Kodaira-Saito vanishing via Higgs bundles in positive characteristic
Let \(X\) be a smooth projective variety defined over an algebraically closed field \(k\). Let \(D\) be a normal crossing divisor and \(L\) an ample line bundle on \(X\). Let \((E, \theta)\) be a semistable logarithmic Higgs bundle on \((X, D)\) with vanishing Chern classes in \(\ell\)-adic cohomology. If \(k\) has positive characteristic the author proves that the \(i\)-th ccohomology group of the de Rham complex of \((E, \theta)\) tensored by \(L\) is vanishing if \(i>\dim X\). Similar theorem holds also in positive characteristic if one assumes that \((X,D)\) is liftable to the ring of length \(2\) Witt vectors \(W_2(k)\). In principle, the author assumes that \(\theta\) is nilpotent, but since every semistable Higgs bundle can be deformed to a semistable one with a nilpotent Higgs field (see, e.g., Corollary 5.7 in the reviewer's paper [Doc. Math. 19, 509--540 (2014; Zbl 1330.14017)]), this assumption is not needed. The author's result implies a special case of Saito's generalization of Kodaira's vanishing theorem for mixed Hodge modules. For \(k={\mathbb C}\) the author proves also a refinement of the semipositivity theorem for complex variations of mixed Hodge structures on \((X,D)\) with unipotent local monodromies around \(D\). He also proves another variant of Saito's vanishing theorem that generalizes the Kawamata-Viehweg vanishing theorem.
scientific article