Non-vanishing theorems for rank two vector bundles on threefolds

2010/05/12 by Edoardo Ballico, Ballico, Edoardo, Paolo Valabrega +3
Mathematics · #14F05 #14J60 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14F05 #msc:14J60

paper · pdf · doi:10.48550/arxiv.1005.2080

18 pages

arxiv created 2010/05/12 · arxiv updated 2010/05/13

Abstract

The paper investigates the non-vanishing of H¹(E(n)), where E is a (normalized) rank two vector bundle over any smooth irreducible threefold X of degree d such that Pic(X) ≅ \ZZ. If ε is the integer defined by the equality ω_X = O_X(ε), and α is the least integer t such that H⁰(E(t)) ≠ 0, then, for a non-stable E (α≤ 0) the first cohomology module does not vanish at least between the endpoints (ε-c₁)/(2) and -α-c₁-1. The paper also shows that there are other non-vanishing intervals, whose endpoints depend on α and also on the second Chern class c₂ of E. If E is stable the first cohomology module does not vanish at least between the endpoints (ε-c₁)/(2) and α-2. The paper considers also the case of a threefold X with Pic(X) ≠ \ZZ but Num(X) ≅ \ZZ and gives similar non-vanishing results.

Related

• Non-vanishing theorems for rank 2 bundles on P³: a simple approach without the speciality lemma
• Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem
• On projective threefolds with two-dimensional space of vanishing cycles
• Vanishing of the top Chern classes of the moduli of vector bundles
• On the cohomology of rank two vector bundles on P2 and a theorem of Chiantini and Valabrega