Autoduality holds for a degenerating abelian variety

Abstract

We prove that certain degenerate abelian varieties that include compactified Jacobians, namely stable semiabelic varieties, satisfy autoduality. We establish this result by proving a comparison theorem that relates the associated family of Picard schemes to the Néron model, a result of independent interest. In our proof, a key fact is that the total space of a suitable family of stable semiabelic varieties has rational singularities.Keywords: Autoduality, Compactified Jacobians, Stable abelic variety Mathematics Subject Classification: Primary 14H40; Secondary 14K30, 14D20In this paper, we prove that certain degenerate abelian varieties satisfy autoduality, a result we now recall. The classical statement of autoduality is a statement about an abelian variety J 0 with an ample divisor 0 that defines a principal polarization. If τ x 0 : J 0 → J 0 denotes translation by a point x 0 , then the morphismfrom J 0 to the (identity component of the) Picard scheme is an isomorphism. The Picard scheme Pic 0 (J 0 /k) is the dual abelian variety, so the fact that (1) is an isomorphism implies that the autoduality theorem holds, i.e., that J 0 is self-dual. Here we construct an isomorphism analogous to (1) in which the abelian variety is replaced by a singular projective variety, namely a stable semiabelic variety. Stable semiabelic varieties are degenerate abelian varieties studied in the moduli theory of abelian varieties in the work of, e.g., Alexeev, Nakamura, and Olsson. An important example of these varieties is the compactified Jacobian of a nodal curve, or moduli space of degree d rank 1, torsion-free sheaves which are required to satisfy a semistability condition when X 0 is reducible. A stable semiabelic variety is acted upon by a natural semiabelian variety J 0 which equals the moduli space of multidegree 0 line bundles when J is a compactified Jacobian.Because J 0 is a (possibly reducible) projective variety, we can form the (identity component of the) Picard scheme Pic 0 (J 0 /k), and Main Theorem of this paper is that there is an isomorphism from J 0 0 to Pic 0 (J 0 /k) that is analogous to (1):Main Theorem (Autoduality) The autoduality theorem holds for stable semiabelic varieties.

Branes on the singular locus of the Hitchin system via Borel and other parabolic subgroups

“…Such autoduality was stated via Fourier-Mukai equivalences by Arinkin [5,6] in the case of integral curves, and by Melo, Rapagnetta and Viviani [48,49] in the case of fine compactified Jacobians. Kass [42] extended the autoduality to the case of coarse compactified Jacobians, which is the one that concerns us, although his construction does not provide a Fourier-Mukai transform.…”