[Paper Review] Relatively stable bundles over elliptic fibrations

C. Bartocci, U. Bruzzo+2 more|arXiv (Cornell University)|Sep 18, 2001

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper generalizes Atiyah's theory of semistable vector bundles on elliptic curves to elliptic fibrations over an arbitrary base scheme using a relative Fourier-Mukai transform. It proves that this transform preserves relative (semi)stability for sheaves of positive relative degree, establishing a duality between moduli functors of sheaves and providing a geometric realization of the relative geometric Langlands correspondence via isomorphisms between moduli spaces and compactified relative Jacobians.

ABSTRACT

We consider a relative Fourier-Mukai transform defined on elliptic fibrations over an arbitrary normal base scheme. This is used to construct relative Atiyah sheaves and generalize Atiyah's and Tu's results about semistable sheaves over elliptic curves to the case of elliptic fibrations. Moreover we show that this transform preserves relative (semi)stability of sheaves of positive relative degree.

Motivation & Objective

• To extend the classical theory of semistable vector bundles on elliptic curves to the relative setting of elliptic fibrations over an arbitrary base scheme.
• To define relative Atiyah sheaves in the context of elliptic fibrations using the relative Fourier-Mukai transform.
• To establish that the relative Fourier-Mukai transform preserves relative (semi)stability for sheaves of positive relative degree.
• To construct an isomorphism between the compactified relative Jacobian and the moduli space of relatively stable sheaves of rank n and degree -1.
• To realize a duality between two moduli functors: one parametrizing extensions of sheaves via relative Atiyah sheaves, and another parametrizing sheaves with torsion subsheaves supported on sections.

Proposed method

• The authors use a relative Fourier-Mukai transform defined on elliptic fibrations $ p: X \to B $, generalizing Mukai's construction for abelian schemes.
• They define relative Atiyah sheaves $ \mathcal{A}_i $ inductively via exact sequences $ 0 \to \mathcal{O}_C \to \mathcal{A}_n \to \mathcal{A}_{n-1} \to 0 $, extended to the relative setting over $ X $.
• The transform is applied to sheaves of positive relative degree, and its compatibility with relative (semi)stability is proven via properties of the compactified relative Jacobian $ \widehat{X} $.
• The duality is established by constructing an isomorphism between the functors $ \mathbf{F}^i_X(n,d) $ and $ \mathbf{H}^i_{\widehat{X}}(d,-n) $, using the relative Fourier-Mukai transform as a bridge.
• The isomorphism between $ \widehat{X} \times_B \mathcal{M}(n,d) $ and $ X \times_B \mathcal{M}(d,-n) $ is realized via the map $ (\varpi^{-1} \circ \iota, \mathbf{S}^0) $, where $ \iota $ is the dualizing involution on $ \widehat{X} $.
• The construction relies on the existence of a section $ e $ of $ p $, which induces a relative polarization $ \Theta $, and on the isomorphism $ \varpi: X \to \widehat{X} $ sending points to twisted ideal sheaves.

Experimental results

Research questions

• RQ1 How can Atiyah’s theory of semistable bundles on elliptic curves be generalized to the relative case of elliptic fibrations over an arbitrary base scheme?
• RQ2 Does the relative Fourier-Mukai transform preserve relative (semi)stability for sheaves of positive relative degree on elliptic fibrations?
• RQ3 Can the moduli space of relatively stable sheaves of rank $ n $ and degree $ -1 $ be naturally identified with the compactified relative Jacobian of torsion-free, rank-1 sheaves of degree $ n $?
• RQ4 What is the precise duality between the functors parametrizing extensions of sheaves via relative Atiyah sheaves and those parametrizing sheaves with torsion subsheaves supported on sections?
• RQ5 How does the relative Fourier-Mukai transform relate to the geometric Langlands correspondence in the relative elliptic setting?

Key findings

• The relative Fourier-Mukai transform preserves relative (semi)stability for sheaves of positive relative degree on elliptic fibrations over an arbitrary base scheme.
• The compactified relative Jacobian $ \widehat{J}_n $ of torsion-free, rank-1 sheaves of degree $ n $ is naturally isomorphic to the moduli space $ \mathcal{M}(n,-1) $ of relatively stable sheaves of rank $ n $ and degree $ -1 $.
• The relative Atiyah sheaf $ \mathcal{A}_i $ is defined via an inductive exact sequence $ 0 \to \mathcal{O}_C \to \mathcal{A}_n \to \mathcal{A}_{n-1} \to 0 $, extended to the relative setting over $ X $.
• The functors $ \mathbf{F}^i_X(n,d) $ and $ \mathbf{H}^i_{\widehat{X}}(d,-n) $ are isomorphic via the relative Fourier-Mukai transform, establishing a duality between extension classes and torsion sheaves on sections.
• The isomorphism $ \widehat{X} \times_B \mathcal{M}(n,d) \cong X \times_B \mathcal{M}(d,-n) $ is realized through the map $ (\varpi^{-1} \circ \iota, \mathbf{S}^0) $, where $ \iota $ is the dualizing involution and $ \mathbf{S}^0 $ induces the duality on moduli spaces.
• The result generalizes Laumon’s geometric Langlands correspondence to the relative elliptic case, realizing the duality between operators $ S $ and $ T_p $ in Drinfeld’s framework.