Abstract

We introduce a strategy to produce exotic rational and elliptic ruled surfaces, and possibly new symplectic Calabi–Yau surfaces, via constructions of symplectic Lefschetz pencils using a novel technique we call breeding. We deploy our strategy to breed explicit symplectic genus-$3$ pencils, whose total spaces are homeomorphic but not diffeomorphic to the rational surfaces ${ℂ\mathbb{P}}^2\#p{\overline{ℂ\mathbb{P}}}^2$ for $p=6,7,8,9$. Similarly, we breed explicit genus-$3$ pencils, whose total spaces are symplectic Calabi–Yau surfaces that have $b_1>0$ and realize all the integral homology classes of torus bundles over tori.

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