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Let $V$ be a complex vector space of dimension $n$, equipped with a Hermitian inner product whose Kahler form we denote by $\omega$. Let's set $P = \bigwedge^{2p} V^*$ and $Q = \bigwedge^{2q} V^*$ for $p,q$ such that $p + q \leq n$. If $u \in P$ and $v \in Q$ are such that $u \wedge v \not= 0$, and they are not both multiples of powers of the Kahler form, then do there exist $z \in u^{\perp}$ and $w \in w^{\perp}$ such that $$ \langle z \wedge v + u \wedge w, u \wedge v \rangle \not= 0 \, ? $$ (Here the inner products are the ones induced by $\omega$ on the exterior powers.)

We do need the condition on multiples on $\omega$, since this inner product is identically zero when $u = \omega^p$ and $v = \omega^q$.

This comes from trying to find the optimal constant $C$ such that $$ |u \wedge v|^2 \leq C |u|^2 |v|^2 $$ for all $u$ and $v$: We can observe that this $C$ is the maximum of the obvious smooth function from $\mathbb{P}(P) \times \mathbb{P}(Q) \to \mathbb{R}$ and hunt for critical points of that function. It's zero on the algebraic subvariety defined by $u \wedge v = 0$ and $([\omega^p],[\omega^q])$ is a critical point of the function. If that inner product is nonzero for some $z,w$ then those are the only critical points, so we find $C$ by evaluating at powers of $\omega$.

If we squint a bit, then $z \wedge v + u \wedge w$ looks kind of like the image of $\mathbb{P}^1 \to \mathbb{P}(H^0(\mathcal{O}(2)))$, so I think I'm asking something about a quartic and constraints on how it lies in projective space, but I haven't been able to turn that into anything useful.

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