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To a complete non-singular [[Algebraic curve|algebraic curve]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s0833801.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s0833802.png" /> one can associate its Jacobian (cf. [[Jacobi variety|Jacobi variety]]). This is an [[Abelian variety|Abelian variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s0833803.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s0833804.png" /> together with a principal polarization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s0833805.png" /> (cf. [[Polarized algebraic variety|Polarized algebraic variety]]). B. Riemann showed in 1857 that algebraic curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s0833806.png" /> depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s0833807.png" /> parameters (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s0833808.png" />). But principally-polarized Abelian varieties of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s0833809.png" /> depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338010.png" /> parameters. Since for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338011.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338012.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338013.png" /> is the moduli space of curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338014.png" /> (i.e. the parameter space of isomorphism classes of such curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338015.png" />, cf. [[Moduli theory|Moduli theory]]) and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338016.png" /> is the moduli space of principally-polarized Abelian varieties of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338017.png" />, there is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338018.png" />, and the problem is to characterize the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338019.png" /> of its image. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338020.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338021.png" />.
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Here only the case where the curve is a complex curve or Riemann surface is considered. If one chooses a symplectic basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338023.png" /> of the homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338024.png" /> and a basis of the space of holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338025.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338026.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338027.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338028.png" /> (Kronecker delta), one obtains the period matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338029.png" />. This matrix lies in the Siegel upper half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338030.png" />, the set of all complex symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338031.png" />-matrices whose imaginary part is positive definite.
+
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 +
{{TEX|done}}
  
The Jacobian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338032.png" /> is given by the complex torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338034.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338035.png" /> is then the divisor of Riemann's [[Theta-function|theta-function]]
+
To a complete non-singular [[Algebraic curve|algebraic curve]]  $  C $
 +
of genus  $  g $
 +
one can associate its Jacobian (cf. [[Jacobi variety|Jacobi variety]]). This is an [[Abelian variety|Abelian variety]]  $  J( C) $
 +
of dimension  $  g $
 +
together with a principal polarization  $  \Theta $(
 +
cf. [[Polarized algebraic variety|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|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} $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338036.png" /></td> </tr></table>
+
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 moduli space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338037.png" /> can be obtained as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338039.png" /> acts naturally on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338040.png" />. Coordinates on (a covering of) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338041.png" /> are provided by the "theta constanttheta constants" , which are the values at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338042.png" /> of the theta-functions
+
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-function]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338043.png" /></td> </tr></table>
+
$$
 +
\theta ( \tau , z)  = \sum _ {m \in \mathbf Z  ^ {g} }
 +
e ^ {\pi i (  ^ {t} m \tau m + 2  ^ {t} mz) } .
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338044.png" />.
+
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
  
The first result is due to W. Schottky for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338045.png" />. He showed in 1888 that a certain polynomial of degree 16 in the theta constants vanished on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338046.png" />, but not everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338047.png" />. J.-I. Igusa showed much later that its [[Zero divisor|zero divisor]] equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338048.png" />.
+
$$
 +
\theta \left [
  
The next step was made by Schottky and H.W.E. Jung in 1909, who constructed expressions in the theta constants that vanish on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338049.png" /> by means of double unramified coverings of curves. (Today this is called the theory of Prym varieties.) These expressions define a certain locus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338050.png" /> (called the Schottky locus) which contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338051.png" />. It is conjectured that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338052.png" />, and this is the Schottky problem in restricted sense. B. van Geemen showed in 1983 that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338053.png" /> is an irreducible component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338054.png" />.
+
for  $  \epsilon  ^  \prime  , \epsilon ^ {\prime\prime } \in ( 1 / 2) \mathbf Z  ^ {g} / \mathbf Z  ^ {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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338055.png" />): the dimension of its singular locus is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338056.png" />. It is therefore natural to consider the locus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338057.png" /> of principally-polarized Abelian varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338058.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338059.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338060.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338061.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338062.png" /> is indeed an irreducible component, as A. Andreotti and A. Mayer showed. However, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338063.png" /> has other components, so this is not strong enough.
+
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|zero divisor]] equals  $  J _ {4} $.
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338064.png" /> there is a mapping to projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338065.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338066.png" /> given by the theta-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338067.png" /> and its image is the Kummer variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338068.png" /> (cf. [[Kummer surface|Kummer surface]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338070.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338071.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338072.png" />, then the three points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338075.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338076.png" /> are collinear, i.e. define a trisecant. (Translated in terms of theta-functions this is Fay's trisecant identity.) It is conjectured that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338077.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338078.png" />.) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338079.png" />, then the Abelian variety is a Jacobian.
+
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} $.
  
One can consider an infinitesimal version of this. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338080.png" />. Then the criterion is: The Abelian variety is a Jacobian if there exist constant vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338083.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338085.png" /> such that
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338086.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
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|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|Korteweg–de Vries equation]]. It is the first of a whole hierarchy of equations.
 
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|Korteweg–de Vries equation]]. It is the first of a whole hierarchy of equations.
Line 29: Line 138:
 
By the Riemann identity
 
By the Riemann identity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338087.png" /></td> </tr></table>
+
$$
 +
\theta ( x+ y) \theta ( x- y)  = \sum _  \sigma
 +
\theta _ {2} [ \sigma ]( x) \theta _ {2} [ \sigma ]( y) ,
 +
$$
  
this translates into the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338088.png" /> satisfies the partial differential equation
+
this translates into the fact that $  u= D _ {1}  ^ {2}  \mathop{\rm log}  \theta $
 +
satisfies the partial differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338089.png" /></td> </tr></table>
+
$$
 +
D _ {1} ( D _ {1}  ^ {3} u + uD _ {1} u+ D _ {2} u)- D _ {3}  ^ {2} u  = 0.
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338090.png" /> map an Abelian variety to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338091.png" />. They also define a mapping from the moduli space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338092.png" /> (a covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338093.png" />) to this projective space, where the image of the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338094.png" /> is the image of the origin of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338095.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338096.png" />. It is conjectured that the intersection of the image of the moduli space and the Kummer variety of an indecomposable Abelian variety has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338097.png" /> and equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338098.png" /> exactly when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s08338099.png" /> is a Jacobian. Similarly for the intersection of the Kummer variety and the tangent space to the moduli space at the point defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380100.png" />. By the heat equation for the theta-function this last conjecture is equivalent to a statement about the set of common zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380101.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380102.png" /> of theta-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380103.png" /> (for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380104.png" />) which vanish with multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380105.png" /> at the origin of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380106.png" />. For a Jacobian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380107.png" /> this set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380108.png" /> consists of the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380109.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380110.png" /> in the Jacobian (with a slight exception for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380111.png" />), as was proved by Welters. Conjecturally, Jacobians are now characterized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380112.png" />.
+
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 [[#References|[a2]]] to a much stronger conjecture which describes the Schottky locus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380113.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380114.png" />, the moduli space of principally-polarized Abelian varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380115.png" /> with a non-zero point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380116.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380117.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380118.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380119.png" /> with the boundary of the compactified moduli space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380120.png" />, while the Novikov conjecture follows by infinitesimalizing this. Donagi proved his conjecture for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083380/s083380121.png" />.
+
These conjectures were refined by R. Donagi in [[#References|[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====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Beauville,  "Le problème de Schottky et la conjecture de Novikov"  ''Astérisque'' , '''152–153'''  (1988)  pp. 101–112  (Sém Bourbaki, Exp. 675)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Donagi,  "The Schottky problem"  E. Sernesi (ed.) , ''Theory of Moduli'' , ''Lect. notes in math.'' , '''1337''' , Springer  (1988)  pp. 84–137</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Mumford,  "Curves and their Jacobians" , Univ. Michigan Press  (1975)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Beauville,  "Le problème de Schottky et la conjecture de Novikov"  ''Astérisque'' , '''152–153'''  (1988)  pp. 101–112  (Sém Bourbaki, Exp. 675)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Donagi,  "The Schottky problem"  E. Sernesi (ed.) , ''Theory of Moduli'' , ''Lect. notes in math.'' , '''1337''' , Springer  (1988)  pp. 84–137</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Mumford,  "Curves and their Jacobians" , Univ. Michigan Press  (1975)</TD></TR></table>

Revision as of 08:12, 6 June 2020


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 [ 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|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|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 . $$ This is known as Novikov's conjecture and was proved by M. Shiota in 1986. Another (more geometric) proof was given by E. Arbarello and C. De Concini. The equation (*) is called the K–P-equation (after Kadomtsev–Petviashvili) and generalizes the [[Korteweg–de Vries equation|Korteweg–de Vries equation]]. It is the first of a whole hierarchy of equations. By the Riemann identity $$ \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)
How to Cite This Entry:
Schottky problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schottky_problem&oldid=13086
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