Syzygies of abelian varieties

Abstract

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 R L = ∞ h=0 H 0 (X, L ⊗h) over the polynomial ring S L = ∞ h=0 Sym h H 0 (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 deg L ≥ 2g(X) + 1, and a theorem of Mattuck, Fujita and Saint-Donat states that if deg L ≥ 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 R L as a graded S L-module (notation as above) looks like 0 → · · · → E p → · · · → E 1 → E 0 → R L → 0 (1) where E 0 = S L ⊕ j S L (−a 0j) (where, since X is embedded by a complete linear system, a 0j ≥ 2 for any j), E 1 = j S L (−a 1j) (where, since the image of X in P(H 0 (L) ∨) is not contained in any hyperplane, a 1j ≥ 2 for any j), and, in general, for p ≥ 1, E p = j S L (−a pj) with a pj ≥ p + 1 for any j. Green introduced the following terminology: L is said to satisfy property N 0 if E 0 = S L. This means that the map S L → R L is surjective, i.e. that the embedded variety X is projectively normal. Moreover L is said to satisfy property N 1 if it satisfies N 0 and a 1j = 2 for any j, i.e. the homogeneous ideal of the embedded variety X is generated by quadrics. Inductively, one says that L satisfies condition N p if it satisfies condition N p−1 and a pj = p+1 for any j. So N 2 means that the relations between the quadrics defining X are generated by linear ones and, for arbitrary p ≥ 2, N p means that the first p − 1 maps of the resolution of the homogeneous ideal are matrices with linear entries. In a word N p means that, up to the p-th step, the resolution (1) is as "regular" as it could possibly be. The aforementioned theorem of Green states that if X is a curve and deg L ≥ 2g(X) + 1 + p, then L satisfies N p. This result stimulated many interesting questions (see [G1], [G2], [L] and [EL]). One of these