Syzygies of Abelian Varieties

Let A be an ample line bundle on an abelian variety X (over an algebraically closed field). A theorem of Koizumi ([Ko], [S]), developing Mumford’s ideas and results ([M1]), states that if m ≥ 3 the line bundle L = A⊗m embeds X in projective space as a projectively normal variety. Moreover, a celebrated theorem of Mumford ([M2]), slightly refined by Kempf ([K4]), asserts that the homogeneous ideal of X is generated by quadrics as soon as m ≥ 4. Such results turn out to be particular cases of a statement, conjectured by Rob Lazarsfeld, concerning the minimal resolution of the graded algebra RL = ⊕∞ h=0H 0(X,L⊗h) over the polynomial ring SL = ⊕∞ h=0 Sym H(X,L). The purpose of this paper is to prove Lazarsfeld’s conjecture. To put such matters into perspective, it is useful to review the case of projective curves. A classical theorem of Castelnuovo states that a curve X , embedded in projective space by a complete linear system |L|, is projectively normal as soon as degL ≥ 2g(X) + 1, and a theorem of Mattuck, Fujita and Saint-Donat states that if degL ≥ 2g(X) + 2, then the homogeneous ideal of X is generated by quadrics. Green ([G1]) unified, re-interpreted and generalized these results to a statement about syzygies. Specifically, given a (smooth) projective variety X and a very ample line bundle L on X , a minimal resolution of RL as a graded SL-module (notation as above) looks like