Blowups of Dirac structures

Abstract

Given a real, twisted Dirac structure $L$ on a smooth manifold $M$, and a closed embedded submanifold $N\subseteq M$ of codimension $>1$, we characterise when $L$ lifts to a smooth, twisted Dirac structure on the real projective blowup of $M$ along $N$. This holds precisely when $N$ is either a submanifold transverse to $L$ (with no further restrictions) or a submanifold invariant for $L$, for which the Lie algebras transverse to $N$ have all of the same constant height $k\geq 0$. We also classify Lie algebras satisfying this Lie-theoretic property. We recover a theorem of Polishchuk, which establishes that a Poisson structure lifts to a Poisson structure on the blowup of a submanifold exactly when the submanifold is invariant and the transverse Lie algebras have constant height $k=0$.