The mirror P=W conjecture for log Calabi–Yau mirror pairs

Let $U$ be a log Calabi–Yau variety and assume that its homological mirror $U^\vee$ is also a log Calabi–Yau variety with the same dimension. Set $d=\dim U=\dim U^\vee$. The perverse mixed Hodge polynomial is

$${\mathrm{PW}}_{M}(u,t,w,p)=\sum_{a,b,r,s}(\dim \mathrm{Gr}_F^a\mathrm{Gr}^W_{s+b}\mathrm{Gr}^P_r(H^s(M)))u^at^sw^bp^r,$$

where the perverse filtration is taken with respect to the affinization map. Mirror P=W conjecture. One has

$${\mathrm{PW}}_{U}(u^{-1}t^{-2},t,p,w)u^dt^d={\mathrm{PW}}_{U^\vee}(u,t,w,p).$$

This conjecture proposes that the Hodge-theoretic weight and perverse filtrations of homological mirror log Calabi–Yau varieties are exchanged under mirror symmetry.