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I have the following question. Consider a $2n$-dimensional almost complex manifold $M$. Assume that on $M$ there exists a complex valued $n$-form $\Omega$ and a $2$-form $\omega$ such that:

  1. $\Omega$ is locally decomposable, i.e. there exists $n$ $1$-forms $\theta_{i}$ such that $\Omega = \theta_{1} \wedge ... \wedge \theta_{n}$.

  2. $\Omega \wedge \omega = 0$.

  3. $|\Omega \wedge \overline{\Omega}| > 0$.

How can one show that $M$ admits an $SU(n)$-structure? I think to define charts and show that the transitions functions are elements of $SU(n)$. But how to define the trivialisations ?

Greetings hapchiu

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Even with the right definition of $\mathrm{SU}(n)$-structure (which, as Ben points out, you don't have), you'll need more hypotheses than you have given. For example, your hypotheses allow $\omega=0$, which you surely don't want. You need (i) $\Omega$ is decomposable, (ii) $\Omega\wedge\bar\Omega\not=0$, and (iii) when you write $\Omega = \theta^1\wedge\cdots\wedge\theta^n$, then $\omega = (i/2)\ g_{k\bar l}\ \theta^k\wedge\overline{\theta^l}$ where $g = (g_{k\bar l})= (\ \overline{g_{l\bar k}}\ )$ is Hermitian positive definite and has determinant $1$. –  Robert Bryant Mar 19 '13 at 19:04
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(cont) Oh, and, with those hypotheses, you don't have to assume that $M$ has an almost complex structure, the form $\Omega$ satisfying the conditions (i) and (ii) define an almost complex structure on $M$, namely, the one for which $\Omega$ has type $(n,0)$. –  Robert Bryant Mar 19 '13 at 19:08

1 Answer 1

An $SU(n)$ structure is not a collection of charts whose transition maps have derivatives valued in $SU(n)$. It is a collection of bases of tangent spaces, forming a principal $SU(n)$-subbundle of the bundle of all bases of tangent spaces. The forms you have specified uniquely determine (by pure linear algebra) a collection of bases of tangent spaces: the bases which are identified by linear isomorphism with the standard basis of $\mathbb{C}^n$ in such a way that these tensors are identified with the standard $SU(n)$-invariant tensors in $\mathbb{C}^n$. For example, you might look at Dominic Joyce's book Riemannian Holonomy Groups and Calibrated Geometry, Oxford University Press, pages 36 to 39, for more information. In fact, $SU(n)$-structures depend locally on $3n^2-1$ functions of $2n$ variables. The objects you were thinking of, given by transition maps with derivatives in $SU(n)$, are much more rigid, and are equivalent to flat $SU(n)$-structures. These are locally unique up to diffeomorphism and complex unimodular affine rigid motion of $\mathbb{C}^n$, so if we don't divide out by the diffeomorphism pseudogroup action, then flat $SU(n)$-structures depend on $2n$ functions of $2n$ variables locally.

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