Final Report Summary - GC4M (Generalized complex 4-manifolds)
This project dealt with the area of mathematics which concerns itself with the search of geometrical structures on manifolds, higher dimensional versions of surfaces, such as a soup bubble or the shell of a doughnut. There are several layers of refinement one can consider when looking for geometric structures on such spaces. If we for a moment make an analogy and think of a manifold as a room, the most basic type of structure one can study, a smooth or a differential structure, would correspond to the plumbing and the walls, which give the room its shape. Next level up, one can study metrics or Poisson structures. Any manifold admits one of those and these would be analogous to wallpaper and flooring. At a more refined level, one finds the structures studied in this project generalised complex structures together with several others, such as complex or symplectic structures. These correspond to the furniture of the room: a couch, a bookshelf or a sink and an oven. This last type of structure is what makes the space suitable for its applications: Symplectic structures are needed for mechanics, complex structures arise naturally in algebraic geometry and generalised complex manifolds are the natural geometric object underlying string theory. If the differential and topological structure is not right, the manifold will simply not admit any symplectic or complex structure, much in the same way that if there is no plumbing, the room can not be a kitchen.
The overall main purpose of this project was to further our understanding of generalised complex structures and provide a clearer picture of the type of space which admits them. Also we tried to draw consequences of the existence of these structures on spaces and expand the knowledge in adjacent areas by applying techniques developed for generalised complex structures to apparently unrelated geometric structures.
Of the six projects listed in the original proposal, we have made progress in the first five. The first project 'Neighborhood theorems and blow-downs' was completely solved by the researcher in collaboration with M. Gualtieri (Toronto University) and lead to several new examples of spaces admitting generalised complex structures. These techniques were picked up by Rafael Torres (Oxford University) to completely solve project 3 'Connected sums for generalised complex manifolds' in consultation with the researcher. Projects 1 and 3 together provided us with a much clearer picture to the question of 'Which spaces admit generalised complex structures' by showing that there are several such spaces. In fact we showed that the set of manifolds which admit generalised complex structures is much bigger than the set of manifolds admitting symplectic or complex structures.
Projects 2, 4 and 5 were connected by an underlying construction known in mathematics as a surgery and which amounts to a change of the 'plumbing' of the room, transforming it, in fact, into another room. For us, this surgery changes a generalised complex manifold into a symplectic one. With this surgery at hand, it became clear that the analogy with complex / symplectic geometry where one has Lefschetz fibrations (project 2) falls apart and one must adapt heavily the concept of a Lefschetz fibration in order to have something similar in the generalised complex world. Yet, the tools developed for this surgery clearly show that the moduli space of generalised complex structures (project 4) is unobstructed. Finally, since one can perform a reverse surgery one also obtains information about the relative Seiberg-Witten invariants of a generalised complex manifold in terms of the usual Seiberg-Witten invariants of an associated manifold. This final part sheds some light into the question of which manifolds do not admit generalised complex structures. This is particularly relevant since one of the conclusions we got from projects 1 and 3 is that far more manifolds admit these structures than originally anticipated and a natural question is that of finding obstructions to the existence of the structure. In plain terms, when we say 'if there is no plumbing the room is not a kitchen', what we do is to identify a basic characteristic of the room (the plumbing) which is needed for it to be possible to furnish it as kitchen. These relative Seiberg-Witten invariants seem to play that role in the generalised complex context. We are still working on producing concrete examples of manifolds which do not admit generalised complex structures, and this part of the project is what sheds light into this question.
Besides work on the projects proposed, the research also worked on new, but related topics. For example, project 1 regarded blow-up and blow-down of generalised complex four-manifolds. In a related topic, in collaboration with Gualtieri, the author extended those results to generalised Kahler manifolds.
Also, in a single authored paper, the researcher extended the theory of generalised complex and generalised Kahler structures to be applicable to strong KT and parallel Hermitian structures. With that parallel, the researcher improved greatly the conceptual understanding of strong KT structures developing several tools that brought it close to what is classically studied in Kahler geometry. Since strong KT structures appear naturally in string theory in the context of supersymmetric sigma models, these results are very likely to have an impact in that field.
Finally, in collaboration with Bursztyn (IMPA, Rio de Janeiro) and Gualtieri, the researcher studied the relation between geometric structures on a manifold and on the moduli space of instantons on a bundle over the manifold (this means, a different space constructed in terms of data in the original space). This way the researcher recovered and extended a previous result by Hitchin.
Overall, the results obtained in this project are likely to have applications to string theory and relate the study of generalised complex structures with the study of four dimensional manifolds, another area of mathematical research.
The overall main purpose of this project was to further our understanding of generalised complex structures and provide a clearer picture of the type of space which admits them. Also we tried to draw consequences of the existence of these structures on spaces and expand the knowledge in adjacent areas by applying techniques developed for generalised complex structures to apparently unrelated geometric structures.
Of the six projects listed in the original proposal, we have made progress in the first five. The first project 'Neighborhood theorems and blow-downs' was completely solved by the researcher in collaboration with M. Gualtieri (Toronto University) and lead to several new examples of spaces admitting generalised complex structures. These techniques were picked up by Rafael Torres (Oxford University) to completely solve project 3 'Connected sums for generalised complex manifolds' in consultation with the researcher. Projects 1 and 3 together provided us with a much clearer picture to the question of 'Which spaces admit generalised complex structures' by showing that there are several such spaces. In fact we showed that the set of manifolds which admit generalised complex structures is much bigger than the set of manifolds admitting symplectic or complex structures.
Projects 2, 4 and 5 were connected by an underlying construction known in mathematics as a surgery and which amounts to a change of the 'plumbing' of the room, transforming it, in fact, into another room. For us, this surgery changes a generalised complex manifold into a symplectic one. With this surgery at hand, it became clear that the analogy with complex / symplectic geometry where one has Lefschetz fibrations (project 2) falls apart and one must adapt heavily the concept of a Lefschetz fibration in order to have something similar in the generalised complex world. Yet, the tools developed for this surgery clearly show that the moduli space of generalised complex structures (project 4) is unobstructed. Finally, since one can perform a reverse surgery one also obtains information about the relative Seiberg-Witten invariants of a generalised complex manifold in terms of the usual Seiberg-Witten invariants of an associated manifold. This final part sheds some light into the question of which manifolds do not admit generalised complex structures. This is particularly relevant since one of the conclusions we got from projects 1 and 3 is that far more manifolds admit these structures than originally anticipated and a natural question is that of finding obstructions to the existence of the structure. In plain terms, when we say 'if there is no plumbing the room is not a kitchen', what we do is to identify a basic characteristic of the room (the plumbing) which is needed for it to be possible to furnish it as kitchen. These relative Seiberg-Witten invariants seem to play that role in the generalised complex context. We are still working on producing concrete examples of manifolds which do not admit generalised complex structures, and this part of the project is what sheds light into this question.
Besides work on the projects proposed, the research also worked on new, but related topics. For example, project 1 regarded blow-up and blow-down of generalised complex four-manifolds. In a related topic, in collaboration with Gualtieri, the author extended those results to generalised Kahler manifolds.
Also, in a single authored paper, the researcher extended the theory of generalised complex and generalised Kahler structures to be applicable to strong KT and parallel Hermitian structures. With that parallel, the researcher improved greatly the conceptual understanding of strong KT structures developing several tools that brought it close to what is classically studied in Kahler geometry. Since strong KT structures appear naturally in string theory in the context of supersymmetric sigma models, these results are very likely to have an impact in that field.
Finally, in collaboration with Bursztyn (IMPA, Rio de Janeiro) and Gualtieri, the researcher studied the relation between geometric structures on a manifold and on the moduli space of instantons on a bundle over the manifold (this means, a different space constructed in terms of data in the original space). This way the researcher recovered and extended a previous result by Hitchin.
Overall, the results obtained in this project are likely to have applications to string theory and relate the study of generalised complex structures with the study of four dimensional manifolds, another area of mathematical research.