Corks with large shadow-complexity and exotic 4-manifolds

Abstract: We construct an infinite family ${ C_{n,k}}{k=1}^{\infty}$ of corks of Mazur type satisfying $2n\leq \mathrm{sc}^{\mathrm{sp}}(C{n,k})\leq O(n^{3/2})$ for any positive integer $n$. Furthermore, using these corks, we construct an infinite family ${(W_{n,k},W'{n,k})}{k=1}^{\infty}$ of exotic pairs of $4$-manifolds with boundary whose special shadow-complexities satisfy the above inequalities. We also discuss exotic pairs with small shadow-complexity.