mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Siegel upper half-space of degree ''g'' (or genus ''g'') (also called the Siegel upper half-plane) is the set of ''g'' × ''g''

symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...

over the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s whose

imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

is positive definite. It was introduced by . It is the

symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geome ...

associated to the

symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gr ...

. The Siegel upper half-space has properties as a

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

that generalize the properties of the

upper half-plane In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...

, which is the Siegel upper half-space in the special case ''g'' = 1. The group of automorphisms preserving the complex structure of the manifold is isomorphic to the symplectic group . Just as the two-dimensional hyperbolic metric is the unique (up to scaling) metric on the upper half-plane whose isometry group is the complex automorphism group = , the Siegel upper half-space has only one metric up to scaling whose isometry group is . Writing a generic matrix ''Z'' in the Siegel upper half-space in terms of its real and imaginary parts as ''Z'' = ''X'' + ''iY'', all metrics with isometry group are proportional to :

d s^2 = \text(Y^ dZ Y^ d \bar).

The Siegel upper half-plane can be identified with the set of tame almost complex structures compatible with a symplectic structure

\omega

, on the underlying

2n

dimensional real vector space

V

, that is, the set of

J \in \mathrm(V)

such that

J^2 = -1

and

\omega(Jv, v) > 0

for all vectors

v \ne 0

.Bowman As a symmetric space of non-compact type, the Siegel upper half space

\mathcal_g

is the quotient :

\mathcal_g = \mathrm(2g,\mathbb)/\mathrm(n),

where we used that

\mathrm(n)=\mathrm(2g,\mathbb)\cap \mathrm(g,\mathbb)

is the maximal torus. Since the isometry group of a symmetric space

G/K

G

, we recover that the isometry group of

\mathcal_g

\mathrm(2g,\mathbb)

. An isometry acts via a generalized Möbius transformation :

Z\mapsto (AZ+B)(CZ+D)^ \text Z\in\mathcal_g, \left(\beginA&B\\ C&D\end\right)\in \mathrm_(\mathbb).

The quotient space

\mathcal_g/\mathrm(2g,\mathbb)

is the moduli space of principally polarized abelian varieties of dimension

g

References

*. * * * Complex analysis Automorphic forms Differential geometry 1939 introductions {{differential-geometry-stub

See also

References