Suppose $X$ is a compact complex manifold acted upon by biholomorphisms by a complex Lie group. Suppose $L$ is an equivariant line bundle which is also ample (in the sense that it admits an equivariant metric whose curvature is a Kahler form). Then is there an equivariant Kodaira embedding into projective space ?
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1$\begingroup$ Yes. Some power $L^n$ is very ample and equivariant, so the group acts on the space of sections $V=H^0(X, L^n)$. Them the embedding $X\to \mathbb{P}(V)$ (or $X\to \mathbb{P}(V^\vee)$, depending on your convention) is equivariant. $\endgroup$– Piotr AchingerCommented Nov 2, 2017 at 9:29
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