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

Abstract

We prove that a projective vertical exact log smooth morphism of fs log analytic spaces with a base of log rank one yields polarized log Hodge structures in the canonical way. Contents §1. Polarized log Hodge structures §2. Main theorems §3. Review of [5] §4. Compatibility of polarizations §5. Degeneration of log Hodge and log de Rham spectral sequences §6. Proof of Theorem 2.6 §7. Log Picard varieties and log Albanese varieties

Classifying spaces of degenerating mixed Hodge structures, IV: The fundamental diagram

“…This proves (2). We prove (3) ), and every point of U is vertical over Y in the sense of 2.2.7. Let u : U → V be the induced map.…”

“…Thus we have H m (X/Y ) := (H Z , H O , F, ι, •, • ). By [3] Theorem 2.5 (1) and ( 2), the Hodge to de Rham spectral sequence…”

“…1.1.8. The following (1)- (3) are main results about properties of log real analytic functions, and are proved in Section 1.6.…”

“…We first show that h extends to a continuous function on U (not only on U triv ). It is sufficient to prove that when U triv ∋ (p, y) → µ and U triv ∋ (p ′ , y ′ ) → µ, we have the following (1)- (3).…”

“…By [3], this condition for y ∈ Y are satisfied if the rank of (M gp Y /O × Y ) y is ≤ 1. By the properness of T [:] → T , the condition (1) is equivalent to the following condition (2).…”