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1. Geography

    1. Problem 1.1.

      1. (a) Does there exist a simply-connected symplectic 4-manifold $X \neq \mathbb{C}P^2$ on the BMY line?
      2. (b) Are there symplectic 4-manifolds on the BMY line that are not complex? (An infinite family on the same lattice point would be the most desirable outcome.)
        1. Remark. Any complex manifold $X \neq \mathbb{C}P^2$ on the BMY line and with holomorphic Euler characteristic $\chi_h(X) > 0$ is a minimal surface of general type, so it has only one Seiberg-Witten basic class up to sign. Moreover, it is a ball quotient, so the fundamental group is a torsion-free cocompact arithmetic subgroup of $\operatorname{PU}(2,1)$. Therefore, one possible approach is to perform Luttinger surgeries along tori in a Kähler surface on the BMY line to vary either the number of SW basic classes or $\pi_1$.
            • Problem 1.2.

              [Palka] Consider a (possibly singular) surface $C \subset \mathbb{C}P^2$ homeomorphic to $S^2$ that either (i) is symplectic or (ii) satisfies the adjunction formula.

              1. (a) What is the maximal number of singularities of $C$? (When $C$ is algebraic, it is conjectured that the answer is four.)
              2. (b) Do the collections of realizable singularities differ for symplectic and complex curves?
              3. (c) What if neither condition (i) or (ii) is satisfied?

                  Cite this as: AimPL: Symplectic four-manifolds through branched coverings, available at http://aimpl.org/symplecticfour.