zbMATH — the first resource for mathematics

Parabolic ample bundles. (English) Zbl 0877.14013
In this paper the notion of ample parabolic bundle is introduced and ample parabolic bundles over a Riemann surface are characterized. Parabolic bundles with respect to an effective divisor were defined by V. B. Mehta and C. S. Seshadri for Riemann surfaces [Math. Ann. 248, 205-239 (1980; Zbl 0454.14006)] and by M. Maruyama and K. Yokogawa for higher dimensional varieties [Math. Ann. 293, No. 1, 77-100 (1992; Zbl 0735.14008)]. A parabolic structure on a torsion free sheaf \(E\) is equivalent to the existence of certain decreasing filtration \(\{E_t\}_{t\in \mathbb{R}}\) of \(E\). For a parabolic bundle \(E_\ast\), one can define a parabolic \(n\)-fold symmetric product \(S^n(E_\ast)\); it is a parabolic sheaf so that one can consider the associated filtration and in particular the subsheaf \(S^n(E_\ast)_0\). The author then defines a parabolic ample sheaf as a parabolic sheaf \(E_\ast\) such that \(S^n(E_\ast)_0\otimes V\) is generated by its global sections for any vector bundle \(V\). When the divisor \(D\) is empty, an ample parabolic sheaf is nothing but a usual ample sheaf. Parabolic sheaves on compact Riemann surfaces are characterized: a parabolic vector bundle \(E_\ast\) over a compact Riemann surface is ample parabolic if and only if the parabolic degree of \(E_\ast\) and that of all its quotient bundles with the induced parabolic structure are all strictly positive numbers. In particular a parabolic semistable bundle over a compact Riemann surface is parabolic ample if and only if its parabolic degree is strictly positive. The paper gives also a vanishing theorem for the cohomology of the sheaves of vector bundle valued differential forms:
If the divisor \(D\) has normal crossings and \(E_\ast\) satisfies certain conditions, then \[ H^q(X,\Omega^p(\log D)\otimes E)=0\text{ for }p+q\geq r+d, \] \(r\) being the rank of \(E\) and \(d\) the dimension of \(X\).
This generalizes a celebrated theorem of J. Le Potier [Math. Ann. 218, 35-53 (1975; Zbl 0313.32037)]. The paper also gives parabolic analogues of vanishing theorems for tensor powers of ample bundles due to J.-P. Demailly [Invent. Math. 91, No. 1, 203-220 (1988; Zbl 0647.14005)] and L. Manivel [J. Reine Angew. Math. 422, 91-116 (1991; Zbl 0728.14011)]. The paper is very clearly written despite of the fact that certain proofs are rather technical.

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14F17 Vanishing theorems in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
Full Text: DOI