[Paper Review] On the cohomology of rank two vector bundles on P2 and a theorem of Chiantini and Valabrega

Philippe Ellia|arXiv (Cornell University)|Jan 3, 2019

Homotopy and Cohomology in Algebraic Topology4 references5 citations

TL;DR

This paper establishes that a normalized rank two vector bundle on ℙ² splits if and only if h¹(E(-1)) = 0, providing a sharp cohomological criterion analogous to a theorem of Chiantini and Valabrega for ℙ³. It proves that h¹(E(k)) ≤ h¹(E(-1)) for all k ∈ ℤ, and classifies all such bundles with h¹(E(-1)) ≤ 4, offering a complete resolution description and identifying stable and unstable cases via Serre's construction and cohomological constraints.

ABSTRACT

We show that a normalized rank two vector bundle, E, on P2 splits if and only if h1(E(-1)) = 0. Using this fact we give another proof of a theorem of Chiantini and Valabrega. Finally we describe the normalized bundles with h1(E(-1)) <= 4.

Motivation & Objective

• To establish a cohomological criterion for splitting of normalized rank two vector bundles on ℙ², analogous to the Chiantini-Valabrega theorem on ℙ³.
• To prove that h¹(E(k)) ≤ h¹(E(-1)) for all integers k, showing that E(-1) has the maximal first cohomology among all twists.
• To classify all normalized, indecomposable rank two vector bundles on ℙ² with h¹(E(-1)) ≤ 4, using Serre's construction and cohomological invariants.
• To recover the Chiantini-Valabrega theorem on ℙ³ as a consequence of the ℙ² result, demonstrating the broader applicability of the cohomological bound.

Proposed method

• Use of Serre's construction to realize indecomposable rank two bundles as extensions of ideal sheaves of zero-dimensional subschemes.
• Application of Riemann-Roch theorem to compute Euler characteristics χ(E(k)) and relate them to cohomology groups h¹(E(k)).
• Use of Grauert-Mülich theorem to analyze the restriction of E to a general line, showing that h¹(E(k)) is non-increasing for k ≤ -1.
• Employment of Serre duality and stability conditions to show that h¹(E(-1)) > 0 for stable bundles, thus proving the splitting criterion.
• Analysis of the minimal twist r with h⁰(E(r)) ≠ 0 to classify bundles via their zero-locus schemes Z and their degrees c₂(E(r)).
• Construction of explicit resolutions for H⁰*(E) in the case h¹(E(-1)) ≤ 4, particularly for r = 2 and deg Z = 6 not contained in a conic.

Experimental results

Research questions

• RQ1 Does a normalized rank two vector bundle E on ℙ² split if and only if h¹(E(-1)) = 0, and is this the best possible cohomological condition?
• RQ2 Can the cohomological bound h¹(E(k)) ≤ h¹(E(-1)) for all k ∈ ℤ be established uniformly for all normalized rank two bundles on ℙ²?
• RQ3 What is the complete classification of normalized, indecomposable rank two vector bundles on ℙ² with h¹(E(-1)) ≤ 4?
• RQ4 Can the Chiantini-Valabrega theorem on ℙ³ be recovered as a consequence of the ℙ² result?
• RQ5 What are the possible resolutions of the module H⁰*(E) for bundles with h¹(E(-1)) ≤ 4, and how do they relate to the geometry of the zero-locus scheme Z?

Key findings

• A normalized rank two vector bundle E on ℙ² splits if and only if h¹(E(-1)) = 0, and this condition is sharp: for E = Ω(1), h¹(E(k)) = 0 for all k ≠ -1, yet E is indecomposable.
• For any normalized rank two bundle E on ℙ², h¹(E(k)) ≤ h¹(E(-1)) holds for all k ∈ ℤ, with equality in the stable case only when h¹(E(-1)) = 0.
• All normalized, indecomposable bundles with h¹(E(-1)) ≤ 4 are classified: they arise via Serre's construction from zero-dimensional subschemes Z of degree c₂(E(r)) with r = min{k | h⁰(E(k)) ≠ 0}.
• For h¹(E(-1)) = 1, the bundle E(r) has a section vanishing at a single point, and E admits a resolution 0 → 𝒪(−b−1) → 𝒪(−a) ⊕ 2𝒪(−b) → E → 0 with a ≤ b.
• For h¹(E(-1)) = 2, E(r) either vanishes along a degree-two subscheme or, in the stable case with c₁ = 0, c₂ = 2, r = 1, it vanishes along a degree-three subscheme not contained in a line.
• The resolution of H⁰*(E) is fully described for all such bundles, including the case r = 2 with deg Z = 6 not on a conic, which admits a resolution 0 → 3𝒪(−4) → 4𝒪(−3) → 𝒪_Z → 0.