Periodic Reporting for period 1 - GENERALIZED (Generalized geometry: 3-manifolds and applications)
Periodo di rendicontazione: 2018-09-01 al 2020-08-31
Generalized geometry is a revolutionary approach to geometric structures with the power of providing suitable mathematical frameworks, unifying theories, and defining interesting new structures. Thus, generalized Kähler geometry is the most convenient language for bihermitian geometry, whereas generalized complex geometry unifies symplectic and complex geometry but there are manifolds that are neither complex nor symplectic and admit a generalized complex structure. This generalized complex structure has type change, the phenomenon that makes generalized geometry unique. Both generalized Kähler and complex geometry are only possible for even-dimensional manifolds. What can generalized geometry offer for odd-dimensional manifolds? What about the case of three-manifolds? What can we do with generalized geometry?
GENERALIZED has dealt with numerous topics related to generalized geometry.
Generalized geometry of type Bn admits analogues of generalized complex structures for odd-dimensional manifolds. The simplest interesting case is that of three-manifolds, in which the type-change locus (provided stability) is a union of circles. First, this type-change locus is proved to be disconnected in the case of a compact manifold, and second, it gives B3-generalized complex structures more flexibility than their classical counterparts, cosymplectic or normal almost contact structures. This project has proved that all Thurston’s geometries admit B3-generalized complex structures, as opposed to cosymplectic and normal almost contact, and the possible cases of manifolds that are neither cosymplectic nor normal almost contact have been narrowed down to one Euclidean case (the Hantzsche-Wendt manifold) and hyperbolic and Sol manifolds not fibering over the circle. The type-change structure on the three-sphere has been unravelled as an open-book decomposition whose binding is the type-change locus (the Hopf link) and the structure has been rebound to define new B3-generalized complex structures on similar open-book decompositions (joint work with J. Porti).
Beyond three-manifolds, the deformation theory of Bn-generalized complex structures has been properly established, in relation with that of usual generalized complex structures, and a new construction, the odd double of a pair of dual Lie algebroids and a derivation, has been introduced. On the other hand, an initial study of generalized contact geometry has led to a thorough study of complex Dirac structures, defining a set of invariants that determine the pointwise structure (order and normalized type together with the real index) and proving a splitting theorem (joint work with D. Agüero and cosupervised by H. Bursztyn). An extra by-product of the approach to contact geometry has been the description of the contact manifold of null geodesics through Engel geometry, as the Cartan deprolongation of the Lorentz prolongation (joint work with A. Marín-Salvador). Finally, for holomorphic Courant algebroids, whose set up is a generalization of Bn-generalized geometry, we have introduced a moment map picture whose zero locus gives the solutions to the Calabi system. This system is a generalization of the classical Calabi problem and the Hull-Strominger system, and its moduli space of solutions has been shown to carry a pseudo-Kähler metric (joint with M. García-Fernández and C. Tipler).
All the results have been or will appear published on the open-access repository arXiv.org thus complying with the EU Responsible Research and Innovation strategy, and the funding of the EU Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 750885 GENERALIZED has been or will be acknowledged.
The results have been disseminated through seven invited talks in workshops and four invited seminar talks, organized in Germany, Spain, Russia, China and Israel. A special session on Geometric Structures has been organized (joint with J. Cirici, at the Barcelona Mathematical Days) and a new online format under the acronym of ONEW (online one-day workshop) has been created, with two editions: higher Dirac structures (joint with H. Bursztyn), and generalized connections and curvature (joint with V. Cortés). A two-week summer course on generalized geometry has been organized at UAB, with 14 students receiving full accommodation and travel grants after a selection process. The notes of the course will become available as a book.
On the other hand, outreach has been an important component of this project, with lectures in events such as the Barcelona Science Festival, the European Researcher’s Night or Math Saturdays, the tutoring of high-school students through the ARGO programme, and the coordination of the project “Can computers do math? An approach to axiomatic geometry”, jointly with M. Masdeu, in the Barcelona International Youth Science Challenge.
Generalized geometry of type Bn admits analogues of generalized complex structures for odd-dimensional manifolds. The simplest interesting case is that of three-manifolds, in which the type-change locus (provided stability) is a union of circles. First, this type-change locus is proved to be disconnected in the case of a compact manifold, and second, it gives B3-generalized complex structures more flexibility than their classical counterparts, cosymplectic or normal almost contact structures. This project has proved that all Thurston’s geometries admit B3-generalized complex structures, as opposed to cosymplectic and normal almost contact, and the possible cases of manifolds that are neither cosymplectic nor normal almost contact have been narrowed down to one Euclidean case (the Hantzsche-Wendt manifold) and hyperbolic and Sol manifolds not fibering over the circle. The type-change structure on the three-sphere has been unravelled as an open-book decomposition whose binding is the type-change locus (the Hopf link) and the structure has been rebound to define new B3-generalized complex structures on similar open-book decompositions (joint work with J. Porti).
Beyond three-manifolds, the deformation theory of Bn-generalized complex structures has been properly established, in relation with that of usual generalized complex structures, and a new construction, the odd double of a pair of dual Lie algebroids and a derivation, has been introduced. On the other hand, an initial study of generalized contact geometry has led to a thorough study of complex Dirac structures, defining a set of invariants that determine the pointwise structure (order and normalized type together with the real index) and proving a splitting theorem (joint work with D. Agüero and cosupervised by H. Bursztyn). An extra by-product of the approach to contact geometry has been the description of the contact manifold of null geodesics through Engel geometry, as the Cartan deprolongation of the Lorentz prolongation (joint work with A. Marín-Salvador). Finally, for holomorphic Courant algebroids, whose set up is a generalization of Bn-generalized geometry, we have introduced a moment map picture whose zero locus gives the solutions to the Calabi system. This system is a generalization of the classical Calabi problem and the Hull-Strominger system, and its moduli space of solutions has been shown to carry a pseudo-Kähler metric (joint with M. García-Fernández and C. Tipler).
All the results have been or will appear published on the open-access repository arXiv.org thus complying with the EU Responsible Research and Innovation strategy, and the funding of the EU Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 750885 GENERALIZED has been or will be acknowledged.
The results have been disseminated through seven invited talks in workshops and four invited seminar talks, organized in Germany, Spain, Russia, China and Israel. A special session on Geometric Structures has been organized (joint with J. Cirici, at the Barcelona Mathematical Days) and a new online format under the acronym of ONEW (online one-day workshop) has been created, with two editions: higher Dirac structures (joint with H. Bursztyn), and generalized connections and curvature (joint with V. Cortés). A two-week summer course on generalized geometry has been organized at UAB, with 14 students receiving full accommodation and travel grants after a selection process. The notes of the course will become available as a book.
On the other hand, outreach has been an important component of this project, with lectures in events such as the Barcelona Science Festival, the European Researcher’s Night or Math Saturdays, the tutoring of high-school students through the ARGO programme, and the coordination of the project “Can computers do math? An approach to axiomatic geometry”, jointly with M. Masdeu, in the Barcelona International Youth Science Challenge.
Generalized geometry and three-manifolds were two disjoint areas whose combination has brought a new setting (Thurston’s geometries, link theory, open-book decompositions) to generalized geometry, new geometric structures (B3-generalized complex structures) to three-manifolds, thus posing new interesting questions (is B3-generalized complex more flexible than cosymplectic and normal almost contact?), and has served as a motivation for several applications.