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Patrick D. Baier in his Ph.D. thesis for proving the theorem 2.1.4 used the following non-trivial fact (in chapter 2 on page 14):

Let $0\neq X\in V$ (here $V$ is of dimension 6), $W^\ast = Ann(X)$ and $\Omega\in\wedge^3 V^\ast$. Then we can find unique elements $\psi\in\wedge^2 V^\ast$ and $\phi\in\wedge^3 V^\ast$ such that $\Omega=\theta\wedge\psi + \phi$ (I think this decomposition is from Hitchin in his paper about geometry of three forms in six dimensions) so that $i_X\Omega=\psi$ (why?). Moreover, there are unique elements $r\in\mathbb{R}$ and $Y\in Ker(\theta)$ (here $\theta\in V^\ast$ with $\theta (X)=1$) such that (why?) we have

1) $i_{(rX)} v = \psi\wedge\phi$

2) $i_Y v_0 = \psi\wedge\psi$

(here $v_0\in\wedge^5 W^\ast$ is the unique element such that $v=\theta\wedge v_0$)

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From the thesis, I see that one is supposed to make some choices of $v \in \Lambda^6 V^*$ and $\theta \in V^*$. $i_X v$ and $\psi \wedge \phi$ are both in $\ker\; i_X \cong \Lambda^5 W^* \subset \Lambda^5 V^*$, so must be proportional. $\ker \theta \to \Lambda^4 W^*, \; Y \mapsto i_Y v_0$ is an isomorphism. – Johannes Nordström Feb 3 '13 at 10:12
Dear @Johannes, why we have $keri_X≅Λ^5W^∗$ – Hassan Jolany Feb 3 '13 at 14:12
The wording of the non-trivial fact appears somewhat garbled to me. Any chance you could edit it to make its statement clearer? – Deane Yang Nov 10 '13 at 17:42
For example, what's $\nu$? – Deane Yang Nov 10 '13 at 18:34
And where in the statement is $W^*$ used? – Deane Yang Nov 10 '13 at 18:41
up vote 1 down vote accepted

I have no idea what the second half of the statement is, but here's what I think the first half says:

If $X \in V$ and $\theta \in V^*$ such that $\langle \theta, X\rangle \ne 0$, then given any nonzero $\Omega \in \Lambda^3V^*$, there exists unique elements $\psi \in \Lambda^2V^*$ and $\phi \in \Lambda^3V^*$ such that the following hold:

  • $\Omega = \theta\wedge\psi + \phi$
  • $i(X)\Omega = \psi$

To prove this, simply extend $\theta$ to a basis of $V^*$, $\theta_1, \dots, \theta_n$, where $\theta_1 = \theta$ and $\theta_2, \dots, \theta_n$ comprise a basis of $X^\perp$ and expand $\psi$ and $\phi$ with respect to the bases induced on $\Lambda^2V^*$ and $\Lambda^3V^*$.

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Dear @Hassan, about the first question: why $\iota_X \Omega=\psi$ please note that $\psi\in\Lambda^2 W^* $ and $\phi\in\Lambda^3 W^* $. So the identity comes from $\theta(X)=1$. About the second question me too I'd like to know why.


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