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The goal of this note is to prove that the mapping class groups of closed orientable surfaces of genus 2 (with punctures) are not Kähler. An application to compactifications of the moduli space of genus g curves (with punctures) is given.doi:10.1090/s0002-9939-06-08636-9 fatcat:m6bjoeggkjhafp73mychp2h72q