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Small exotic 4-manifolds and symplectic Calabi–Yau surfaces via genus-3 pencils

### R. İnanç Baykur

Vol. 5 (2022), No. 1, 185–221
##### 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 ${ℂℙ}^{2}#p{\overline{ℂℙ}}^{2}$ 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.

##### Keywords
Lefschetz pencil, exotic 4-manifold, symplectic Calabi–Yau
##### Mathematical Subject Classification
Primary: 57R55
Secondary: 57K20, 57K43