MODULI SPACES OF FLAT TORI AND ELLIPTIC HYPERGEOMETRIC FUNCTIONS

In the genus one case, we make explicit some constructions of Veech [80] on flat surfaces and generalize some geometric results of Thurston [77] about moduli spaces of flat spheres as well as some equivalent ones but of an analyticocohomological nature of Deligne and Mostow [11], on the monodromy of Appell- Lauricella hypergeometric functions. In the dizygotic twin paper [20], we follow Thurston’s approach and study moduli spaces of flat tori with cone singularities and prescribed holonomy by means of geometrical methods relying on surgeries on flat surfaces. In the present memoir, we study the same objects making use of analytical and cohomological methods, more in the spirit of Deligne-Mostow’s paper. Our starting point is an explicit formula for flat metrics with cone singularities on elliptic curves, in terms of theta functions. From this, we deduce an explicit description of Veech’s foliation: at the level of the Torelli space of n-marked elliptic curves, it is given by an explicit affine first integral. From the preceding result, one determines exactly which leaves of Veech’s foliation are closed subvarieties of the moduli space M1;n of n-marked elliptic curves. We also give a local explicit expression, in terms of hypergeometric elliptic integrals, for the Veech map by means of which is defined the complex hyperbolic structure of a leaf. Then we focus on the n = 2 case: in this situation, Veech’s foliation does not depend on the values of the cone angles of the flat tori considered. Moreover, a leaf which is a closed subvariety of M₁₂ is actually algebraic and is isomorphic to a modular curve Y₁(N) for a certain integer N ≥ 2. In the considered situation, the leaves of Veech’s foliation are CH₁-curves. By specializing some results of Mano and Watanabe [54], we make explicit the Schwarzian differential equation satisfied by the CH₁-developing map of any leaf and use this to prove that the metric completions of the algebraic ones are complex hyperbolic conifolds which are obtained by adding some of its cusps to Y₁(N). Furthermore, we explicitly compute the conifold angle at any cusp c X₁(N), the latter being 0 (i.e., c is a usual cusp) exactly when it does not belong to the metric completion of the considered algebraic leaf.