Proceedings of the International Congress of Mathematicians
Berkeley, California, USA, 1986
Periods of Abelian Integrals, Theta Functions,
and Differential Equations of KdV Type
E. ARBARELLO
The generalized Siegel upper half plane Hg c C9^1^2
is the set of complex
g X g symmetric matrices with positive definite imaginary part. Given a matrix
r E Mg one can construct the complex torus XT = OF/ZQ + rZ9 together with
Riemann's theta function
exp 2iri < - fprp + lpz \.
6(z, r) = ^
pezs
{
'
Consider now a compact Riemann surface C of genus g. Fix a symplectic basis a\,..., ag, bi,..., bg of Hi (C, Z) and a basis u)i,..., ug of the vector space
of homomorphic differentials on G having the property that J a . u)j = Sij. As
Riemann noticed, the matrix r(G) of the "ö-periods" is symmetric and has positive imaginary part. The matrices of Mg that are obtained in this way form a
(3<7 — 3)-dimensional analytic subvariety Ig of )ig which is called the Jacobian
locus. To each point r(C) of Ig corresponds the Jacobian variety J(C) = XT^c)
of C, together with the theta function 6(z,r(C)).
The problem of finding explicit equations for Ig inside Mg, goes back to Riemann, Schottky, and Jung [R], [S], [S-J].
In his study of solutions of nonlinear equations of Korteveg de Vries type,
Krichever [K] proved the following fact:
Fix a compact Riemann surface C of genus g > 0 and consider its theta
function 0(z), then there exist three vectors W^ ,W^ ,W^ inC9, withW^
^
0, such that, for every z E C9 the function
u(x, y, t\z) = ^
log0(zWM 4- yW{2) + tW™ + z)
(1)
satisfies the so-called Kodomcev-Petviashvili equation
Suyy = —(ut~
Suux - 2uxxx).
(2)
S. P. Novikov [N] then conjectured that a matrix r E Mg, which is not in
block form, is a Riemann matrix if and only if the corresponding theta function
© 1987 International Congress of Mathematicians 1986
623
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E. ARB ARELIX)
satisfies the K.P. equation, in the sense we just explained. A strong indication
of the validity of this conjecture was given by Dubrovin in [DU2].
Following work of Mumford [M2] and Mulase [M], Novikov's conjecture has
been recently proved by Shiota [S].
A more geometrical approach to the same problem originated from the following "trisecant formula," discovered by Fay [FY]. Fix a compact Riemann surface
C of genus g > 0. Look at the Abel-Jacobi map of G into its Jacobian and denote
by T the isomorphic image of G under this map. Fay's trisecant formula says
that given any three points a, ß, 7 on T and any point f on ^(T — a — ß — 7),
there exist constants ci and c2 such that
0(z - a)9(z + 2^ + a) = Cl6(z - ß)0(z + 2ç + ß) + c20(z - i)0(z + 2^ + 7). (3)
As Mumford noticed [M4], when a, ß, 7 tend to 0 on V, one then gets the
equation
D\e-e-
w\e • e + Z(D\O)2 - S(D2O)2
(4)
+ 3D%0 • 0 + 3Di0 • D30 - 3DiD30 -0 + d40-0 = O
where d4 is a suitable constant and where Di, D2, Ds are the constant vector
fields on J(C) obtained as follows. Let W^\ W^2\ W^ be, respectively, the
tangent, the normal, and the binormal vector to V at 0, and set
3= 1
°^
A computation shows that the K.P. equation (2) is equivalent to (4) via the
substitution (1).
One can then express Novikov's conjecture as follows.
(5) THEOREM ([S] AND [AD2]). LetX = XT be an irreducible principally
polarized abelian variety (i.p.p.a.v.) of dimension g>l. Then X is the Jacobian
of a compact Riemann surface of genus g if and only if there exist a constant d4
and constant vector fields Di ^ 0 , D2, Ds on X such that the equation (4) is
satisfied.
We shall sketch a proof of this theorem following the lines of [AD2]. Let
0 C X be the theta divisor. It is well known that the 29 functions
0[n](z,r)=
] T exp2wi{t(p + n)T(p + n) + 2t(p + n)z},
pezo
nE\Z9/Z9,
form a basis of H°(X, 26) and that, as Riemann discovered,
0(z + ç)0(*-i)=
E
Ô[n](z)6[n}(ç).
(6)
Fay's identity (3) can then be geometrically interpreted in terms of the Kummer
map T : X - • P N , JV = 29 - 1, given by T(ç) = [•••, 0[n](f),...]. In fact when
AT is a Jacobian, using (6), the trisecant formula becomes
7(S + a)=ci~0(s + ß) + c2~0(s + 1),
\fse\(T-a-ß-i).
(7)
PERIODS OF ABELIAN INTEGRALS
625
This says that the points 0 (ç + a), 0 (ç + ß), 0 (£ + 7) lie on a line which is
then a trisecant to the Kummer variety 0 (X).
For fixed triple of points a, ß, 7 on any principally polarized abelian variety
X, Gunning [G] introduces the subscheme Va$n C X defined by
Kßn = ( f e X: t(ç + a)A 7{ç + ß) A ~0(ç + 7) = 0},
and proves that: an i.p.p.a.v. X is a Jacobian if and only if there is a positive
dimensional component of 2Vaß1 passing through the point a - ß and if there is
no complex multiplication mapping ß — a and 7 — a into 0.
Inspired by this criterion and motivated by the relation between the trisecant
formula and the K.P. equation, Welters [W2] gives an infinitesimal version of
this criterion by substituting the points a, ß, 7 with a second order germ of
curve
(8)
£^WW£ + WW£2
at the origin in X. Welters's criterion is the following:
An i.p.p.a.v. X is a Jacobian if and only if there exist constant vector fields
Di ^ 0, D2 on X such that the subscheme
V = {ÇEX: 7 ( f ) A Pi 7(f) A (Dl + D2)l(()
= 0}
has a positive dimensional component passing through the origin.
As Welters remarks, the subscheme V coincides at the origin of X, up to
second order, with the germ (8). Therefore, the condition of Welters's criterion
can be expressed by saying that there exist vectors W^\W^,...
such that the
formal germ ((e) = J2ili W^e* is contained in V, i.e.,
7 ( f (e)) A Di 7 ( f (e)) A S 8 7 ( f (ff)) = 0
(Ä2 = Dl + D2).
(9)
Now set
d
3=1
A8 = As(D1,...,Da)=
°%
rrVi^i1---^
£
2l-f2î 2 H
)-sia=3
A computation contained in [ADI] shows that, up to a renormalization of the
£Vs, (9) is equivalent to the existence of constants d{, i = 1,2,..., with di —
d2 = ds = 0, such that
£*+i^
^>o
?(({e)) + Dit(ç(e)) - eÂ2t(ç(e)) = 0.
(10)
J
Equating to zero the coefficients of es in this relation and using (6), the condition
(10) is equivalent to the equations
A 5 Ai - A 5 A 2 + J2
i=l
d
i+iA*-i
0(* + f)0(s-f)lc=o = O,
*>i.
(11)
626
E. ARBARELLO
The left-hand side of (11) only depends on the knowledge of
Di,...,D3,
d±,..., d3+i and we denote it by P8(z). Clearly Pi(2) and P2(z) are identically
zero and, as expected, the equation P3 = 0 is the K.P. equation. Welters's
criterion is then translated in the following (cf. [ADI]).
An i.p.p.a.v. X is a Jacobian if and only if there exist constant
vector fields Di ^ 0, D2, Ds,... and complex numbers e/4, d s , . . .
such that P8(z) = 0, s > 1.
(12)
(It is interesting to notice that equations (11) are all part of the so-called K.P.
hierarchy (see [D] and [ADI]).
The next step in the proof of (5) consists in restricting equation (11) to the
subscheme DiO C X defined by the simultaneous vanishing of 0 and Di0. The
first observation is that P8(z) is a section of Ox(20). This section can be written
as
p3(z) = P'9(z) + 2DiD30 • 0 - 2Di0 • Ds0 + d 5 + 1 0 • 0
where Pf8(z) only depends on the knowledge of D\,..., D8-i, d4,...,d8. Looking
at the cohomology sequences given by restricting 0(26) to 6 and Di&, one can
prove the following statement. Given Di,..., D8-i, d±,..., d8, there exist D8
and d s +i such that P8(z) = 0 if and only if Pfs\ 2^e = 0 or, equivalently, if and
only if one can find a meromorphic solution in C9 to the equation
Dlh = P'J02.
(13)
The existence of D8 and rfs+i is thus translated into analytical terms. Via a
cohomological argument one then shows that to find a global solution of (13) it
is in fact sufficient to find local solutions near every point of
U = C9\{z E C9 : 0(z) = Dx0(z) = 0}.
The solution of the local problem is easy. First of all one sees that the local
integrability condition is given by Y(P'3/02) = 0 where
r = 0 4 (\D\ +D2 - DiDs
+4Di(D2\og0)Di),
then a direct computation shows that the integrability condition holds under
the assumption that P3 = • • • = P3_t = 0. The conclusion is that, under this
assumption, there exists D8 and d s +i such that P8 = 0. The theorem follows,
by induction, from (12).
REFERENCES
[ADI] E. Arbarello and C. De Concini, On a set of equations characterizing
Riemann
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[BD] A. Beauville and O. Debarre, Une relation entre deux approches du problème de
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