Hodge Structures

  • Published 2013


The upper half-plane H is the quotient of SL2(R) by its maximal compact subgroup SO(2). More generally, Siegel upper-half spaceHg is the quotient of Sp2g(R) by its maximal compact subgroup U(g). Hg is a complex manifold, with an action of Sp2g(R) by holomorphic automorphisms. Thus if Γ ⊂ Sp2g(R) is a discrete subgroup, then Γ\Hg has a chance at being an algebraic variety. In fact, if Γ ⊂ Sp2g(Z) is a congruence subgroup, then this is in fact the case; the theorem of Baily-Borel shows that Γ\Hg is an open subset of a projective variety, known as a Siegel modular variety. Just how general is this process? If we start with a group such as SLn(R), and take its quotient by (say) the compact subgroup SO(n), the result can be characterized as the set of positive definite quadratic forms of rank n and determinant 1. Its dimension is n(n+1)/2−1. For n = 3, this is an odd number, so there can’t be a complex structure on SL3(R)/ SO(3). For arithmetic subgroups Γ ⊂ SL3(Z), one can still form the quotient Γ\ SL3(R)/SO(3) to get an interesting manifold, but it won’t be an algebraic variety. What about SL2(C)? That group has a complex structure at least. SL2(C) acts on the set of 2× 2 hermitian matrices M , by the action g ·M = gMg∗, and preserves the determinant. The determinant, considered as a quadratic form on the space of hermitian matrices, has signature (1, 3). Thus we get a homomorphism SL2(C)→ O(1, 3), whose kernel is just {±I}. The Lorentz group O(1, 3) acts transitively on the set H of 4-tuples (t, x, y, z) such that t−x−y−z = 1, t > 0; the stabilizer of (1, 0, 0, 0) is O(1)×O(3) which is the maximal compact subgroup. Hence the symmetric space for the complex Lie group SL2(C) is actually H, a hyperbolic 3-manifold!

Cite this paper

@inproceedings{2013HodgeS, title={Hodge Structures}, author={}, year={2013} }