The present project consists in a series of closely interrelated research problems in the theory of differential and complex varieties, linked together by the common theme of curvature. A wide class of results are expected, ranging from the extension in a suitable form of fundamental results from the smooth Riemannian geometry to the singular context, to the infinitesimal and global description of several moduli spaces of fixed curvature (sub-)manifolds, and from the investigation of the consequences of Kobayashi hyperbolicity in complex geometry, to the possibility of providing explicit locally complete families of projective hyper-Kaehler manifolds of Kummer type and so a new approach to Torelli's theorem for these. Advanced and innovative techniques will be exploited and developed, including the use of entropy bounds, of Ehresmann's geometric structures, of microanalytic techniques for singular elliptic operators, of complex Monge-Ampere equations and of differential graded and L-infinity algebras. Both the expected results and the techniques developed in order to prove them will predictably have a major impact on the subject, advancing the state of the art in the understanding of the local and global geometry of differential and complex varieties.
The proposed research is certainly ambitious and the expected results will considerably advance the current understanding of the geometry of differential and complex varieties. Moreover, the innovative techniques that will be developed to attack and solve the research problems will provide remarkable progress beyond the state of the art in the subject.
Here is a detailed list of the expected achievements.
1) The classical theory of convergence for Riemannian manifolds will be generalized to singular spaces by means of entropy bounds as flexible replacements for classical curvature bounds.
2) A characterization of tori among all Riemannian manifolds with dominant Abel-Jacobi map and almost minimal asymptotic volume, under suitable Ricci curvature and diameter bounds will be given. This will give at one time a topological analogue of Burago-Ivanov's theorem on about periodic metrics on tori, and a new proof of Gromov almost non-negative curvature. conjecture.
3) A characterization of hyperbolic manifolds among all irreducible 3-manifolds under a volume-entropy pinching condition and a partial solution the minimal asymptotic volume problem for nilmanifolds will be given.
4) The spectrum of the magnetic Laplacian will be related to the basic geometric invariants of the base variety and to the combinatorics of the flux of its potential around each homology class. The case of harmonic potentials is of particular interest in mathematical physics because it is directly related to the so-called Aharonov-Bohm effect.
5) The cohomology of the spaces of translation surfaces (i.e. Riemann surfaces with Abelian differentials) with assigned singularities an constant zero curvature will be investigated, and
an extension of Harer's results on the cohomology of moduli spaces of Riemann surfaces will be obtained.
6 ) The space of metrics in each conformal class of constant K=1 curvature surfaces will be analyzed, proving its finiteness and investigating the geometry of the map that sends a metric to its underlying complex structure.
7) Complex Monge-Ampere type equations will be used to translate curvature negativity conditions into information about Ricci curvature. In particular, the work of Wu and Yau will be extended to the singular setting, as well as to the merely hermitian case, paving the way toward a differential-geometric approach for the Kobayashi conjecture.
8) A cubic analogue of Getzler construction of the higehr Deligne groupoid will be described together with a higher analogue of Baker-Campbell-Hausdorff series. This will greatly enhance the ability of making explicit computations in formal deformation theory.
9) Palamodov's resolvent will be extended from analytic varieties to separated Noetherian schemes, providing a differential graded algebra description of the infinitesimal deformation of such schemes, by use of Reedy model structures theory.
10) The differential graded Lie algebra governing infinitesimal deformations of a pair consisting of a smooth projective variety and a divisor will be described , generalizing classical results by Horikawa. A Bogomolov-Tian-Todorov-type theorem for pairs consisting of a smooth projective variety and a possibly singular hypersurface will be derived, generalizing results obtained by Iacono for smooth hypersurfaces.
11) A a strongly homotopy Lie algebra governing infinitesimal deformations of a Psi-manifold will be described, providing a detailed description of formal and smooth deformations of the Psi manifold. As particular subcases, the theory of formal and smooth deformations of complex closed submanifolds and of phi-calibrated closed submanifolds, where phi is a parallel form in a Riemannian manifold, will be recovered.
12) By using deformation techniques a` la Witten, a Fredholm self-adjoint extension of the spin-c Dirac operator on the regular part of the singular quotient will be exhibited, and the index of this operator will be related to the trivial component of the equivariant index on the original manifold.
13) A complete description of the (expected) codimension g cycle in the moduli space of principally polarized Abelian varieties defined by the vanishing of the gradient of an odd theta function at z=0 will be given, and an investigation of the space of vector valued modular forms will be pursued.
14) An explicit description of certain locally complete families of abelian fourfolds of Weil type will be used to provide explicit locally complete families of projective Hyper-Kaehler manifolds of Kummer type and, more ambitiously, a new approach to Torelli's Theorem for Hyper-Kaehler manifolds of Kummer type will be given.