# 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 $g \leq 5$.

How to Cite This Entry:
Schottky problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schottky_problem&oldid=49575
This article was adapted from an original article by G. van der Geer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article