The Teichmüller space of genus curves carries the Hodge bundle , the total space of which maps into the first cohomology space via the period map (i. e., a holomorphic 1-form maps into its cohomology class). Haupt’s (or Haupt–Kapovich) theorem describes the image in terms of the integral structure on and the intersection pairing . It might be stated as follows.
Theorem (O. Haupt 1920, M. Kapovich 2000). Let be a genus topological surface, and a cohomology class on it. Suppose that:
- One has ;
- The submodule is either dense or is a lattice of covolume strictly greater than .
Then there exists a complex structure on s. t. one has —that is, belongs to the range of the period map . ∎
This is actually if and only if condition, but the converse is much easier to show. The second part of the second condition emerges in the following way: if is a holomorphic representative of its cohomology class, and the module is a lattice, then the multivaluate integral , , gives rise to a holomorphic map from to the elliptic curve , and since , this map must have degree at least two.
Kapovich’s proof goes as follows. Consider the action of the mapping class group on the Teichmüller space, and, via pullbacks, on the total space of . It also acts on the cohomology through its quotient , and the period map is equivariant. Therefore its image is the union of orbits. If one restricts to the normalized classes with , the image lies within the unit hyperboloid in the cohomology, which is isomorphic as a homogeneous space to . By C. Moore’s theorem, the -action on this quotient is ergodic. On the other hand, one can show that the period map is open away from the zero section. Invariance under ergodic action implies that this open set is dense (in particular, contains any dense orbit—which is in this case the orbit of a class with dense module ).
Hence the objective is to classify the non-dense orbits, and to understand, which of those realize. Here the Ratner theory comes into play:
Theorem (M. Ratner, 1991–95). Let be a semisimple Lie group, its subgroup generated by unipotent elements, and a lattice. Then for any element there exists an intermediate Lie subgroup , such that the connected component of the closure of the -orbit equals to the orbit , and the intersection is a lattice. ∎
The Ratner theorem applies to the -action on the quotient . Kapovich considers the list of maximal subgroups in containing . It is either a subgroup isomorphic to , or a stabilizer of a line; Kapovich treats these cases separately and, with certain involvement, proves that whenever such a class can be realized, it reduces to one of the cases from the statement of the theorem.
This, however, might be simplified, if one takes into consideration the -action on the total space of the Hodge bundle. Let me remind its construction: a point in is a pair of a genus curve with a holomorphic 1-form on it. Thinking of it as of a Riemannian surface, let us cut it along straight segments between the zeroes of , so what rests would be connected and simply connected domain. The form in this domain is nowhere zero, and hence its integral is a well-defined holomorphic mapping from it to . Its image is a polygon with pairs of parallel sides, if we glue them, we get our surface back, and would be recovered as the restriction of . Now consider as (this is a horrible thing to ask an algebraic geometer, yet this is what is to be done), and allow the group act on such polygonal data. This gives an -action on the pairs , since one might check that the result does not depend on the choice of cuts. The subgroup acts as multiplication of the 1-form by complex numbers of unit norm; hence the projection of the orbits of this action to the Teichmüller space are isomorphic to the unit disks , known as the Teichmüller curves.
It is clear from the construction that the period map is equivariant w. r. t. the -action. Its orbits are mapped into the orbits of the -action on , which in its turn are the fibers of the projection
Hence the image of the period map is the preimage of a certain -invariant subset in the base of this fibration, which is the Grassmannian variety of symplectic 2-planes in the real symplectic -space.
And if one applies the Ratner theory at this point, the list of possible orbits reduces drastically: the intermediate group must be either the whole , in which case the orbit is dense, or the group , which means that the orbit is discrete. The point in the symplectic Grassmannian corresponding to the class is the plane spanned by (or, more precisely, its real part). Ratner theory asserts that in the case of a discrete orbit the lattice intersects the subgroup in a lattice; and by a theorem of Margulis, whenever , a lattice in such a product splits into a product of lattices in the factors (for a case of Hilbert modular lattice is also possible). In terms of the symplectic Grassmannian, it means precisely that the symplectic plane is spanned by the lattice vectors, i. e. defined over . This lattice vectors correspond to the periods of , that is, the case of a discrete orbit is exactly the case of discrete module . So the conclude the proof we need to show that for any there exists a degree map from some genus curve to an elliptic curve. Yet this is trivial, since one can always consider the degree ramified cover of an elliptic curve with simple ramification points (that is, any ramification point has preimage).