Wu, Lei Vanishing and injectivity for Hodge modules and \(\mathbb{R}\)-divisors. (English) Zbl 1496.14012 Mich. Math. J. 71, No. 2, 373-399 (2022). Let \(X\) be a projective manifold, \(D\) a simple normal crossing divisor, \(B\) a \(\mathbb{R}\)-divisor supported on \(D\). Let \(V=(\mathcal{V}, F_\bullet, \mathbb{V}_{\mathbb{Q}})\) be a polarizable \(\mathbb{Q}\)-variation of Hodge structures defined over \(U=X\setminus D\). Denote by \(\mathcal{V}^{\le B}\) the lower canonical Deligne lattice of index \(B\). Denote by \(S^{\le B}(V)\) the first term in the subbundle filtration \(F_\bullet^{\le B}=\mathcal{V}^{\le B}\cap j_*F_\bullet\), where \(j: X\setminus D\to X\) is the open embedding.Let \(L\) and \(E\) be divisors on \(X\). Assume that \(L-B\) is nef and big, and \(E\) is effective. Along with some applications, in this paper, the author shows that \[ H^i(X, S^{\le B}\otimes\omega_X(L))\hookrightarrow H^i(X, S^{\le B}(V)\otimes\omega_X(L+E)) \] is injective and \[ H^i(X, S^{\le B}\otimes\omega_X(L))=0 \] for \(i>0\), which generalize the injectivity and vanishing theorems in [L. Wu, Trans. Am. Math. Soc. 369, No. 11, 7719–7736 (2017; Zbl 1401.14056)]. The proofs employ the \(E_1\)-degeneration of the Hodge-to-de-Rham spectral sequence for a twisted Hodge module. Reviewer: Fei Ye (New York) Cited in 1 Document MSC: 14D07 Variation of Hodge structures (algebro-geometric aspects) 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14F17 Vanishing theorems in algebraic geometry Keywords:vanishing theorem; injectivity; Hodge modules Citations:Zbl 1401.14056 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] U. Angehrn and Y. T. Siu, Effective freeness and point separation for adjoint bundles, Invent. 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