Siegel modular variety

They are named after Carl Ludwig Siegel, the 20th-century German number theorist who introduced the varieties in 1943.

[5] The Siegel modular variety Ag, which parametrize principally polarized abelian varieties of dimension g, can be constructed as the complex analytic spaces constructed as the quotient of the Siegel upper half-space of degree g by the action of a symplectic group.

Complex analytic spaces have naturally associated algebraic varieties by Serre's GAGA.

[1] The Siegel modular variety Ag(n), which parametrize principally polarized abelian varieties of dimension g with a level n-structure, arises as the quotient of the Siegel upper half-space by the action of the principal congruence subgroup of level n of a symplectic group.

[1][6] Furthermore, it was shown by Yung-Sheng Tai, Eberhard Freitag, and David Mumford that Ag is of general type when g ≥ 7.

A 2D slice of a Calabi–Yau quintic. One such quintic is birationally equivalent to the compactification of the Siegel modular variety A 1,3 (2). [ 1 ]