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Let $(M,\omega)$ be a Kähler manifold with a pre-quantum Line bundle $L$ and $\text {Aut}(M,L)$ means the group biholomorphisms of $M$ which lift to holomorphic bundles maps $L\to L$. My question is for the flag variety $G/P$ when $\frac{\text{Aut}(G/P,L)}{S^1}$ is discrete?

PS; I think $\text{Aut}(G/P,L)$ is equal to $\text{Aut}(G/P)$ but I am not sure.

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Without knowing anything in particular about the $S^1$-action, your condition seems to me unlikely to be satisfied very often. Let $G=SL_2(\mathbb{C})$ and let $P$ be the standard Borel of upper-triangular matrices. Given any weight $n\in\mathbb{Z}$ of the standard maximal torus $T$, we can construct an associated line bundle $$L=\mathcal{O}(n):=G\times_{P}\mathbb{C}(n)$$ over $G/P=\mathbb{P}^1$. This is a $G$-equivariant line bundle, and the $G$-action alone tells you that $Aut(G/P,L)$ has dimension greater than $1$. So, it seems that $Aut(G/P,L)/S^1$ cannot possibly be discrete.

Furthermore, the bundles $\mathcal{O}(n)$, $n\in\mathbb{Z}$, are (up to isomorphism) all of the holomorphic line bundles on $G/P=\mathbb{P}^1$. So, it seems your condition cannot be satisfied for the $G$ and $P$ I have chosen. For this reason, I am skeptical as to whether it holds for higher-dimensional partial flag varieties.

ADDED: I think I have a general idea for a proof. Let $G$ be connected, simply-connected and semisimple. First note that every holomorphic line bundle $L$ over $G/P$ is actually isomorphic to an associated line bundle $G\times_{P}\mathbb{C}(\alpha)$, where $\alpha$ is a root orthogonal to every simple root defining $P$. However, $L=G\times_P\mathbb{C}(\alpha)$ is a $G$-equivariant line bundle. This $G$-action is a lift of the $G$-action on the base $G/P$, so the $G$-action alone tells you $Aut(G/P,L)$ has dimension greater than $1$. As above, this seems to imply $Aut(G/P,L)/S^1$ cannot be discrete.

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  • $\begingroup$ I am looking for a condition such that $\frac{\text{Aut}(G/P,L)}{S^1}$ become discrete. $\endgroup$
    – user21574
    Commented May 29, 2014 at 14:21
  • $\begingroup$ I think I have an argument to the effect that this never happens. I may have made a mistake, though. $\endgroup$ Commented May 29, 2014 at 14:26
  • $\begingroup$ Dear Peter, I always enjoy of your solutions :), would you please explain for me why "This $G$-action is a lift of the $G$-action on the base $G/P$, so the $G$-action alone tells you $\text{Aut}(G/P,L)$ has dimension greater than $1$. As above, this seems to imply $\text{Aut}(G/P,L)/S^1$ cannot be discrete." $\endgroup$
    – user21574
    Commented May 29, 2014 at 14:28
  • $\begingroup$ In fact, if $ G/P\cong \mathbb CP^n$ then $\text{Aut}(G/P,L)/S^1\cong PGL(n+1,\mathbb C)$ $\endgroup$
    – user21574
    Commented May 29, 2014 at 14:33
  • $\begingroup$ The kernel of the $G$-action on $G/P$ is contained in $P$. So, if you find any Lie subalgebra $\mathfrak{h}\subseteq\mathfrak{g}$ meeting $\mathfrak{p}$ trivially and integrate it to a closed subgroup $H$, then the action map gives you a subgroup of $Aut(G/P)$ with dimension equal to that of $H$. I imagine you can choose $H$ to have dimension greater than $1$ in most examples. In these cases, you really have a subgroup of $Aut(G/P,L)$ of dimension greater than $1$, meaning your quotient cannot be discrete. $\endgroup$ Commented May 30, 2014 at 14:12

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