Essays in Mathematics and its Applications by Panos M. Pardalos & Themistocles M. Rassias

Author:Panos M. Pardalos & Themistocles M. Rassias
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

4.9 On Kählerian and Hyperbolic Moduli Spaces

The Abel-Jacobi-Albanese construction needs a choice of a complex structure in the target torus covered by . This can be compensated by considering the moduli space of all such structures, where, however, some caution is needed, since some complex tori, e.g. for , admit infinite groups of complex automorphisms.

Accordingly, the moduli space of the complex -tori , that is an orbispace which is locally at a point corresponding to equals the quotient of a complex analytic space of deformations of the complex structure in by the automorphism group of , has a pretty bad singularity at this .

To remedy this, one fixes a polarization i.e. a translation invariant non-singular -form on a torus and considers the moduli space of the isomorphism classes of the invariant complex structures where this serves as the imaginary part of an invariant Hermitian metric.

The resulting moduli space of Kählerian tori is a non-compact locally symmetric Hermitian orbi-space of finite volume, that is a quotient of a Hermitian symmetric space by a discrete isometry group , also denoted .

These tori themselves, parametrized by , make the universal family, say where the fibers represent the isomorphism classes of all these .

The Abel-Jacobi-Veronese theorem says in this language that

every continuous map map from a compact Kähler manifold to a fiber , , which induces an isomorphism , is homotopic to a holomorphic map with the image in a single fiber , .

The universal orbi-covering space of has a natural structure of a holomorphic vector bundle over , where this bundle carries a invariant flat -linear (but not -linear) connection.

On the other hand, since is topologically contractible as well as Stein, this bundle is holomorphically (non-canonically) isomorphic to the trivial bundle .

The Galois group of the covering map is the semidirect product , for the monodromy action of on and where the action of on is -affine on the fibers

This monodromy action on is of the kind we met in Sect. 2.3 (where we discussed/conjectured the super-stability of such actions) and the Siu theorem for equivariant map can be reformulated in “dynamics” terms as well.

Namely, let be a complex analytic space with a discrete action of and be an -equivariant continuous map for a homomorphism . Regard the trivial bundle as that induced from and, thus, lift the action of on to a continous fiber-wise -linear action of on .

If the map is holomorphic, then so is this lifted action, and, whenever Siu theorem applies, the continuous action of on is equivariantly homotopic to a holomorphic action.

(The moduli space and its finite orbi-covers contain lots of compact Hermitian totally geodesic subspaces, e.g. some locally isometric the complex hyperbolic spaces . The Siu theorem applies, for example, if the image of is contained in the fundamental (sub)group of such a and .)

Besides the action of , the map induces an action of on by parallel translations in each fiber , where the two actions together define an action of the semidirect product where acts on as .

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