Hermitian-Einstein metrics and Chern number inequalities on parabolic stable bundles over Kähler manifolds (Q5929580)

Let \(\overline X\) be a compact Kähler manifold of dimension \(n\) with a Kähler metric \(\omega\), and \(D\) a divisor in \(\overline X\) with normal crossings. Let \(X= \overline X\setminus D\), then the restriction of \(\omega\) to \(X\) gives a Kähler metric on \(X\).NEWLINENEWLINENEWLINEThe author first gives the notion of parabolic structures of a holomorphic vector bundle \(E\) over \(\overline X\), with respect to \(D\), and the parabolic degree of \(E\) using the parabolic structure as well as the parabolic stability for \(E\). Let \(E'= E|_X\).NEWLINENEWLINENEWLINEThe author proves that the parabolic stability is essentially equivalent to the existence of a Hermitian-Einstein metric on \(E'\) with respect to \(\omega_\alpha\) for some \(0<\alpha< 2\), where \(\omega_\alpha\) is a Kähler metric on \(X\) constructed from \(\omega\) and \(\alpha\). Furthermore, the author gives a Chern number inequality for a parabolic stable bundle \(E\). Using the vanishing theorem of Hironaka, the author generalizes the above results in the case when we do not assume that the irreducible components \(D_i\) of \(D\) meet transversely.