In mathematics, an abelian surface is a 2-dimensional abelian variety.
One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic via the Riemann bilinear relations. Essentially, these are conditions on the parameter space of period matrices for complex tori which define an algebraic subvariety. This subvariety contains all of the points whose period matrices correspond to a period matrix of an abelian variety.
The algebraic ones are called abelian surfaces and are exactly the 2-dimensional abelian varieties. Most of their theory is a special case of the theory of higher-dimensional tori or abelian varieties. Finding criteria for a complex torus of dimension 2 to be a product of two elliptic curves (up to isogeny) was a popular subject of study in the nineteenth century.
Invariants: The plurigenera are all 1. The surface is diffeomorphic to S1×S1×S1×S1 so the fundamental group is Z4.
Hodge diamond:
1
2
2
1
4
1
2
2
1
Examples: A product of two elliptic curves. The Jacobian variety of a genus 2 curve.
In mathematics, an abeliansurface is a 2-dimensional abelian variety. One-dimensional complex tori are just elliptic curves and are all algebraic, but...
in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group...
is abelian Abelianisation Abelian variety, a complex torus that can be embedded into projective space Abeliansurface, a two-dimensional abelian variety...
In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form ∫ z 0...
genus 1 (an abeliansurface) has Kodaira dimension 0; the product of a curve of genus 1 with a curve of genus at least 2 (an elliptic surface) has Kodaira...
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is...
Hurwitz surface, and its automorphism group is isomorphic to the unique simple group of order 168, which is the second-smallest non-abelian simple group...
Cayley nodal cubic surface, a certain cubic surface with 4 nodes Cayley's ruled cubic surface Clebsch surface or Klein icosahedral surface Fermat cubic Monkey...
unclassifiable). K3 surfaces can be considered the simplest algebraic varieties whose structure does not reduce to curves or abelian varieties, and yet...
of an abeliansurface with automorphism, and then blowing up to make the surface smooth. The resulting surfaces include some special K3 surfaces and rational...
projective space, and are the abelian varieties. The actual projective embeddings are complicated (see equations defining abelian varieties) when n > 1, and...
This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves. c. 1000 Al-Karaji writes on congruent numbers...
differentials. In either case the definition has its origins in the theory of abelian integrals. The dimension of the space of differentials of the first kind...
defining an abelian variety, and on algebraic surfaces. His books Abelian Varieties (with C. P. Ramanujam) and Curves on an Algebraic Surface combined the...
mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic...
Humbert surface, studied by Humbert (1899), is a surface in the moduli space of principally polarized abeliansurfaces consisting of the surfaces with a...
taking the quotient of R2 by a lattice, i.e. a free Abelian subgroup of rank 2. These closed surfaces have no isometric embeddings in E3. They do nevertheless...
period vectors). This gave the first glimpse of an abelian variety of dimension 2 (an abeliansurface): what would now be called the Jacobian of a hyperelliptic...
K3 surface S with Picard number 22 and Artin invariant at most 2 is a Kummer surface, meaning the minimal resolution of the quotient of an abelian surface...
defined to be an abelian variety of some rank g whose endomorphism ring has rank (2g)2. Supersingular K3 surface. Certain K3 surfaces in non-zero characteristic...
curves appearing in the SIDH construction, giving an abeliansurface (more generally, an abelian variety), and computing a specially crafted isogeny defined...
surface S, then S is orientable if and only if H1(S) has a trivial torsion subgroup. More precisely, if S is orientable then H1(S) is a free abelian group...
is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with...
Quasi-abelian category Abelian group Abelianization Metabelian group Non-abelian group Abelian extension Abelian integral AbeliansurfaceAbelian variety...
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