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We develop the foundations of higher geometric stacks in complex analyticgeometry and in non-archimedean analytic geometry. We study coherent sheavesand prove the analog of Grauert's theorem for derived direct images underproper morphisms. We define analytification functors and prove the analog ofSerre's GAGA theorems for higher stacks. We use the language of infinitycategory to simplify the theory. In particular, it enables us to circumvent thefunctoriality problem of the lisse-\'etale sites for sheaves on stacks. Ourconstructions and theorems cover the classical 1-stacks as a special case.

Higher analytic stacks and GAGA theorems