Abstract
Abstract We show that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can always be computed in terms of a topological index.
Highlights
Introduction and summary of resultsIt is usually hard to compute the cohomology of a holomorphic line bundle L over a complex manifold X
In this paper we study certain classes of non-singular complex projective surfaces for which we show that the region in the Picard lattice where the dimension of the zeroth cohomology is given by a topological index can be extended to the entire e ective cone
For surfaces such as del Pezzos and toric surfaces whose fans have convex support, the statement holds true thoughout the entire Picard lattice, leading to closed form expressions for the dimension of the zeroth cohomology and for all higher cohomologies. The validity of these results is derived from the existence of a map taking e ective divisor classes to nef divisor classes while preserving the dimension of the zeroth cohomology
Summary
It is usually hard to compute the cohomology of a holomorphic line bundle L over a complex manifold X. In the situation when L is ample and X a smooth complex projective variety, Kodaira’s vanishing theorem ensures that all cohomology groups Hi(KX ⊗ L) with indices i > are automatically zero, where KX is the canonical line bundle of X In this case, the dimension of the zeroth cohomology of KX ⊗ L can be computed as a topological index. For surfaces such as del Pezzos and toric surfaces whose fans have convex support, the statement holds true thoughout the entire Picard lattice, leading to closed form expressions for the dimension of the zeroth cohomology and for all higher cohomologies. The map D → D , where Dis de ned by
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
