[Paper Review] ADE bundles over surfaces with ADE singularities

Yunxia Chen, Naichung Conan Leung|arXiv (Cornell University)|Sep 22, 2012

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper constructs ADE Lie algebra bundles and their minuscule representation bundles over compact complex surfaces with ADE singularities and pg=0, showing they can be deformed to descend to the singular surface via holomorphic structures on the minimal resolution. The key result is that such bundles descend if and only if certain cohomological conditions on exceptional curves are satisfied, linking geometric configurations of (−1)-curves to representation theory.

ABSTRACT

Given a complex projective surface with an ADE singularity and p_{g}=0, we construct ADE bundles over it and its minimal resolution. Furthermore, we descibe their minuscule representation bundles in terms of configurations of (reducible) (-1)-curves.

Motivation & Objective

• To construct ADE Lie algebra bundles over compact complex surfaces with ADE singularities and pg=0.
• To describe minuscule representation bundles in terms of configurations of (−1)-curves in the minimal resolution Y.
• To determine conditions under which these bundles can descend from the resolution Y to the singular surface X.
• To relate geometric data (−1)-curves and exceptional divisors) to representation-theoretic data (minuscule representations and symmetric forms).

Proposed method

• Use the minimal resolution π:Y→X of a surface X with ADE singularity to lift ADE root systems from the exceptional divisor’s dual graph.
• Construct a holomorphic Lie algebra bundle E₀ᵍ over Y as O_Y^⊕n ⊕ ⨁_{α∈Φ} O_Y(α), with Φ the ADE root system.
• Deform the holomorphic structure using φ∈Ω^{0,1}(Y,⨁_{α∈Φ⁺} O_Y(α)) such that ∂̄_φ = ∂̄ + ad(φ) defines a new holomorphic structure on E₀ᵍ.
• Prove that the deformed bundle E_φᵍ is trivial on each exceptional curve C_i iff [φ_{C_i}|_{C_i}] ≠ 0 in H¹(C_i, O_{C_i}(C_i)) ≅ ℂ.
• For minuscule representations V, construct the bundle L₀^{(ℊ,V)} = ⨁_{l: π(l)=C₀} O_Y(l) from (−1)-curves l above a (−1)-curve C₀ in X.
• Show that L₀^{(ℊ,V)} admits a holomorphic structure ∂̄_φ compatible with the action of E_φᵍ, and that it descends to X iff the cohomological condition holds.

Experimental results

Research questions

• RQ1 Under what conditions can an ADE Lie algebra bundle over the minimal resolution Y of a surface with ADE singularity descend to the singular surface X?
• RQ2 How are minuscule representation bundles of ADE Lie algebras realized geometrically via configurations of (−1)-curves in Y?
• RQ3 What role do cohomological classes [φ_{C_i}|_{C_i}] ∈ H¹(C_i, O_{C_i}(C_i)) play in the descent of holomorphic bundles from Y to X?
• RQ4 Can the Lie algebra bundle structure be preserved under deformation of holomorphic structures parametrized by φ ∈ Ω^{0,1}(Y,⨁_{α∈Φ⁺} O_Y(α))?
• RQ5 How is the symmetric tensor structure on minuscule representation bundles related to the geometry of (−1)-curves and the ADE root system?

Key findings

• The ADE Lie algebra bundle E₀ᵍ over Y can be deformed to a holomorphic bundle E_φᵍ via φ ∈ Ω^{0,1}(Y,⨁_{α∈Φ⁺} O_Y(α)) such that ∂̄_φ preserves the Lie algebra structure.
• The deformed bundle E_φᵍ descends to X if and only if [φ_{C_i}|_{C_i}] ≠ 0 in H¹(C_i, O_{C_i}(C_i)) for all i.
• For minuscule representations V, the bundle L₀^{(ℊ,V)} of line bundles over (−1)-curves l with π(l)=C₀ admits a holomorphic structure ∂̄_φ compatible with the E_φᵍ-action.
• When V is the standard representation, there exists a holomorphic symmetric multi-linear form f:⨂^r L_φ^{(ℊ,V)} → O_Y(D) with r=2,3,4 for ℊ=A_n,D_n,E_6,E_7 such that E_φᵍ ≅ aut₀(L_φ^{(ℊ,V)}, f).
• The first Chern class of E_φᵍ is zero, and c₂(E_φᵍ) = dim(ℊ) − rank(ℊ), showing the bundles are nontrivial.
• The construction is not unique: different choices of φ can yield non-isomorphic holomorphic structures, as shown by replacing φ_{α} with φ_{α}+ψ for ψ≠0 in H¹(Y,O_Y(α)).