The perverse filtration and the Lefschetz hyperplane theorem
Let \(Y\) be a quasi-projective variety and \(K\) be a \(\Sigma\)-constructible complex on \(Y\) with respect to a given stratification \(\Sigma\) of \(Y\). The theory of \(-\)structures endows the hypercohomology groups \(H(Y,K)\) with the so-called perverse filtration \(P^pH(Y,K)\). Given a sequence \(Y\supset Y_{-1}\supset\dots\supset Y_{-n}\) of closed subvarieties of \(Y\), called an \(n\)-flag, basic sheaf theory endows \(H(Y,K)\) with the so-called flag filtration given by \[ F^pH(Y,K)= \text{Ker}(H(Y,K)\to H(Y_{p-2},K|_{y_{p-1}}). \] The main result of this paper is that if the \(n\)-flag is obtained using \(n\) hyperplanes in sufficiently general position, then these two filtrations coincide: \[ P^pH^j(Y,K)= F^{p+j}H^j(Y,K). \]
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