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We answer negatively an open problem of Illusie on the DR-decomposability ofthe log de Rham complex of the special fiber of a semi-stable reduction overthe Witt ring. We also show that $E_1$ degeneration of the Hodge to log de Rhamspectral sequence does not imply DR-decomposability of semi-stable varieties.

Some examples of DR-indecomposable special fibers of semi-stable reductions over Witt rings