The Michigan Mathematical Journal

Splitting Criteria for Vector Bundles Induced by Restrictions to Divisors

Mihai Halic

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Abstract

In this article we obtain criteria for the splitting and triviality of vector bundles by restricting them to partially ample divisors. This allows us to study the problem of splitting on the total space of fibre bundles. The statements are illustrated with examples.

For products of minuscule homogeneous varieties, we show that the splitting of vector bundles can be tested by restricting them to subproducts of Schubert 2-planes. By using known cohomological criteria for multiprojective spaces, we deduce necessary and sufficient conditions for the splitting of vector bundles on products of minuscule varieties.

The triviality criteria are particularly suited to Frobenius split varieties. We prove that a vector bundle on a smooth toric variety, whose anticanonical bundle has stable base locus of codimension at least three, is trivial precisely when its restrictions to the invariant divisors are trivial, with trivializations compatible along the various intersections.

Article information

Source
Michigan Math. J., Volume 67, Issue 2 (2018), 227-251.

Dates
Received: 19 July 2016
Revised: 18 January 2018
First available in Project Euclid: 24 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1521856929

Digital Object Identifier
doi:10.1307/mmj/1521856929

Mathematical Reviews number (MathSciNet)
MR3802253

Zentralblatt MATH identifier
06914762

Subjects
Primary: 14C20: Divisors, linear systems, invertible sheaves 14F17: Vanishing theorems [See also 32L20] 14M17: Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15] 14M25: Toric varieties, Newton polyhedra [See also 52B20]

Citation

Halic, Mihai. Splitting Criteria for Vector Bundles Induced by Restrictions to Divisors. Michigan Math. J. 67 (2018), no. 2, 227--251. doi:10.1307/mmj/1521856929. https://projecteuclid.org/euclid.mmj/1521856929


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