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Recovery of vanishing cycles by log geometry. (English) Zbl 1015.14005
From the introduction: K. Kato and C. Nakayama [Kodai Math. J. 22, No. 2, 161-186 (1999; Zbl 0957.14015)] constructed a ringed space $$(X^{\log},{\mathcal O}_X^{\log})$$ over a given fs log analytic space $$(X,{\mathcal M}_X)$$ and proved a log version of the Riemann-Hilbert correspondence on them. In the case where the fs log analytic space $$(X,{\mathcal M}_X)$$ is the one corresponding to a divisor $$D$$ with normal crossings on a complex manifold $$X$$, i.e., $${\mathcal M}_X: =\{f\in{\mathcal O}_X\mid f$$ is invertible outside $$D_{\text{red}}\}$$, the projection $$\tau_X:X^{\log}\to X$$ is nothing but the real oriented blowing-up of $$X$$ along $$D_{\text{red}}$$.
Let us consider a relative case. Let $$f:X\to\Delta$$ be a proper surjective flat morphism of a complex manifold onto an open disc such that $$f$$ is smooth over the punctured disc $$\Delta^*: =\Delta-\{0\}$$ and that the central fiber $$X_0: =f^{-1} (0)$$ is a reduced divisor with simple normal crossings. Let $$Y$$ be a divisor on $$X$$, flat with respect to $$f$$. We assume that $$X_0+Y$$ is also a divisor with simple normal crossings. Then, by the paper cited above, we can construct a map $$f^{\log}: X^{\log}\to\Delta^{\log}$$ and a subspace $$Y^{\log}$$ of $$X^{\log}$$ over the given ones and we have a commutative diagram: $\begin{tikzcd} (X^{\log}, Y^{\log})\ar[r,"\tau_X"]\ar[d,"f^{\log}" '] & (X,Y) \ar[d,"f"]\\ \Delta^{\log} \ar[r,"\tau_\Delta" '] & \Delta \rlap{\, .}\end{tikzcd}\tag{1}$ The main result in the present paper is that the family $\overset\circ f^{\log}: (X^{\log}-Y^{\log}{)} \Delta^{\log} \tag{2}$ of open spaces is locally piecewise $$C^\infty$$ trivial over the base $$\Delta^{\log}$$. As a consequence, we see that the family (2) is the one which recovers the vanishing cycles, in the most naive sense, of the degenerating family $${\overset\circ f}: (X-Y)\to \Delta$$.
This implies, in particular, that $$L_Z:=R^q \left(\overset\circ f^{\log}\right)_*\mathbb{Z}$$ is a locally constant sheaf of $$\mathbb{Z}$$-modules on $$\Delta^{\text{log}}$$. On the other hand, J. Steenbrink and S. Zucker [Invent. Math. 80, 489-542 (1985; Zbl 0626.14007)] showed that $${\mathcal V}:=R^qf_* \Omega^\bullet_{X/ \Delta}(\log (X_0+Y))$$ is a free $${\mathcal O}_\Delta$$-module with the Gauss-Manin connection $$\Delta$$. We thus have ${\mathcal V} \simeq (\tau_\Delta)_* ({\mathcal O}_\Delta^{\text{log}}\otimes_CL_C) \text{ on }\Delta$ under the log version of the Riemann-Hilbert correspondence established in the cited paper by Kato and Nakayama. As a corollary, we have two types of integral structure of the degenerate variation of mixed Hodge structure on $${\mathcal V}$$.

##### MSC:
 14D07 Variation of Hodge structures (algebro-geometric aspects) 14C25 Algebraic cycles 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 32G20 Period matrices, variation of Hodge structure; degenerations
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