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A shortcut in Kapovich’s proof of Haupt’s theorem

The Teichmüller space T_g of genus g curves carries the Hodge bundle \Omega T_g, the total space of which maps into the first cohomology space V = H^1(S_g,\mathbb{C}) via the period map (i. e., a holomorphic 1-form maps into its cohomology class). Haupt’s (or Haupt–Kapovich) theorem describes the image per(\Omega T_g) \subset V in terms of the integral structure on V = H^1(S, \mathbb{Z}) \otimes \mathbb{C} and the intersection pairing \omega \colon \Lambda^2V \to \mathbb{C}. It might be stated as follows.

Theorem (O. Haupt 1920, M. Kapovich 2000). Let S be a genus g>1 topological surface, and \alpha \in H^1(S, \mathbb{C}) a cohomology class on it. Suppose that:

  1. One has \sqrt{-1}\int_S\alpha\wedge\bar{\alpha} > 0;
  2. The submodule \Gamma_\alpha = \left\{\int_\gamma\alpha \mid \gamma \in H_1(S,\mathbb{Z})\right\} \subset \mathbb{C} is either dense or is a lattice of covolume strictly greater than \sqrt{-1}\int_S\alpha\wedge\bar{\alpha}.

Then there exists a complex structure I on S s. t. one has \alpha \in H^{1,0}(S,I)—that is, \alpha belongs to the range of the period map per\colon\Omega T_g \to V. ∎

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 \alpha is a holomorphic representative of its cohomology class, and the module \Gamma_\alpha is a lattice, then the multivaluate integral \int_{x_0}^x\alpha, x_0, x\in S, gives rise to a holomorphic map from S to the elliptic curve \mathbb{C} \mod \Gamma_\alpha, and since g>1, this map must have degree at least two.

Kapovich’s proof goes as follows. Consider the action of the mapping class group \mathrm{MCG}(g) on the Teichmüller space, and, via pullbacks, on the total space of \Omega T_g. It also acts on the cohomology through its quotient \mathrm{Sp}(2g,\mathbb{Z}), and the period map is equivariant. Therefore its image is the union of orbits. If one restricts to the normalized classes \alpha with \sqrt{-1}\int_S\alpha\wedge\bar{\alpha} = 1, the image lies within the unit hyperboloid in the cohomology, which is isomorphic as a homogeneous space to \mathrm{Sp}(2g,\mathbb{R})/\mathrm{Sp}(2g-2,\mathbb{R}). By C. Moore’s theorem, the \mathrm{Sp}(2g,\mathbb{Z})-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 \alpha with dense module \Gamma_\alpha \subset \mathbb{C}).

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 G be a semisimple Lie group, U \subset G its subgroup generated by unipotent elements, and \Gamma \subset G a lattice. Then for any element x = gU \in G/U there exists an intermediate Lie subgroup U \subseteq H \subseteq G, such that the connected component of the closure \overline{\Gamma gU} of the \Gamma-orbit equals to the orbit H^gx \subset G/U, and the intersection \Gamma \cap H^g \subset H^g is a lattice. ∎

The Ratner theorem applies to the \mathrm{Sp}(2g,\mathbb{Z})-action on the quotient \mathrm{Sp}(2g,\mathbb{R})/\mathrm{Sp}(2g-2,\mathbb{R}). Kapovich considers the list of maximal subgroups in \mathrm{Sp}(2g,\mathbb{R}) containing \mathrm{Sp}(2g,\mathbb{R}). It is either a subgroup isomorphic to \mathrm{Sp}(2,\mathbb{R})\times\mathrm{Sp}(2g-2,\mathbb{R}), 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 \mathrm{SL}(2,\mathbb{R})-action on the total space of the Hodge bundle. Let me remind its construction: a point in \Omega T_g is a pair (C,\alpha) of a genus g 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 \alpha, so what rests would be connected and simply connected domain. The form \alpha in this domain is nowhere zero, and hence its integral \int_{x_0}^x\alpha is a well-defined holomorphic mapping from it to \mathbb{C}. Its image is a polygon with pairs of parallel sides, if we glue them, we get our surface C back, and \alpha would be recovered as the restriction of dz. Now consider \mathbb{C} as \mathbb{R}^2 (this is a horrible thing to ask an algebraic geometer, yet this is what is to be done), and allow the group \mathrm{SL}(2,\mathbb{R}) act on such polygonal data. This gives an \mathrm{SL}(2,\mathbb{R})-action on the pairs (C,\alpha), since one might check that the result does not depend on the choice of cuts. The subgroup \mathrm{SO}(2,\mathbb{R}) \subset \mathrm{SL}(2,\mathbb{R}) acts as multiplication of the 1-form \alpha 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 \mathrm{SL}(2,\mathbb{R})/\mathrm{SO}(2,\mathbb{R}), known as the Teichmüller curves.

It is clear from the construction that the period map is equivariant w. r. t. the \mathrm{SL}(2,\mathbb{R})-action. Its orbits are mapped into the orbits of the \mathrm{Sp}(2,\mathbb{R})-action on \mathrm{Sp}(2g,\mathbb{R})/\mathrm{Sp}(2g-2,\mathbb{R}), which in its turn are the fibers of the projection

    \[\frac{\mathrm{Sp}(2g,\mathbb{R})}{\mathrm{Sp}(2g-2,\mathbb{R})} \to \frac{\mathrm{Sp}(2g,\mathbb{R})}{\mathrm{Sp}(2,\mathbb{R})\times\mathrm{Sp}(2g-2,\mathbb{R})}.\]

Hence the image of the period map is the preimage of a certain \mathrm{Sp}(2g,\mathbb{Z})-invariant subset in the base of this fibration, which is the Grassmannian variety of symplectic 2-planes in the real symplectic 2g-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 \mathrm{Sp}(2g,\mathbb{R}), in which case the orbit is dense, or the group \mathrm{Sp}(2,\mathbb{R}) \times \mathrm{Sp}(2g-2,\mathbb{R}), which means that the orbit is discrete. The point in the symplectic Grassmannian corresponding to the class \alpha is the plane spanned by \langle \alpha, \bar{\alpha}\rangle (or, more precisely, its real part). Ratner theory asserts that in the case of a discrete orbit the lattice \mathrm{Sp}(2g, \mathbb{Z}) intersects the subgroup \mathrm{Sp}(2,\mathbb{R}) \times \mathrm{Sp}(2g-2,\mathbb{R}) in a lattice; and by a theorem of Margulis, whenever g>2, a lattice in such a product splits into a product of lattices in the factors (for g=2 a case of Hilbert modular lattice is also possible). In terms of the symplectic Grassmannian, it means precisely that the symplectic plane \langle \alpha,\bar{\alpha}\rangle^{\mathrm{Gal}(\mathbb{C}\colon\mathbb{R})} is spanned by the lattice vectors, i. e. defined over \mathbb{Q}. This lattice vectors correspond to the periods of \alpha, that is, the case of a discrete orbit is exactly the case of discrete module \Gamma_\alpha \subset \mathbb{C}. So the conclude the proof we need to show that for any d>1 there exists a degree d map from some genus g curve to an elliptic curve. Yet this is trivial, since one can always consider the degree d ramified cover of an elliptic curve with 2g-2 simple ramification points (that is, any ramification point has d-1 preimage).

References

  • Misha Kapovich. Periods of abelian differentials and dynamics, preprint, 2000 (PDF, 238.0 kB)
  • rodion n. déev. Haupt–Kapovich theorem revisited, preprint, 2020 (arXiv:2010.15359, revision to appear)
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