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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.

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