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First off, Prof. Figueroa-O'Farrill is correct in noting that you omitted the $G$-equivariance condition for the connection, which reduces to $G$-invariance in the case where $G$ is abelian (in particular, when $G=\mathbb{C}^*$).

1. The choice of normalization conventions really just comes down to how you identify the Lie algebra of $\mathbb{C}^*$ with $\mathbb{C}$. Śniatycki's convention amounts to identifying the $\mathrm{U}(1)$-generator (i.e. the vector field $2\pi\frac{\partial}{\partial \theta}$ in the Lie algebra of $\mathbb{C}^*$) with $1\in\mathbb{C}$, whereas the seemingly more logical choice would be to identify it with $2\pi i\in\mathbb{C}$. Śniatycki does this because it's also Kostant's convention (who got it from Weil I believe).

2. The relationship between your $\alpha$ and Śniatycki's $\alpha$ should be just $$\alpha_{\mathrm{Blake}} = 2\pi i~\alpha_{\mathrm{Śniatycki}}$$

3. The condition for the existence of a line bundle $L$ over $X$ with connection $\alpha$ and $\alpha$-compatible Hermitian pairing $\langle\cdot,\cdot\rangle$ can formulated in terms of the curvature $\omega$ of $\alpha$, defined by $d\alpha = \pi^*\omega$, where $\pi$ is the bundle projection. The big advantage of defining the connection $\alpha$ à la Kostant/Śniatycki is that the condition becomes that $\omega$ is integral, i.e. gives an integer when integrated over closed two-cycles in the base manifold $X$. For this reason, the $2\pi i$ is a common normalization in the theory of Chern classes.

From personal experience, normalization conventions can drive you mad, especially when comparing results from different authors. For example, Guillemin and Sternberg seem to favor the convention that $\frac{\partial}{\partial \theta}\in\textrm{Lie algebra of }\mathbb{C}^* \leftrightarrow 1\in\mathbb{C}$. So best of luck :)

1

First off, Prof. Figueroa-O'Farrill is correct in noting that you omitted the $G$-equivariance condition for the connection, which reduces to $G$-invariance in the case where $G$ is abelian (in particular, when $G=\mathbb{C}^*$).

1. The choice of normalization conventions really just comes down to how you identify the Lie algebra of $\mathbb{C}^*$ with $\mathbb{C}$. Śniatycki's convention amounts to identifying the $\mathrm{U}(1)$-generator (i.e. the vector field $2\pi\frac{\partial}{\partial \theta}$ in the Lie algebra of $\mathbb{C}^*$) with $1\in\mathbb{C}$, whereas the seemingly more logical choice would be to identify it with $2\pi i\in\mathbb{C}$. Śniatycki does this because it's also Kostant's convention (who got it from Weil I believe).

2. The relationship between your $\alpha$ and Śniatycki's $\alpha$ should be just $$\alpha_{\mathrm{Blake}} = 2\pi i~\alpha_{\mathrm{Śniatycki}}$$

3. The condition for the existence of $\alpha$-compatible Hermitian pairing can formulated in terms of the curvature $\omega$ of $\alpha$, defined by $d\alpha = \pi^*\omega$, where $\pi$ is the bundle projection. The big advantage of defining the connection $\alpha$ à la Kostant/Śniatycki is that the condition becomes that $\omega$ is integral, i.e. gives an integer when integrated over closed two-cycles in the base manifold $X$. For this reason, the $2\pi i$ is a common normalization in the theory of Chern classes.

From personal experience, normalization conventions can drive you mad, especially when comparing results from different authors. For example, Guillemin and Sternberg seem to favor the convention that $\frac{\partial}{\partial \theta}\in\textrm{Lie algebra of }\mathbb{C}^* \leftrightarrow 1\in\mathbb{C}$. So best of luck :)