×

Parabolic ample bundles. II: Connectivity of zero locus of a class of sections. (English) Zbl 0929.14006

[For part I of this paper see I. Biswas, Math. Ann. 307, No. 3, 511-529 (1997; Zbl 0877.14013).]
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 \(E\) over a proper manifold with respect to a divisor \(D\) is equivalent to the existence of a certain decreasing filtration \(\{E_t\}_{t\in \mathbb R}\) of \(E\) whose last sheaf is \(E\otimes \mathcal O(-D)\) together with a system of parabolic weights.
Ample parabolic bundles were defined by I. Biswas in part I of this paper who characterized ample parabolic bundles over a Riemann surface and generalized in this context vanishing theorems formerly known for vector bundles.
In this note, the authors prove for ample parabolic bundles an analogue to a connectivity result for the sections of an ample bundles due to W. Fulton and R. Lazarsfeld [Acta Math. 146, 271-283 (1981; Zbl 0469.14018)]. More precisely, the authors associate with every ample parabolic bundle \(E_\ast\) on a complex connected smooth projective manifold \(X\), a natural subspace of its global sections, namely the space of those sections which do not intersect nontrivially the part of the parabolic filtration that correspond to nonzero parabolic weights. For such a section, the corresponding zero locus is nonempty if \(\dim X\geq \text{rank } E_\ast\) and connected if \(\dim X> \text{rank } E_\ast\). It is assumed that the parabolic divisor \(D\) has only normal crossings and that the subsheaves of the parabolic filtration are vector bundles.
The authors generalize the connectivity result proven for sections to the case of morphisms between parabolic bundles with the same parabolic divisor \(D\).
The note also contains a final section where new examples of ample parabolic bundles are constructed.
[For part III of this paper see I. Biswas and S. Subramanian, Proc. Indian. Acad. Sci., Math. Sci. 109, No. 1, 41-46 (1999; see the following review; Zbl 0929.14007)].

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14C20 Divisors, linear systems, invertible sheaves
PDFBibTeX XMLCite
Full Text: DOI