In
mathematics, the Torelli theorem, named after
Ruggiero Torelli, is a classical result of
algebraic geometry over the
complex number field, stating that a
non-singular
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...
projective
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
(
compact Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
) ''C'' is determined by its
Jacobian variety
In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian var ...
''J''(''C''), when the latter is given in the form of a
principally polarized abelian variety. In other words, the
complex torus
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ...
''J''(''C''), with certain 'markings', is enough to recover ''C''. The same statement holds over any
algebraically closed field. From more precise information on the constructed
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
of the curves it follows that if the canonically principally polarized Jacobian varieties of curves of genus
are ''k''-isomorphic for ''k'' any
perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds:
* Every irreducible polynomial over ''k'' has distinct roots.
* Every irreducible polynomial over ''k'' is separable.
* Every finite extension of ''k' ...
, so are the curves.
This result has had many important extensions. It can be recast to read that a certain natural
morphism, the
period mapping In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures.
Ehresmann's theorem
Let be a holomorphic submersive morphism. For a point ''b'' of ''B'', we denote ...
, from the
moduli space of curves of a fixed
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
, to a moduli space of
abelian varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...
, is
injective (on
geometric point
This is a glossary of algebraic geometry.
See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry.
...
s). Generalizations are in two directions. Firstly, to geometric questions about that morphism, for example the
local Torelli theorem. Secondly, to other period mappings. A case that has been investigated deeply is for
K3 surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected al ...
s (by
Viktor S. Kulikov,
Ilya Pyatetskii-Shapiro,
Igor Shafarevich and
Fedor Bogomolov) and
hyperkähler manifold
In differential geometry, a hyperkähler manifold is a Riemannian manifold (M, g) endowed with three integrable almost complex structures I, J, K that are Kähler with respect to the Riemannian metric g and satisfy the quaternionic relations I^2 ...
s (by
Misha Verbitsky,
Eyal Markman and
Daniel Huybrechts).
Automorphisms of Hyperkähler manifolds
/ref>
Notes
References
*
*
*
Algebraic curves
Abelian varieties
Moduli theory
Theorems in complex geometry
Theorems in algebraic geometry
{{algebraic-geometry-stub