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I would be interested to know the answer to the above question for the constructible bounded derived category on complex analytic or complex algebraic manifolds (or some other context). A reference would be helpful.

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According to Dimca's "Sheaves in topology" Proposition 4.2.10, the (perverse) nearby and vanishing cycle functors commute non-canonically with the Verdier duality functor if you have field coefficients. For a proof he refers to Brylinski's "Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques" which I don't have available.

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  • $\begingroup$ What do you mean by perverse nearby cycle? Only on perverse sheaves? $\endgroup$
    – asv
    Commented Nov 25, 2015 at 12:39
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    $\begingroup$ I mean the version which includes a degree shift of -1 compared to the "naive" definition of nearby and vanishing cycles. With this degree shift the functors send perverse sheaves to perverse sheaves and commute with Verdier duality. Otherwise they just commute with Verdier duality up to a degree shift. $\endgroup$
    – Dan Petersen
    Commented Nov 25, 2015 at 12:43

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