We construct viscosity solutions to the special Lagrangian equation that are Lipschitz but not $C^1$, and have nonminimal gradient graphs.

We define an integer-valued virtual count of embedded pseudoholomorphic curves of two times a primitive homology class and arbitrary genus in symplectic Calabi–Yau 3-folds, which can be viewed as an extension of Taubes’s Gromov invariant. The construction depends on a detailed study of bifurcations of moduli spaces of embedded pseudoholomorphic curves which is partially motivated by Wendl’s recent solution of Bryan and Pandharipande’s superrigidity conjecture.

For X a smooth, proper complex variety we show that for $p\gg 0$, the restriction of the mod p cohomology $H^i(X,{\mathbb{F}}_p)$ to any Zariski open has dimension at least $h_X^{0,i}$. The proof uses the prismatic cohomology of Bhatt and Scholze.

We use this result to obtain lower bounds on the p-essential dimension of covers of complex varieties. For example, we prove the p-incompressibility of the mod p homology cover of an abelian variety, confirming a conjecture of Brosnan for sufficiently large p. By combining these techniques with the theory of toroidal compactifications of Shimura varieties, we show that for any Hermitian symmetric domain X, there exist p-congruence covers that are p-incompressible.