The general framework of this research project is to investigate algebraic and/or topological properties of manifolds (or even families of manifolds) under some assumptions about their curvature. An example of such kind of assumptions is, for instance, requiring the (sectional, or Ricci, or scalar) curvature to be constant or, more generally, to have a fixed sing.
When the manifolds in question are compact complex Kähler, one important issue is to establish the connection among the negativity of the curvature, the hyperbolicity (in the complex sense, namely Kobayashi hyperbolicity), and the positivity of the canonical bundle (which roughly corresponds to the negativity of the Ricci curvature) of the manifold as well as of its subvarieties. None of these different aspects is trivially a consequence of the others, and a huge amount of work has been done in the past few decades to understand it. One research line of this project exactly deals with this sort of questions, and aims at giving (at least partial) answers to some of the central problems in the subject.
On a different side, the analysis of locally rigid geometric structures on manifolds, such as metrics of constant curvature or affine structures, often translates into the study of their moduli spaces and their monodromy representations. The other main research line of this project is to investigate the topology of such moduli spaces and of their monodromy maps, and to compare them with related well-known moduli spaces of Riemann surfaces, or moduli spaces of quasi-Fuchsian 3-manifolds.
We try to give in what follows a flavor of the impact of the researches we propose for this project.
As mentioned above, Kobayashi hyperbolicity of compact Kähler or complex projective manifolds has an incredible number of conjectural equivalent properties in related areas of mathematics. In particular, a better insight of hyperbolicity from a curvature viewpoint might furnish new angles of attack for the understanding of Diophantine properties of hyperbolic projective manifolds defined over a number field. These kind of interactions have already given spectacular results in the past decades: let us cite for instance A. Moriwaki's result [Mor95] about finiteness of rational points for projective manifolds defined over a number field whose cotangent bundle is ample (a property which is in turn implied by the negativity of the holomorphic bisectional curvature) and globally generated.
It is well-known that Kobayashi's conjecture on ampleness of the canonical bundle of hyperbolic projective manifolds is strictly related, and conjecturally equivalent, to the non-hyperbolicity of Calabi-Yau manifolds. More than this, Calabi-Yau manifolds are expected to always possess rational curves (which can be seen as a strong algebraic form of hyperbolicity). The presence of such rational curves has big relevance in (super)string theory, where the ¿hidden dimensions¿ constitute precisely a Calabi-Yau manifold (with some extra structure). Any advance on Kobayashi's conjecture (even just in dimension three and four) would therefore have immediate consequences in particular for a better understanding of the new vacua described by F-theory.
Spherical surfaces with conical singularities are the lowest-dimensional example of Kähler-Einstein manifolds (beside flat and hyperbolic surfaces). The study of the topological and geometric properties of their moduli spaces would shed some light on moduli spaces of KE metrics in general. Also, some Morse theory techniques are available only for dimension 1 at the moment. So developing them in higher dimension would be of considerable progress.
Even in the simple case of G=PSL(2,R), the dynamics of the mapping class group of S that acts on the space of representations of the fundamental group of S in G is not well-understood. Studying the case of a punctured surface and moving the peripheral monodromy is a way to get access to the non-maximal components of the representations space for S closed, to better understand the dynamics of the MCG action and to attack a famous conjecture by Goldman (the action of MCG on non-maximal components is ergodic).
Since so little is known about moduli of affine structures, the proposed study is almost a foundational one. Strong conjectures like Chern's and Auslander's contrast with how little we know. For this reason, it seem absolutely necessary and useful to construct a moduli space of such structures, studying local invariants and the topological property of such moduli space.
After Ahlfors-Bers double uniformization theorem, Thurston rediscovered a rich geometry around quasi-Fuchsian manifolds and surfaces bent in quasi-Fuchsian 3-manifolds.
Mess did something similar with surfaces bent in the anti-de Sitter space. Both constructions have become a classical tool to investigate many problems concerning couples of Riemann surfaces, and the complex geometry of Teichmuller space.