Limiting mixed Hodge structures on the relative log de Rham cohomology groups of a projective semistable log smooth degeneration (Q6116728)

A morphism from a complex manifold to a polydisc is said to be semistable, if it is locally isomorphic to a product of semistable degenerations over the unit disc [\textit{T. Fujisawa}, Limits of Hodge structures in several variables. II'', Preprint, \url{arXiv:1506.02271}, Lemma 3.3]. The notion of semistable log smooth degeneration is an abstraction of the central fiber of a semistable morphism in the context of log geometry. Namely, a semistable log smooth degeneration is a log complex analytic space \((X, \mathcal{M}_X)\) over the log point \((\ast, \mathbb{N}^k)\) (i.e. a morphism of log complex analytic space \(f : (X, \mathcal{M}_X) \longrightarrow (\ast,\mathbb{N}^k)\)), which is locally isomorphic to the central fiber of a semistable morphism to the \(k\)-dimensional polydisc in the category of log complex analytic spaces. One of the main results of this paper is the following. Theorem 0.1. Let \(f : (X, \mathcal{M}_X) \longrightarrow (\ast, \mathbb{N}^k)\) be a projective semistable log smooth degeneration. Then the relative log de Rham cohomology groups \(\mathrm{H}^n(X, \Omega_{X/\ast}(\log (\mathcal{M}_X/\mathbb{N}^k)))\) admit a \text{limiting} mixed Hodge structure, whose Hodge filtration \(F\) is induced from the stupid filtration (filtration bête in [\textit{P. Deligne}, Publ. Math., Inst. Hautes Étud. Sci. 40, 5--57 (1971; Zbl 0219.14007), (1.4.7)]) on \(\Omega_{X/\ast}(\log (\mathcal{M}_X/\mathbb{N}^k))\). Here, a \(\mathbb{Q}\)-mixed Hodge structure \(((V_{\mathbb{Q}}, W), (V_{\mathbb{C}}, W, F))\) is called a \textit{limiting} mixed Hodge structure, if there exists a nilpotent endomorphism \(N\) of \(V_{\mathbb{Q}}\) with \(W=W(N)[k]\) for some \(k \in \mathbb{Z}\), where \(W(N)\) denotes the monodromy weight filtration of \(N\) [\textit{M. Green} and \textit{P. Griffiths}, Lond. Math. Soc. Lect. Note Ser. 427, 88--133 (2016; Zbl 1365.14012), p. 90]. Theorem 0.1 is deduced from the following theorem: Theorem 0.2. On \(\mathrm{H}^n(X, \Omega_{X/\ast}(\log (\mathcal{M}_X/\mathbb{N}^k)))\), we can construct a finite increasing filtration \(L(I)\) for all \(I \subset \{1,2,\dots,k\}\) and nilpotent endomorphisms \(N_1, \dots, N_k\) such that the following is satisfied: \begin{itemize} \item[(0.2.1)] By setting \(L=L(\{1,2,\dots,k\})\), the triple \((\mathrm{H}^n(X, \Omega_{X/\ast}(\log (\mathcal{M}_X/\mathbb{N}^k))), L[n], F)\) underlies a \(\mathbb{Q}\)-mixed Hodge structure. \item[(0.2.2)] \(L(I)\) coincides with the monodromy weight filtration of the nilpotent endomorphism \(N_I(c_I)=\sum_{i \in I}c_iN_i\) for all \(c_I=(c_i)_{i \in I} \in (\mathbb{R}_{>0})^I\). \end{itemize} The case of \(I=\{1,2,\dots,k\}\) in (0.2.2) together with (0.2.1) implies Theorem 0.1. Moreover, (0.2.2) claims that the monodromy weight filtration of \(N_I(c_I)\) is independent of the choice of \(c_I \in (\mathbb{R}_{>0})^I\). The following theorem states the relation between the filtrations \(L\) and \(L(I)\). Theorem 0.3. On \(\mathrm{H}^n(X, \Omega_{X/\ast}(\log (\mathcal{M}_X/\mathbb{N}^k)))\), the filtration \(L\) is the monodromy weight filtration of \(N(c)=\sum_{i=1}^{k}c_iN_i\) relative to \(L(I)\) for all \(c=(c_i)_{i=1}^k \in (\mathbb{R}_{>0})^k\).