[Paper Review] Derived Categories of Twisted Sheaves on Elliptic Threefolds

Andrei Căldăraru|ArXiv.org|Dec 11, 2000

Algebraic structures and combinatorial modelsMathematics8 references20 citations

TL;DR

This paper establishes a derived equivalence between the derived category of coherent sheaves on a generic elliptic threefold without a section and the derived category of twisted sheaves (modules over an Azumaya algebra) on a small resolution of its relative Jacobian. The key contribution is constructing a Fourier-Mukai-type equivalence using a twisted sheaf when a universal sheaf fails to exist due to semistable sheaves and singularities.

ABSTRACT

We construct an equivalence between the derived category of sheaves on an elliptic threefold without a section and a derived category of twisted sheaves (modules over an Azumaya algebra) on any small resolution of its relative Jacobian.

Motivation & Objective

• To resolve the failure of the moduli problem for rank 1, degree 0 sheaves on elliptic threefolds without a section, which prevents the existence of a universal sheaf.
• To construct a derived equivalence between the derived category of coherent sheaves on such an elliptic threefold and a derived category of twisted sheaves on a small resolution of its relative Jacobian.
• To extend the classical Fourier-Mukai equivalence to non-fine moduli problems by incorporating Azumaya algebras and Brauer classes.
• To demonstrate that this equivalence is a global phenomenon, not induced by automorphisms of the Jacobian, by showing no automorphism maps the Brauer class to its power.

Proposed method

• Construct the relative Jacobian $ J \to S $ as the moduli space of semistable sheaves of rank 1, degree 0 on fibers of the elliptic fibration $ f: X \to S $, which admits a natural section.
• Resolve the singularities of $ J $ via an analytic small resolution $ \bar{J} $, since algebraic small resolutions may not exist.
• Identify the obstruction to the existence of a universal sheaf as a Brauer class $ \alpha \in \mathrm{Br}(\bar{J}) $, arising from non-uniqueness of local universal sheaves.
• Use the fact that $ X^k $, the relative moduli space of rank 1, degree $ k $ sheaves on fibers of $ X \to S $, is a fine moduli problem and hence has a universal sheaf.
• Establish an equivalence $ \mathbf{D}_{\mathrm{coh}}^b(X) \cong \mathbf{D}_{\mathrm{coh}}^b(X^k) $ via the universal sheaf on $ X \times_S X^k $, and identify $ X^k $ with a twisted sheaf on $ \bar{J} $ via the Brauer class $ \alpha^k $.
• Prove that the Brauer class $ \beta $ corresponding to $ X^k $ satisfies $ \beta = \alpha^k $ in $ \mathrm{Br}(\bar{J}) $, leading to the final derived equivalence $ \mathbf{D}_{\mathrm{coh}}^b(\bar{J}, \alpha) \cong \mathbf{D}_{\mathrm{coh}}^b(X) $.

Experimental results

Research questions

• RQ1 Can a derived equivalence be established between an elliptic threefold without a section and a derived category of twisted sheaves on a resolution of its relative Jacobian?
• RQ2 What is the obstruction to the existence of a universal sheaf on the relative moduli space when the moduli problem is not fine?
• RQ3 How does the Brauer class $ \alpha \in \mathrm{Br}(\bar{J}) $, arising from the non-fine moduli problem, relate to the geometry of the original elliptic threefold $ X $?
• RQ4 Is the derived equivalence between $ X $ and $ \bar{J} $ with twist $ \alpha $ induced by an automorphism of $ \bar{J} $, or is it a genuinely global phenomenon?
• RQ5 What are the implications of this equivalence for Calabi-Yau threefolds, particularly in relation to the Torelli problem and mirror symmetry?

Key findings

• The derived category of coherent sheaves on a generic elliptic threefold $ X $ without a section is equivalent to the derived category of twisted sheaves on a small resolution $ \bar{J} $ of its relative Jacobian, with twist given by a Brauer class $ \alpha $.
• The Brauer class $ \alpha \in \mathrm{Br}(\bar{J}) $ corresponds to the original fibration $ X \to S $, and has order 5 when $ X $ admits a 5-section, as per Ogg-Shafarevich theory.
• The equivalence $ \mathbf{D}_{\mathrm{coh}}^b(X) \cong \mathbf{D}_{\mathrm{coh}}^b(\bar{J}, \alpha) $ is not induced by an automorphism of $ \bar{J} $, as no automorphism maps $ \alpha $ to $ \alpha^2 $, proving it is a global phenomenon.
• The derived equivalence lifts to $ X^k $, the relative moduli space of rank 1, degree $ k $ sheaves, yielding $ \mathbf{D}_{\mathrm{coh}}^b(X) \cong \mathbf{D}_{\mathrm{coh}}^b(\bar{J}, \alpha^k) $, and $ X^k $ is Calabi-Yau if $ X $ is.
• The equivalence implies $ K_{X^k} = 0 $ and $ H^{2,0}(X^k) = 0 $, confirming that $ X^k $ is Calabi-Yau, providing counterexamples to the Torelli problem for Calabi-Yau threefolds.
• The construction provides a concrete example of a Fourier-Mukai transform involving a derived category of modules over an Azumaya algebra, extending classical results to non-fine moduli problems.