Generalized complex structure: Difference between revisions

Revision as of 18:01, 7 September 2006

Revision as of 18:26, 7 September 2006

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==Maximal isotropic subbundles==

===Classification===

There is a one-to-one correspondence between maximal isotropic [[subbundle]] of '''T'''<math>\oplus</math>'''T'''^* and pairs ('''E''',''ε'') where '''E''' is a subbundle of of '''T''' and ''ε'' is a 2-form. This correspondence extends straightforwardly to the complex case.

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To see that ''L''('''E''',''ε'') is isotropic, notice that if ''Y'' is a section of '''E''' and ''ξ'' restricted to '''E'''^* is ''ε(X)'' then ''ξ(Y)=ε(X,Y)'', as the part of ''ξ'' orthogonal to '''E'''^* annihilates ''Y''. Thesefore if ''X+ξ'' and ''Y+η'' are sections of '''T'''<math>\oplus</math>'''T'''^* then

::<math>\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X))=\frac{1}{2}(\epsilon(Y,X)+\epsilon(X,Y))=0</math>

and so ''L''('''E''',''ε'') is isotropic. Furthermore ''L''('''E''',''ε'') is maximal because there are ''dim''('''E''') (complex) dimensions of choices for '''E''', and ''ε'' is unrestricted on the [[complement (mathematics)|complement]] of '''E'''^*, which is of (complex) dimension ''n-dim''('''E'''). Thus the total (complex) dimension in ''n''. Gualtieri has proven that all maximal isotropic subbundles are of the form ''L''('''E''',''ε'') for some '''E''' and ''ε''.

===Type===

The '''type''' of a maximal isotropic subbundle ''L''('''E''',ε) is the (complex) dimension of the subbundle that annihilates '''E'''. Equivalently it is ''n'' minus the (complex) dimension of the [[projection (mathematics)|projection]] of ''L''('''E''',ε) onto the tangent bundle '''T'''.

==Canonical bundle==