We prove that various classical tree forcings—for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing—preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes are added to thin projective transitive relations by these forcings.

中文翻译：

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通过树强制来保持投影确定性的水平

我们证明了各种经典的树强迫（例如麻袋强迫，Mathias强迫，紫菜强迫，Miller强迫和Silver强迫）保留了每个实数都具有清晰的因而具有确定性的确定性的说法。然后，我们通过内部模型理论的方法得出此结果，以获得逐级保存的投影确定性（PD）。假设PD，我们进一步证明了射影泛型绝对性成立，并且这些强制没有将新的等价类添加到稀薄的射影传递关系中。