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A problem studied in GIT is the descending of vector bundles (or more in general coherent sheaves) to quotients. It is a result of Kempf that whenever we have a vector bundle over a quasiprojective scheme defined over $\mathbb{C}$ (or any algebraically closed field of characteristic zero) $$\pi:E\to X$$ and a reductive group $G$ acting linearly on $E$ and $X$ with the property that $$\pi(g\cdot v)=g\cdot \pi(v)$$ Let $X//G$ be the GIT quotient, then $E$ descends to a vector bundle $$\bar{E}\to X//G$$ if for any $x\in X^{ss}$ (closed point) $Stab(x)$ acts trivially on $E_x$. My problem is the following: i am interested to know if there is a similar condition which ensures that a projective bundle descends to the quotient. Thank you in advance.

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  • $\begingroup$ You can get a condition by combining the result cited in the question with the observation that projective $G$-bundle on $X$ is the same as a $\widetilde{G}$-bundle on $X$, where $\widetilde{G}$ is a central extension of $G$ by $\mathbf{G}_m$. $\endgroup$
    – skd
    May 2, 2020 at 20:46

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