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I'm trying to learn about the Lefschetz decomposition but am having a very basic problem: For the fundamental form $K$ of a Kahler metric on a complex manifold $M$, the corresponding Lefschetz operator $L$ is defined by $$ L:\Omega^k(M) \to \Omega^{k+2}(M), ~~~~~~~ \omega \mapsto K \wedge \omega. $$ From basic exterior algebra we must have $K \wedge K = 0$. Thus, to my eyes, we should have $$ L^2(\omega) = L(K \wedge \omega) = K \wedge (K \wedge \omega) = (K \wedge K) \wedge \omega = 0 \wedge \omega = 0. $$ However, the repeated Lefschetz operator is a central feature in Kahler theory. What am I missing?

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closed as off-topic by abx, Stefan Waldmann, Peter Michor, Ricardo Andrade, Qfwfq Nov 13 '14 at 16:30

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If $K$ is the Kaehler form then $K \wedge K$ is not zero, as $K$ has even degree. –  Paul Reynolds Mar 28 '12 at 14:23
If $K$ is a form of degree $p$, then we have $K \wedge K = (-1)^p K \wedge K$ by anticommutativity. When $p$ is odd this entails that $K \wedge K = 0$, but gives no information when $p$ is even, which is for example the case for the fundamental form of a hermitian metric. –  Gunnar Þór Magnússon Mar 28 '12 at 14:23
12 seconds apart. Well played, Paul, well played. ;) –  Gunnar Þór Magnússon Mar 28 '12 at 14:25
Yes, of course!! If the form is a simple product $f$d$g \wedge $d$h$, then it's square is zero, but when it's a sum of such products the same is not true. Thanks for your help, and sorry for the stupid question. –  Ago Szekeres Mar 28 '12 at 14:29
Well, it's not really a stupid question. Many of us had the same question, when we first learned about Kahler metrics. So in that sense it is an elementary question, and maybe more appropriate for math.stackexchange.com than here. But never mind. –  Deane Yang Mar 28 '12 at 15:34

1 Answer 1

(Just so this question has an answer.)

If $\alpha \in \Omega^k(M)$ and $\beta \in \Omega^l(M)$, $\alpha\wedge\beta = (-1)^{kl}\beta\wedge\alpha$.

So if one of $\alpha$ or $\beta$ has even degree (i.e. $k$ or $l$ is even), $\alpha\wedge\beta = \beta\wedge\alpha$; if both $\alpha$ and $\beta$ have odd degree, then $\alpha\wedge\beta = -\beta\wedge\alpha$.

In particular, if $\alpha$ has odd degree, $\alpha\wedge\alpha = -\alpha\wedge\alpha$ so $\alpha\wedge\alpha = 0$. Note, if $\alpha$ has even degree, the above discussion gives the tautology $\alpha\wedge\alpha = \alpha\wedge\alpha$.

In the case of the Lefschetz operator, we have $L\circ L : \Omega^k(M) \to \Omega^{k+4}(M)$ given by $$(L\circ L)(\alpha) = L(K\wedge\alpha) = K\wedge(K\wedge\alpha) = (K\wedge K)\wedge\alpha$$ where $K$ is the fundamental form. As $K$ is a two-form, it has even degree, so we cannot deduce that it is zero. However, as Deane Yang points out below, we know that $K\wedge K$ is nowhere zero as $K^n$ is a volume form.

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Actually we can say more about the exterior product of $K$ with itself. $K^n$ is the volume form and therefore is nonzero everywhere. –  Deane Yang Nov 13 '14 at 5:21
@DeaneYang: Good point. –  Michael Albanese Nov 13 '14 at 14:10

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