DR-decomposability of a smooth variety is an important property which leads to the first algebraic proof of Kodaira vanishing and degeneration of Hodge-de Rham spectral sequence. In the case of the semi-stable variety, which appears at the boundary of the moduli of manifolds, the DR-decomposability is in mixed characteristic is posted by L.Illusie as a problem in a 1996 survey. In this talk, I will report my recent work on this problem (joint with Mao Sheng). It turns out that Illusie’s problem generally has a negative answer. Moreover, we show that the DR-decomposability is not equivalent to the degeneration of the Hodge-de Rham spectral sequence.

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