Torus fibers and the weight filtration

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We show that if ( X , Y ) (X,Y) is a simple normal crossings log Calabi–Yau pair, then there is a real torus of dimension equal to the codimension of the smallest stratum of Y Y which can be used to construct W 2 k − 1 H k ( X ∖ Y ; Q ) W_{2k-1}\mathrm {H}^k(X \setminus Y;\mathbb {Q}) for all k k . We use this to show that a P=W type result holds for pairs ( X , Y ) (X,Y) consisting of a rational surface X X and a nodal anticanonical divisor Y Y .