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We establish sharp deviation inequalities for the linear statistics of the 2DCoulomb gas. These imply sub-Gaussian inequalities, where the variance is givenby the Dirichlet norm. The proofs use complex geometry and potential theory onRiemann surfaces and apply more generally to beta-ensembles, which also includeinteger Quantum Hall states on Riemann surfaces. In a sequel of the paper wegive applications to large and moderate deviation principles, local laws atmesoscopic scales, quantitative Bergman kernel asymptotics. In a series ofcompanion papers applications to concentration of measure, Monte-Carlo methodsfor numerical integration and random matrices are given and relations to Kahlergeometry are explored.

Sharp deviation inequalities for the 2D Coulomb gas and Quantum hall states, I