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Let $(X,D_X)$ and $(S, D_S)$ be smooth normal crossings pairs over $\mathbb C$; i.e. smooth schemes of finite type over $\mathbb C$ with a normal crossings divisor. If $f:X \to S$ is a proper, flat and log-smooth (with respect to the log structures given by the divisors) such that $f^{-1}(D_S)=D_X$ then $\Omega_{X/S}^{log}$ is a vector bundle on $X$, and the Gauss-Manin sheaves $$ \mathcal{H}^i = R^if_*(\Omega^{log, \bullet}_{X/S}) $$ have a connection with singular points at $D_S$, so they are vector bundles on $S\smallsetminus D_S$. Is there some criteria (in terms of $f$) to determine when they are vector bundles on $S$?

I know that this is the case if $f:X \to S$ is a compactified family of abelian varieties (Section 6 in Faltings-Chai) or a stable curve.

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    $\begingroup$ I think these are the canonical extensions of their restrictions to $S\setminus D_S$, and these are always locally free. Will try to find a reference. $\endgroup$
    – Piotr Achinger
    Commented Mar 27 at 15:19

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If I am reading the assumptions correctly, this should follow from Theorem 6.4 in the paper

Luc Illusie, Kazuya Kato and Chikara Nakayama Quasi-unipotent Logarithmic Riemann-Hilbert Correspondences J. Math. Sci. Univ. Tokyo 12 (2005), 1–66.

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  • $\begingroup$ This answers my question. Thank you so much $\endgroup$
    – Aitor Iribar Lopez
    Commented Mar 27 at 16:34

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