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$L^r_M = i_* \circ \hat{L}^{r-1}_M \circ i^*$ by the projection formula and the Poincare dua...

This is a question arising when I am reading M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772. …
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