## Final Report Summary - DEFFOL (Complex manifolds, foliations by complex leaves and their deformations)

My proposal consisted in opening new lines of research on moduli problems in Complex Analytic Geometry and CR Geometry. The general setting is the following. Starting with a fixed object, here a compact smooth manifold, one considers the different structures of a fixed type (for example complex structures, CR structures, or foliations with complex leaves, ...) on it. And one tries to turn the set of parameters describing these different structures up to isomorphism into a nice analytic object (manifold, or analytic space, ...) of finite-dimension. If one succeeds, the resulting set of parameters is called a moduli space. The problem can be handled globally (when considering all the structures of a given type) or locally (for structures close to a fixed given one), leading to a local or global moduli space.

For complex structures, there exists a complete answer to the local problem, due to Kuranishi, following works of Kodaira and Spencer. There always exists a finite-dimensional local moduli space. For other structures, the requirement of finite-dimensionality causes many problems. And the global problem is wide open even for complex structures.

Coming back to my proposal, firstly, it proposed a complete method to construct generalized finite-dimensional local moduli spaces for certain CR manifolds and foliated manifolds, for which there is no classical local moduli space of finite-dimension. Secondly, it aimed at a better geometric understanding of the Kuranishi space, that is «the» local moduli space of a compact complex manifold. Thirdly, it suggested a new definition of rigidity for deformations of complex structures and foliations.

The first objective was fully achieved. I showed that, for a certain class of CR-structures that I call polarized, we may define a weaker equivalence relation than isomorphism and, using this relation to identify two structures, there always exists a finite-dimensional analytic local moduli space. It may be emphasized, that, generally, for these structures, there is an infinite number of parameters up to isomorphism. Moreover, this new relation, as well as this class of CR structures have a nice geometric interpretation that I could develop. Finally, a subclass of these polarized structures appear as a generalization of Sasakian manifolds, and I could prove some general results on these new objects.

The second objective was replaced after the first year by a much more ambitious goal: to construct a global moduli space of complex structures on a fixed compact smooth manifold. By the end of the fellowship, I was able to complete this new objective. Mixing techniques of gluings for Kuranishi spaces and the theory of groupoids and stacks, I proved that the set of global parameters describing the complex structures with bounded dimension of their automorphism group, fits into what should be called an Artin analytic stack, and thus has an analytic structure.

Due to the new developments of the second objective, the third one was not treated.

Beyond the intrinsic value of the mere results, the techniques and concepts introduced to achieve task 1 should allow to deal with moduli problems for more structures, giving a new and effective frame to consider this type of problems.

Moreover, the main result on task 2 should be seen as of a foundational nature. It gives a natural analytic structure on the global set of parameters of complex structures on a fixed smooth manifold, so that it becomes possible, for example, to look for homotopy and cohomology properties of this space, and to compute examples, as the moduli stack of Hopf surfaces.

I am confident that both new lines of research will grow substantially in the next few years.

For complex structures, there exists a complete answer to the local problem, due to Kuranishi, following works of Kodaira and Spencer. There always exists a finite-dimensional local moduli space. For other structures, the requirement of finite-dimensionality causes many problems. And the global problem is wide open even for complex structures.

Coming back to my proposal, firstly, it proposed a complete method to construct generalized finite-dimensional local moduli spaces for certain CR manifolds and foliated manifolds, for which there is no classical local moduli space of finite-dimension. Secondly, it aimed at a better geometric understanding of the Kuranishi space, that is «the» local moduli space of a compact complex manifold. Thirdly, it suggested a new definition of rigidity for deformations of complex structures and foliations.

The first objective was fully achieved. I showed that, for a certain class of CR-structures that I call polarized, we may define a weaker equivalence relation than isomorphism and, using this relation to identify two structures, there always exists a finite-dimensional analytic local moduli space. It may be emphasized, that, generally, for these structures, there is an infinite number of parameters up to isomorphism. Moreover, this new relation, as well as this class of CR structures have a nice geometric interpretation that I could develop. Finally, a subclass of these polarized structures appear as a generalization of Sasakian manifolds, and I could prove some general results on these new objects.

The second objective was replaced after the first year by a much more ambitious goal: to construct a global moduli space of complex structures on a fixed compact smooth manifold. By the end of the fellowship, I was able to complete this new objective. Mixing techniques of gluings for Kuranishi spaces and the theory of groupoids and stacks, I proved that the set of global parameters describing the complex structures with bounded dimension of their automorphism group, fits into what should be called an Artin analytic stack, and thus has an analytic structure.

Due to the new developments of the second objective, the third one was not treated.

Beyond the intrinsic value of the mere results, the techniques and concepts introduced to achieve task 1 should allow to deal with moduli problems for more structures, giving a new and effective frame to consider this type of problems.

Moreover, the main result on task 2 should be seen as of a foundational nature. It gives a natural analytic structure on the global set of parameters of complex structures on a fixed smooth manifold, so that it becomes possible, for example, to look for homotopy and cohomology properties of this space, and to compute examples, as the moduli stack of Hopf surfaces.

I am confident that both new lines of research will grow substantially in the next few years.