Characterizing Jacobians via flexes of the Kummer variety.

*(English)*Zbl 1098.14020The classical Riemann-Schottky problem consists of characterizing the locus of Jacobian varieties inside the space of abelian varieties. During the last decades, approaches of different flavors for solving this problem have been developed. For general surveys we address the reader the papers [E. Arbarello, C. De Concini, in: Global geometry and mathematical physics, Lect. 2nd Sess. CIME, Montecatini Terme/Italy 1988, Lect. Notes Math. 1451, 95–137 (1990; Zbl 0726.14029)] and [G. van der Geer, in: Arbeitstag. Bonn 1984, Proc. Meet. Max-Planck-Inst. Math., Bonn 1984, Lect. Notes Math. 1111, 385–406 (1985; Zbl 0598.14027)].

One of these approaches, that based on differential equations, fructified and yielded the famous Shiota characterization of Jacobians in terms of the KP equation [T. Shiota, Invent. Math. 83, 333–382 (1986; Zbl 0621.35097)]. Another approach is based on geometric properties of the Kummer varieties. Recall that given a principally polarized abelian variety (p.p.a.v.) its Kummer variety is defined as its image of the morphism given by the second order theta functions. From the works of R. C. Gunning [Invent. Math. 66, 377–389 (1982; Zbl 0485.14009)] and G. E. Welters [Ann. Math. (2) 120, 497–504 (1984; Zbl 0574.14027)] it follows that Jacobian varieties are those p.p.a.v. whose Kummer variety admits a length 3 formal jet of flexes. Similarly to Shiota’s result that claims that only the first equation of the KP hierarchy suffices to characterize Jacobians, Welters conjectured whether the existence of only one trisecant (instead of a family) would characterize Jacobians. There are three cases of Welter’s conjecture and they correspond to the three possible configurations of the intersection points of the Kummer variety and the trisecant: (i) all three points coincide, (ii) two of them coincide; (iii) all three intersection points are distinct. The first of these cases is proven in the present paper.

Indeed, the authors prove that an indecomposable abelian variety \(X\) is the Jacobian of a curve if and only if there exists a point \(a=2b\in X-\{0\}\) such that \(<a>\) is irreducible and the image of \(2b\) in the Kummer variety is a flex. The proof is based on a detailed study of flows on the p.p.a.v. preserving the theta divisor and the fact that the wave function can be understood as the germ of the Baker-Akhiezer function. Finally, it is worth mentioning two subsequent preprints of the second author. In the first one [I. Krichever, Integrable linear equations and the Riemann-Schottky problem, math. AG/0504192) some restrictions of the given characterization are removed while in the second one [I. Krichever, Characterizing Jacobians via trisecants of the Kummer variety, math.AG/0605625] the remaining two cases are studied.

One of these approaches, that based on differential equations, fructified and yielded the famous Shiota characterization of Jacobians in terms of the KP equation [T. Shiota, Invent. Math. 83, 333–382 (1986; Zbl 0621.35097)]. Another approach is based on geometric properties of the Kummer varieties. Recall that given a principally polarized abelian variety (p.p.a.v.) its Kummer variety is defined as its image of the morphism given by the second order theta functions. From the works of R. C. Gunning [Invent. Math. 66, 377–389 (1982; Zbl 0485.14009)] and G. E. Welters [Ann. Math. (2) 120, 497–504 (1984; Zbl 0574.14027)] it follows that Jacobian varieties are those p.p.a.v. whose Kummer variety admits a length 3 formal jet of flexes. Similarly to Shiota’s result that claims that only the first equation of the KP hierarchy suffices to characterize Jacobians, Welters conjectured whether the existence of only one trisecant (instead of a family) would characterize Jacobians. There are three cases of Welter’s conjecture and they correspond to the three possible configurations of the intersection points of the Kummer variety and the trisecant: (i) all three points coincide, (ii) two of them coincide; (iii) all three intersection points are distinct. The first of these cases is proven in the present paper.

Indeed, the authors prove that an indecomposable abelian variety \(X\) is the Jacobian of a curve if and only if there exists a point \(a=2b\in X-\{0\}\) such that \(<a>\) is irreducible and the image of \(2b\) in the Kummer variety is a flex. The proof is based on a detailed study of flows on the p.p.a.v. preserving the theta divisor and the fact that the wave function can be understood as the germ of the Baker-Akhiezer function. Finally, it is worth mentioning two subsequent preprints of the second author. In the first one [I. Krichever, Integrable linear equations and the Riemann-Schottky problem, math. AG/0504192) some restrictions of the given characterization are removed while in the second one [I. Krichever, Characterizing Jacobians via trisecants of the Kummer variety, math.AG/0605625] the remaining two cases are studied.

Reviewer: Francisco José Plaza Martín (Salamanca)