[Paper Review] Tannakian Categories attached to abelian Varieties

Rainer Weissauer|ArXiv.org|Apr 10, 2007

Algebraic Geometry and Number Theory2 references3 citations

TL;DR

This paper constructs a super-Tannakian category from the derived category of $\overline{\mathbb{Q}}_l$-sheaves on an abelian variety $X$, quotiented by translation-invariant complexes, and shows that the category of multipliers—complexes closed under convolution—forms a rigid, semisimple tensor category equivalent to the representation category of a pro-supergroup $G(X)$. The key result identifies $G(X,Y)$ for a principally polarized abelian variety as $Sp(2g-2,\overline{\mathbb{Q}}_l)/\mu_{g-1}[2]$ or $Sl(2g-2,\overline{\mathbb{Q}}_l)/\mu_{g-1}$, linking the Tannakian group to the geometry of Jacobians.

ABSTRACT

Starting from certain perverse sheaves on an abelian variety, including the intersection cohomology sheaves of curves and smooth ample divisors, we construct a semisimple super-Tannakian category.

Motivation & Objective

• To construct a Tannakian category from the derived category of $\overline{\mathbb{Q}}_l$-sheaves on an abelian variety modulo translation-invariant complexes.
• To identify the category of multipliers—complexes closed under convolution—as a rigid, semisimple tensor category.
• To show that the Tannakian group of the category of multipliers on a principally polarized abelian variety is isomorphic to $Sp(2g-2,\overline{\mathbb{Q}}_l)/\mu_{g-1}[2]$ or $Sl(2g-2,\overline{\mathbb{Q}}_l)/\mu_{g-1}$, depending on the curve's hyperellipticity.
• To conjecture that a principally polarized abelian variety arises as a Jacobian if and only if its Tannakian group is one of these two groups.

Proposed method

• Define the quotient category $\overline{D}_{c}^{b}(X,\overline{\mathbb{Q}}_l)$ by factoring out translation-invariant complexes.
• Introduce the notion of a 'multiplier' as a perverse sheaf whose convolution preserves the quotient category $\overline{Perv}(X)$.
• Establish that the category $\overline{M}(X)$ of multipliers is a rigid, semisimple, $\overline{\mathbb{Q}}_l$-linear tensor category.
• Use the fiber functor $\omega$ to realize $\overline{M}(X)$ as the category of representations of a pro-supergroup $G(X)$.
• Apply the Tannakian formalism to show $\overline{M}(X)$ is equivalent to $Rep(G(X), \varepsilon)$, with $G(X)$ a projective limit of supergroups.
• For a principal polarization $Y$, compute the Tannakian group $G(X,Y)$ of the subcategory generated by $\delta_Y$, showing it is $Sp(2g-2,\overline{\mathbb{Q}}_l)/\mu_{g-1}[2]$ or $Sl(2g-2,\overline{\mathbb{Q}}_l)/\mu_{g-1}$.

Experimental results

Research questions

• RQ1 When is the Tannakian group of the category of multipliers on an abelian variety isomorphic to $Sp(2g-2,\overline{\mathbb{Q}}_l)/\mu_{g-1}[2]$ or $Sl(2g-2,\overline{\mathbb{Q}}_l)/\mu_{g-1}$?
• RQ2 What conditions ensure that a divisor $Y$ in an abelian variety $X$ is a multiplier, i.e., closed under convolution in the quotient category?
• RQ3 How does the Tannakian group $G(X,Y)$ of a principally polarized abelian variety relate to the geometry of the associated curve?
• RQ4 Can the Tannakian group $G(X,Y)$ uniquely characterize the abelian variety as a Jacobian of a curve?
• RQ5 What role does the weight filtration and purity play in the construction of the Tannakian category for abelian varieties in characteristic zero or positive characteristic?

Key findings

• The category $\overline{M}(X)$ of multipliers on an abelian variety $X$ is a rigid, semisimple, $\overline{\mathbb{Q}}_l$-linear tensor category, equivalent to the representation category of a pro-supergroup $G(X)$.
• The Tannakian group $G(X,Y)$ for a principal polarization $Y$ is isomorphic to $Sp(2g-2,\overline{\mathbb{Q}}_l)/\mu_{g-1}[2]$ if the associated curve is hyperelliptic, and to $Sl(2g-2,\overline{\mathbb{Q}}_l)/\mu_{g-1}$ otherwise.
• The representation $W = \omega(\delta_Y)$ of $G(X,Y)$ is the unique irreducible representation of highest weight occurring in the $(g-1)$-st exterior power of the standard representation of $G(X,Y)$.
• The intersection cohomology sheaf $\delta_Y$ of a smooth ample divisor $Y$ is a multiplier, and its convolution preserves the quotient category $\overline{Perv}(X)$.
• The convolution product on $\overline{D}_{c}^{b}(X,\overline{\mathbb{Q}}_l)$ is well-defined and respects the tensor structure, enabling the Tannakian formalism.
• The conjecture is supported by the fact that the Tannakian group $G(X,Y)$ matches the known group for Jacobians of curves, suggesting a characterization of Jacobians via their Tannakian groups.