Counting curves with local tangency constraints

Abstract: We construct invariants for any closed semipositive symplectic manifold which count rational curves satisfying tangency constraints to a local divisor. More generally, we introduce invariants involving multibranched local tangency constraints. We give a formula describing how these invariants arise as point constraints are pushed together in dimension four, and we use this to recursively compute all of these invariants in terms of Gromov--Witten invariants of blowups. As a key tool, we study analogous invariants which count punctured curves with negative ends on a small skinny ellipsoid.