On the cohomology of parabolic line bundles (Q1910671)

Let \(X\) be a smooth projective variety over \(\mathbb{C}\) of dimension \(n\). Let \(D= \sum^d_{i=1} D_i\) be a divisor of normal crossing with decomposition into irreducible components. Fix rational numbers \(\{\alpha_1, \dots, \alpha_d\}\) with \(0< \alpha_i <1\). Assume that the Poincaré dual of the \(\mathbb{Q}\)-divisor \(\sum^d_{i=1} \alpha_iD_i\) is in the image of \(H^2 (X,\mathbb{Z})\) in \(H^2 (X,\mathbb{Q})\). Such data constitute a parabolic bundle in the sense of \textit{M. Maruyama} and \textit{K. Yokogawa} [Math. Ann. 293, No. 1, 77-100 (1992; Zbl 0735.14008)]. Let \(P(X)\) be a component of the moduli space of parabolic bundles of parabolic degree zero (which simply is a component of the Picard group of \(X\) consisting of line bundles with first Chern class \(-\sum^d_{i=1} \alpha_i [D_i]\), where \([D_i]\) is the Poincaré dual of \(D_i)\). Let \(\text{Pic}^0 (X)\) be the abelian variety consisting of isomorphism classes of topologically trivial line bundles. The group \(\text{Pic}^0 (X)\) acts on \(P(X)\) using tensor product, and \(P(X)\) is an affine group for \(\text{Pic}^0 (X)\). Define the subvariety \(T^i_m: =\{L \in P(X) \mid \dim H^i (X,L)\geq m\} \subset P(X)\). We prove the following theorem: Any irreducible component of \(T^i_m\) is a translation of an abelian subvariety of \(\text{Pic}^0 (X)\) by a point of \(P(X)\) for the above action. The special case of the above theorem where \(D\) is empty was proved by \textit{M. Green} and \textit{R. Lazarsfeld} [J. Am. Math. Soc. 4, No. 1, 87-103 (1991; Zbl 0735.14004)].