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I want to how an equivariant 2-form and equivariant 3- form look like i,e.,

Let M be a complex Manifold say $ S^4 \subset C^3$ and a compact lie group $S^1$ acting on it via the action $\exp{i\theta}. (z_1, z_2, z_3)= (\exp{i\theta}.z_1, \exp{i\theta}.z_2, \exp{i\theta}.z_3)$. So in this particular example I want to calculate 2-form and 3-forms which respect this action (If we denote the action by $\rho$ then for differential form $\omega, \rho^{*} \omega = \omega$) and horrizontal ($i_{\zeta X}(\omega)= 0$), known as basic forms also.

I am guessing that the final answer should be of the form: 2-form which is horizontal and invariant under action = $f_1^g d_g f_2^g\wedge d_g f_3^g$ for some $f_1^g, f_2^g, f_3^g$ are complex smooth functions on M and invariant under the action and the differential $d_g$ is invariant under the action and horizontal. Similar for 3-forms also.

Approach: I tried calculating an equivariant 3 form as an element in $(\Omega^{3}(M) )_{inv} \bigoplus (\mathcal{G}^{*} \otimes \Omega^{1}(M))_{inv}$ but I think I didn't understand the second part of the summand.

Anyone, please help me with this computation. Thanks in advance.

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