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2

Let $(M,\omega)$ be a symplectic manifold. The first idea behind geometric quantization is that there exists a $S^1$-principal fiber bundle $\pi : Y \to M$, equipped with a connection $\lambda$ with curvature $\omega$, that is, $d\lambda = \pi^*(\omega)$. Let $\xi$ be the vector field generating the $S^1$ action, then, for all $y \in Y$,  T_yY = ...

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I assume that the algebraic group $G$ is smooth and connected, and that you are asking about equivariance for the natural action of $G$ on $H$. There is a quotient morphism $q:G\to H$ that is $G$-equivariant and faithfully flat. Thus, to prove that a $G$-equivariant coherent sheaf on $H$ is locally free, it suffices to check that the pullback ...

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Let $X$ be the wedge of infinitely many circles (equipped with the CW topology). Every vector bundle $\xi$ over $X$ is a summand of a trivial bundle, namely it is $\xi\oplus\xi$ is trivial because any vector bundle over a circle has this property (alternatively, one could appeal to the fact that $X$ is homotopy equivalent to a smooth manifold, if the number ...

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This is just to supply some details to what Rafe Mazzeo wrote. Let $g_{i}$ be metrics and $^{\left( i\right) }\nabla$ be connections on $E$ and on $M$ for $i=1,2$. Since $\bar{\Omega}$ is compact, the uniform equivalence of norms reduces to a local coordinate chart $(U,\{x^{i}\})$ over which the bundle $E$ is trivialized. In the following, the constant $C$ ...

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The fourth power of the Hodge bundle is isomorphic to the $E_8$ conformal block bundle at level one, and for that bundle (or rather its projectivization) you have the Hitchin/KZ/WZW connection, which is projectively flat.

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There's no flat connection because the Chern classes are nonzero rationally. The Chern classes of the bundles $\mathcal H^k$ were actually discussed just yesterday.

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The suggestion in my comment turns out to be impossible, and this leads to counterexamples to the OP's question, already for $\text{dim}(X)=2$. Let $U$ be a $2$-dimensional vector space over a field $k$. Let $\mathbb{P}(U^\vee)$ denote $\text{Proj} \text{Sym}^\bullet_k U$, i.e., $\mathbb{P}^1_k$. To be very precise, there is a short exact sequence on ...

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