On the slice genus of quasipositive knots in indefinite 4-manifolds

Let $X$ be a closed indefinite $4$-manifold with $b_+(X) = 3 ; ({\rm mod} ; 4)$andwithnon-vanishingmod$2$ Seiberg–Witten invariants. We prove a new lower bound on the genus of a properly embedded surface in $X\setminus B^4$ representing a given homology class and with boundary a quasipositive knot $K \subset S^3$. In the null-homologous case our inequality implies that the minimal genus of such a surface is equal to the slice genus of $K$. If $X$ is symplectic then our lower bound differs from the minimal genus by at most $1$ for any homology class that can be represented by a symplectic surface. Along the way, we also prove an extension of the adjunction inequality for closed $4$-manifolds to classes of negative self-intersection without requiring $X$ to be of simple type.