1 O ct 1 99 9 TORELLI THEOREM VIA FOURIER - MUKAI TRANSFORM

We show that the Fourier transform on the Jacobian of a curve interchanges " δ functions " on the curve and the theta divisor. The Torelli theorem is an immediate consequence. 1. Statement of the theorem. 1.1. We live over an algebraically closed base field k. Let J be an abelian variety equipped with a principal polarization θ : J ∼ → J • = Pic 0 (J), so we have the corresponding Fourier transform F on the derived category of quasi-coherent sheaves D(J, O) (see [6]). Let Θ be the theta divisor in J. Notice that Θ is defined up to translation, and any non-trivial translation does not preserve Θ. So we may consider Θ as a canonically defined algebraic variety equipped with a J-torsor of embeddings j : Θ ֒→ J; we call these j's standard embeddings. Denote by Θ ns the open subset of smooth points of Θ. For a standard embedding j let j ns : Θ ns ֒→ J be its restriction to Θ ns. Our Θ carries a canonical involution x → x ν ; this is the unique involution such that for any standard embedding j the embedding j ν : x → −j(x ν) is also standard. For a line bundle L on Θ or Θ ns set L ν := ν * L. The pull-back j * F of an O J-module F does not change if we translate both j and F by the same element of J. Thus the image of j ns * : Pic(J) → Pic(Θ ns) is a canonically defined subgroup of Pic(Θ ns) (it does not depend on j). Denote by A(J) the corresponding quotient group. Let T ⊂ Pic(Θ ns) be the subset of line bundles L such that (i) L · L ν = ω Θ ns (ii) A(J) is generated by the image of L. Remark. Since the tangent bundle to J is trivial, one has ω Θ ns = j ns * O J (j(Θ)). Thus, if T is non-empty then ν acts on A(J) as −1.