Parabolic ample bundles.

*(English)*Zbl 0877.14013In 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.

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.

Reviewer: D.Hernández Ruipérez (Salamanca)