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We prove that various classical tree forcings -- for instance Sacks forcing,Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preservethe statement that every real has a sharp and hence analytic determinacy. Wethen lift this result via methods of inner model theory to obtainlevel-by-level preservation of projective determinacy (PD). Assuming PD, wefurther prove that projective generic absoluteness holds and no new equivalenceclasses classes are added to thin projective transitive relations by theseforcings.

Preserving levels of projective determinacy by tree forcings