₗ‑modules whose graded pieces have finite tor‑dimension. Within this framework, strictly injective and strictly flat resolutions are established, allowing the definition of the usual derived functors (Rf₍*₎, Lf⁎, RHom, ⊗ᴸ) for bifiltered objects.

Using this machinery, the author builds the ℓ‑adic bifiltered ESZ complex \