Vanishing theorems for parabolic Higgs bundles (Q2278865)

Let \(D\) be a reduced divisor with simple normal crossings on a smooth complex projective variety \(X\). The authors consider parabolic Higgs bundles \((E, \theta, F(E))\) with trivial parabolic Chern classes, where \(E\) is a vector bundle on \(X\), \(F(E)\) is a parabolic structure on \(E\) over \(D\) and \(\theta: E \to E \otimes \Omega^1_X(\log D)\) is a Higgs field satisfying suitable conditions. The authors assume that a parabolic vector bundle is Zariski locally a sum of parabolic line bundles. Since \(\theta \wedge \theta = 0\) one gets a ``de Rham'' complex \[DR(E,\theta) := E \to E \otimes \Omega^1_X(\log D) \to E \otimes \Omega^2_X(\log D) \to \cdots \, .\] The main result of the paper (a generalised Kodaira vanishing theorem) says that if the parabolic Higgs bundle is slope semistable and \(L\) is an ample line bundle, then \[ \mathbb{H}^i (X, DR(E,\theta)\otimes L) = 0 \text{ for }i > \dim X.\] The authors deduce this as a corollary to a more general result saying that if \(M\) is a nef vector bundle such that \(M(- \Delta)\) is ample for some \(\mathbb{Q}\)-divisor supported on \(D\), then for a slope semistable parabolic Higgs bundle one has \[\mathbb{H}^i (X, DR(E,\theta)\otimes M(-D)) = 0 \text{ for } i >\dim X + \mathrm{ rank }M.\] The main result implies the Kodaira-Saito vanishing theorem for complex polarised variations of Hodge structures on \(X \setminus D\) (as such a variation determines a parabolic Higgs bundle). Another interesting result deduced from the vanishing theorem is that if \(E = E_+ \oplus E_{-}\) such that \(\theta(E) \subset E_{-} \otimes \Omega^1_X(\log D)\), then \(E_+\) is nef.