# Dual Lefschetz Operator and Contraction with the Fundamental Form

Let $M$ be a Kahler manifold, with metric $g$, fundamental form $\omega$, and dual Lefschetz operator $\Lambda$. Now $\Lambda$, and contraction with $\omega$, both map the two forms $\Omega^2(M)$ to $0$-forms, ie smooth functions. Are they equal? I think this is almost certainly true, but I can't see a clean argument.

Do I need Kahler here? I would guess this works for all complex manifolds.

• The Kahler form lets you turn a 2-form into a linear endomorphism of the tangent bundle. Taking the trace of that endomorphism gives the value of the dual Lefschetz operator on the 2-form. I don't know of a clean proof, but some calculations in a basis (this is just linear algebra, no Kahler condition is needed) should do the trick. – Gunnar Þór Magnússon Dec 17 '11 at 11:34
• Hi Gunnar. Thanks for your answer, but I don't see how Kahler a Kahler form turns a 2-form into an endomorphism of the tangent bundle. – Ago Szekeres Dec 17 '11 at 16:22
• Dear Ago, I should perhaps have been more precise. Consider a $(1,1)$-form $u$ on $X$. Such a form may be viewed as a sesquilinear form on the holomorphic tangent bundle $T_X$, or equivalently, a $\mathbb C$-linear morphism $T_X \to T_X^*$. The metric $\omega$ is another such morphism, but $\omega$ is invertible. Thus we can consider the endomorphism $A := \omega^{-1} \circ u$ of the holomorphic tangent bundle $T_X$. We now find that $\Lambda u = tr(A) = tr(\omega^{-1} u)$. – Gunnar Þór Magnússon Dec 24 '11 at 11:00
• Ah mince, viewing $u$ as a morphism we should have $u : T_X \to \overline T_X^*$. I forgot the conjugation. – Gunnar Þór Magnússon Dec 24 '11 at 11:01

The question asks whether $\boxed{ \Lambda \alpha = g(\omega,\alpha)}$ for any $\alpha \in \Omega^2(X)$.
This is indeed true, as explained by Gunnar, but here is a simple way of seeing it. Firstly, since we're just using linear operators we can work fiber-wise (hence indeed no closedness of $\omega$ is needed, and whether we work in the context of $TM$ or general vector bundles is irrelevant). Remember that by definition $g(\Lambda \eta, \mu) = g(\eta, L \mu)$ where $L$ is the wedge product with $\omega$.
Using this in our case: let $\alpha \in \Lambda^2 E_x$, then we can rewrite $\boxed{\Lambda \alpha}$ as $g(\Lambda \alpha, 1) = g(\alpha, L \; 1) = \boxed{ g(\alpha, \omega) }$.
• can this proof be used also for arbitrary $k$-forms? It doesn't seem to require modifications – jj_p May 21 '14 at 17:17
• No, since I used that $\Lambda \alpha$ is a number to write $\Lambda \alpha = g(\Lambda \alpha,1)$. (Also, it would not be clear what is meant by $g(\alpha, \omega)$ with $\alpha$ a $k$-form and $\omega$ a $2$-form.) – Ruben Verresen May 22 '14 at 21:07
• Still, the statement should hold true also for $k$-forms, right? – jj_p May 23 '14 at 6:32
• Not the statement that I put in the box, no. All we know (at least without doing any more effort) is that $g(\Lambda \alpha, \beta ) = g(\alpha, L\beta) = g(\alpha, \omega \wedge \beta)$ (which are sadly nothing but the definitions!) The equation $\Lambda \alpha = g(\alpha, \omega)$ doesn't make sense in general, since if $\alpha$ is a $k$-form, what would the RHS mean? – Ruben Verresen May 23 '14 at 6:48
• I was a bit imprecise, by statement I meant 'contraction with $\omega$ is the same as operating with $L$ on a $k$-form': can one try to see this by using the first equation in your last post? – jj_p May 23 '14 at 7:55