On a twisted de Rham complex
Let \(\widetilde f:\widetilde X\to\mathbb{P}^1\) be a morphism from a smooth projective variety \(\widetilde X\) over \(\mathbb{C}\) to the projective line and let \(f:X\to\mathbb{A}^1\) be its restriction over the affine line \(\mathbb{A}^1\). A result of \textit{M. Kontsevitch} and \textit{S. Barannikov} says that the hypercohomology spaces on \(X\) of the complexes \((\Omega_X^\bullet, d-df\wedge)\) and \((\Omega_X^\bullet, df\wedge)\) have the same (finite) dimension. Motivated by this result, the author of the present paper proves the following generalization: Given a mixed Hodge module \((M,F)\) on \(X\), the hypercohomology spaces on \(X\) of the complexes \((\Omega_X^\bullet \otimes_{{\mathcal O}_X}M,\;\nabla-df\wedge)\) and \((\Omega^\bullet_X\otimes_{{\mathcal O}_X} \text{gr}^FM,\;\text{gr}^F\nabla-df \wedge)\) have the same (finite) dimension. An analytic version of this result concerning the vanishing cycle functor is also established.
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