Schottky problem

To a complete non-singular algebraic curve $ C $ of genus $ g $ one can associate its Jacobian (cf. Jacobi variety). This is an Abelian variety $ J( C) $ of dimension $ g $ together with a principal polarization $ \Theta $( cf. Polarized algebraic variety). B. Riemann showed in 1857 that algebraic curves of genus $ g $ depend on $ 3g- 3 $ parameters (for $ g > 1 $). But principally-polarized Abelian varieties of dimension $ g $ depend on $ {g( g+ 1)/2 } $ parameters. Since for $ g \geq 4 $ one has $ g( g+ 1)/2 > 3g- 3 $, the question arises which principally-polarized Abelian varieties are Jacobians. This question was posed by Riemann, but is known as the Schottky problem. More precisely, if $ {\mathcal M} _ {g} $ is the moduli space of curves of genus $ g $( i.e. the parameter space of isomorphism classes of such curves of genus $ g $, cf. Moduli theory) and if $ {\mathcal A} _ {g} $ is the moduli space of principally-polarized Abelian varieties of dimension $ g $, there is a mapping $ {\mathcal M} _ {g} \rightarrow {\mathcal A} _ {g} $, and the problem is to characterize the closure $ J _ {g} $ of its image. For $ g \leq 3 $ one has $ J _ {g} = {\mathcal A} _ {g} $.

Here only the case where the curve is a complex curve or Riemann surface is considered. If one chooses a symplectic basis $ \alpha _ {1} \dots \alpha _ {g} $, $ \beta _ {1} \dots \beta _ {g} $ of the homology $ H _ {1} ( C, \mathbf Z ) $ and a basis of the space of holomorphic $ 1 $- forms $ \omega _ {1} \dots \omega _ {g} $ on $ C $ such that $ \int _ {\alpha _ {i} } \omega _ {j} = \delta _ {ij } $( Kronecker delta), one obtains the period matrix $ \tau = ( \tau _ {ij } )= ( \int _ {\beta _ {i} } \omega _ {j} ) $. This matrix lies in the Siegel upper half-space $ {\mathcal H} _ {g} $, the set of all complex symmetric $ ( g \times g ) $- matrices whose imaginary part is positive definite.

The Jacobian $ J( C) $ is given by the complex torus $ \mathbf C ^ {g} / \Lambda _ \tau $, where $ \Lambda _ \tau = \mathbf Z ^ {g} + \tau \mathbf Z ^ {g} $, and $ \Theta $ is then the divisor of Riemann's theta-function

$$ \theta ( \tau , z) = \sum _ {m \in \mathbf Z ^ {g} } e ^ {\pi i ( ^ {t} m \tau m + 2 ^ {t} mz) } . $$

The moduli space $ {\mathcal A} _ {g} $ can be obtained as $ {\mathcal H} _ {g} / \mathop{\rm Sp} ( 2g, \mathbf Z ) $, where $ \mathop{\rm Sp} ( 2g, \mathbf Z ) $ acts naturally on $ {\mathcal H} _ {g} $. Coordinates on (a covering of) $ {\mathcal A} _ {g} $ are provided by the "theta constanttheta constants" , which are the values at $ z= 0 $ of the theta-functions

$$ \theta \left [ \begin{array}{c} \epsilon ^ \prime \\ \epsilon ^ {\prime\prime } \\ \end{array} \right ] ( \tau , z) = \sum _ {m \in \mathbf Z ^ {g} } e ^ {\pi i ( ^ {t} ( m+ \epsilon ^ \prime ) \tau ( m+ \epsilon ^ \prime ) + 2 ^ {t} ( m+ \epsilon ^ \prime )( z+ \epsilon ^ {\prime\prime } )) } $$

for $ \epsilon ^ \prime , \epsilon ^ {\prime\prime } \in ( 1 / 2) \mathbf Z ^ {g} / \mathbf Z ^ {g} $.

The first result is due to W. Schottky for $ g = 4 $. He showed in 1888 that a certain polynomial of degree 16 in the theta constants vanished on $ J _ {4} $, but not everywhere on $ {\mathcal A} _ {4} $. J.-I. Igusa showed much later that its zero divisor equals $ J _ {4} $.

The next step was made by Schottky and H.W.E. Jung in 1909, who constructed expressions in the theta constants that vanish on $ J _ {g} $ by means of double unramified coverings of curves. (Today this is called the theory of Prym varieties.) These expressions define a certain locus $ S _ {g} $( called the Schottky locus) which contains $ J _ {g} $. It is conjectured that $ S _ {g} = J _ {g} $, and this is the Schottky problem in restricted sense. B. van Geemen showed in 1983 that $ J _ {g} $ is an irreducible component of $ S _ {g} $.

Since the Schottky problem asks for a characterization, different approaches might lead to different answers. One approach uses the fact that the theta divisor of a Jacobian is singular (for $ g \geq 4 $): the dimension of its singular locus is $ \geq g- 4 $. It is therefore natural to consider the locus $ N _ {g} ^ {m} \subset {\mathcal A} _ {g} $ of principally-polarized Abelian varieties $ ( A, \Theta ) $ of dimension $ g $ with $ \mathop{\rm dim} ( \mathop{\rm Sing} ( \Theta )) \geq m $. Then $ J _ {g} \subset N _ {g} ^ {g- 4 } $, and $ J _ {g} $ is indeed an irreducible component, as A. Andreotti and A. Mayer showed. However, $ N _ {g} ^ {g- 4 } $ has other components, so this is not strong enough.

Another way to try to distinguish Jacobians and general principally-polarized Abelian varieties uses trisecants. Below, only principally-polarized Abelian varieties that are indecomposable, i.e. not products, are considered. For such an Abelian variety $ ( X= \mathbf C ^ {g} / \Lambda _ \tau , \Theta ) $ there is a mapping to projective space $ \Phi = \Phi _ {2 \Theta } : X \rightarrow \mathbf P ^ {N} $ with $ N= 2 ^ {g} - 1 $ given by the theta-functions $ \theta [ \begin{array}{c} \epsilon ^ \prime \\ 0 \end{array} ]( 2 \tau , 2z) $ and its image is the Kummer variety $ K( X, \Theta ) $( cf. Kummer surface). If $ X = J( C) $ and $ a, b, c, d \in C $ and if $ r \in X $ is such that $ 2r= a+ b- c- d $, then the three points $ \Phi ( r) $, $ \Phi ( r- b+ c) $ and $ \Phi ( r- b+ d) $ in $ \mathbf P ^ {N} $ are collinear, i.e. define a trisecant. (Translated in terms of theta-functions this is Fay's trisecant identity.) It is conjectured that if $ ( X, \Theta ) $ is an indecomposable principally-polarized Abelian variety whose Kummer variety admits a trisecant, then it is a Jacobian. (It was proved by A. Beauville and O. Debarre that the existence of a trisecant implies $ \mathop{\rm dim} {\textrm{ Sing } } ( \Theta ) \geq g- 4 $.) A weakened version of this is proved by R.C. Gunning and G. Welters. A simplified form reads: If the Kummer variety admits a continuous family of trisecants and $ \mathop{\rm dim} {\textrm{ Sing } } ( \Theta ) = g- 4 $, then the Abelian variety is a Jacobian.

One can consider an infinitesimal version of this. Let $ \theta _ {2} [ \sigma ]( \tau , z) = \theta [ \begin{array}{c} \sigma \\ 0 \end{array} ]( 2 \tau , 2z) $. Then the criterion is: The Abelian variety is a Jacobian if there exist constant vector fields $ D _ {1} $, $ D _ {2} $, $ D _ {3} $ on $ X $ and $ d \in \mathbf C $ such that

$$ \tag{* } \left ( \left ( D _ {1} ^ {4} - D _ {1} D _ {3+} { \frac{3}{4} } D _ {2} ^ {2} + d \right ) \theta _ {2} [ \sigma ] \right ) ( \tau , 0) = 0 \ {\textrm{ for all } } \sigma . $$

This is known as Novikov's conjecture and was proved by M. Shiota in 1986. Another (more geometric) proof was given by E. Arbarello and C. De Concini. The equation (*) is called the K–P-equation (after Kadomtsev–Petviashvili) and generalizes the Korteweg–de Vries equation. It is the first of a whole hierarchy of equations.

By the Riemann identity

$$ \theta ( x+ y) \theta ( x- y) = \sum _ \sigma \theta _ {2} [ \sigma ]( x) \theta _ {2} [ \sigma ]( y) , $$

this translates into the fact that $ u= D _ {1} ^ {2} \mathop{\rm log} \theta $ satisfies the partial differential equation

$$ D _ {1} ( D _ {1} ^ {3} u + uD _ {1} u+ D _ {2} u)- D _ {3} ^ {2} u = 0. $$

Yet another approach (due to van Geemen and G. van der Geer) and gives a connection to the approach using the K–P-equation. The theta-functions $ \theta _ {2} [ \sigma ] ( \tau , z) $ map an Abelian variety to $ \mathbf P ^ {N} $. They also define a mapping from the moduli space $ {\mathcal A} _ {g} ( 2, 4) $( a covering of $ {\mathcal A} _ {g} $) to this projective space, where the image of the class of $ ( X, \Theta ) $ is the image of the origin of $ X $ under $ \Phi $. It is conjectured that the intersection of the image of the moduli space and the Kummer variety of an indecomposable Abelian variety has dimension $ \leq 2 $ and equals $ 2 $ exactly when $ ( X, \Theta ) $ is a Jacobian. Similarly for the intersection of the Kummer variety and the tangent space to the moduli space at the point defined by $ ( X, \Theta ) $. By the heat equation for the theta-function this last conjecture is equivalent to a statement about the set of common zeros $ F _ {X} \subset X $ of the space $ \Gamma _ {00 } \subset \Gamma ( X, O( 2 \Theta )) $ of theta-functions $ \sum _ \sigma a _ \sigma \theta _ {2} [ \sigma ]( \tau , z) $( for fixed $ \tau $) which vanish with multiplicity $ \geq 4 $ at the origin of $ X= \mathbf C ^ {g} / \Lambda _ \tau $. For a Jacobian $ X= J( C) $ this set $ F _ {X} $ consists of the image $ \{ {x- y } : {x, y \in C } \} \subset J( C) $ of $ C- C $ in the Jacobian (with a slight exception for $ g= 4 $), as was proved by Welters. Conjecturally, Jacobians are now characterized by $ F _ {X} \neq \{ 0 \} $.

These conjectures were refined by R. Donagi in [a2] to a much stronger conjecture which describes the Schottky locus $ {\mathcal R} {\mathcal S} _ {g} $ in $ {\mathcal R} {\mathcal A} _ {g} $, the moduli space of principally-polarized Abelian varieties $ ( A, \Theta , p) $ with a non-zero point $ p $ of order $ 2 $ on $ A $. This conjecture is very strong: it implies a strengthened version of the Novikov conjecture and all the conjectures of van Geemen and van der Geer. These last ones are obtained by intersecting $ {\mathcal R} {\mathcal S} _ {g} $ with the boundary of the compactified moduli space $ \overline{ {{\mathcal R} {\mathcal A} }}\; _ {g} $, while the Novikov conjecture follows by infinitesimalizing this. Donagi proved his conjecture for $ g \leq 5 $.

References

[a1]	A. Beauville, "Le problème de Schottky et la conjecture de Novikov" Astérisque , 152–153 (1988) pp. 101–112 (Sém Bourbaki, Exp. 675)
[a2]	R. Donagi, "The Schottky problem" E. Sernesi (ed.) , Theory of Moduli , Lect. notes in math. , 1337 , Springer (1988) pp. 84–137
[a3]	G. van der Geer, "The Schottky problem" F. Hirzebruch (ed.) J. Schwermer (ed.) S. Suter (ed.) , Arbeitstagung Bonn 1984 , Lect. notes in math. , 1111 , Springer (1985) pp. 385–406
[a4]	D. Mumford, "Curves and their Jacobians" , Univ. Michigan Press (1975)