Let $\Sigma$ is a compact Riemann surface with the trivial action of SU(2) and let $\mathbb{P}^1$ be equipped with the standard SU(2) action. Then $X=\Sigma \times \mathbb{P}^1$ has an SU(2) action. My question is ``What are all SU(2) invariant Kahler metrics on X?" I conjecture that they are of the form $\Omega = \pi_1^{*} \omega + a \pi_2 ^{*} \omega _{FS}$ where $\omega$ is a metric on $X$, $a>0$ is a constant, $\omega_{FS}$ is the Fubini-study metric and $\pi_i$ are projection maps.
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$\begingroup$ Start by asking a weaker question, namely, what are the smooth $SU(2)$ actions on $X\times\mathbb{P}^1$. $\endgroup$– Liviu NicolaescuCommented Nov 5, 2015 at 18:25
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1$\begingroup$ I am giving the SU(2) action. It acts trivially on X and in the usual manner on $\mathbb{P}^1$. $\endgroup$– VamsiCommented Nov 5, 2015 at 18:30
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1 Answer
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Yes, your conjecture is true. You can prove this as follows. Let $\omega$ be an $SU(2)$-invariant Kahler metric of $X$.
1) Show that at any point of $X$ the tangent spaces at both factors $\mathbb{P}^1$ and $\Sigma$ are perpendicular.
As a corollary there are smooth functions $f,g$ on $X$ such that $$ \omega = f \, \pi_1^* \omega_1 + g \, \pi_2^* \omega_{FS}$$ where $\omega_1$ is any Kahler metric on $\Sigma$.
2) Show that $g$ is constant and that $f = \pi_1^* h$ is the pullback of a smooth function $h$ on $\Sigma$. Use that $\mathrm{d} \omega = 0 $ and the $SU(2)$-invariance of $\omega$. This is a local computation by using local coordinates of $\Sigma$ and $\mathbb{P}^1$.
3) Now $\omega = \pi_1^* \omega_2 + a \, \pi_2^* \omega_{FS} $ as you conjectured, with $\omega_2 = h \omega_1$ which is a Kahler metric on $\Sigma$.
Let me know if need more details about 1) or 2).
