Blowups of Dirac structures

TL;DR

This paper characterizes when twisted Dirac structures on a manifold can be lifted to its blowup along a submanifold, linking geometric conditions with Lie algebra properties, and recovers a known Poisson lifting theorem.

Contribution

It provides a complete characterization of Dirac structure liftability to blowups, extending previous results and classifying relevant Lie algebras.

Findings

Lifting occurs when the submanifold is either transverse or invariant with constant height Lie algebras.

Classified Lie algebras satisfying the invariance and height conditions.

Reproduced Polishchuk's theorem on Poisson structure liftability.

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 $k=0$.