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Is there any nontrivial example of Generalized complex groupoid?

By trivial, I mean all the classes of symplectic groupoids/ Abelian varieties as well as their products.

What I mean is that, is there any example of a groupoid in the category of generalized complex (GC) manifolds, in the sense of Hitchin. GC geometry is a way to unify complex geometry and symplectic geometry in to one realm. The definition could be found in wiki.

The space of arrows is a generalized complex manifold, and the source and target maps are generalized complex morphisms.

I can only come up with trivial examples, which are symplectic/complex, so are not real examples reflecting the nature of the theory.

-Any complex manifold is a GC manifold, so any abelian variety is a GC groupoid.

-Any symplectic manifold is a GC manifold, so any symplectic groupoid is also a GC groupoid.

-any product of these examples is also a GC groupoid.

But I can not find out any other example.

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Could you clarify what you mean? – David Roberts Jan 7 '11 at 0:29
Please read the "how to ask" page, which has a link on the top of this page. – S. Carnahan Jan 7 '11 at 4:37

There are three examples which I'm aware of.

  1. GC Lie groups: these are Lie groups equipped with GC structures compatible with the multiplication map. Holomorphic Poisson Lie groups are an example of this, but there are others. For example, the known examples of generalized Kahler structures on compact even-dimensional semisimple Lie groups (we just need a bi-invariant metric, not all hypotheses are necessary) consist of two commuting GC structures, one of which is multiplicative in the above sense, and the other of which is a GC homogeneous space over the GC Lie groupd defined by the first. This situation will be familiar to those in Poisson Lie Group theory, and this is joint work in progress with Jiang-Hua Lu. David is of course correct in his statement that GC actions of GC Lie groups would then define GC action groupoids.

  2. B-symplectic groupoids as described in These are, first and foremost, symplectic groupoids, but they have an extra 2-form making them GC Lie groupoids.

  3. Any holomorphic Poisson groupoid is an example of a generalized complex groupoid. For example, if Z is a Poisson manifold, then $Z\times Z$ is a Poisson groupoid and hence a generalized complex groupoid.

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Thanks for confirming my (somewhat educated) guess, Marco! – David Roberts Jan 8 '11 at 23:33

OK, I see what you mean now. How about this: take a complex Lie group $G$ which acts by symplectomorphisms on a symplectic manifold $(M,\omega)$. Then there is a Lie groupoid $M \times G \rightrightarrows M$ - the action groupoid - which actually lifts to be internal to generalised complex manifolds, if (edit:because) the product of a complex manifold and a symplectic manifold is a generalised complex manifold.

(Thanks to Marco G for confirming this last fact)

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I'm not very comfortable with GC geometry, but if I'm not mistaken Poisson-Lie groups might be group objects in the generalized complex world, so if you're given a Poisson space with a Poisson-Lie action (as come up very often in integrable systems) that would qualify as a nontrivial example of a GC groupoid..

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