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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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