Assistant Professor
Department of Mathematics
University of Sherbrooke
Symplectic geometry; differentiable stacks; shifted symplectic geometry; Dirac geometry; complex geometry; hyperkähler geometry; geometric quantization; holomorphic Poisson geometry; Lie groupoids; Lie algebras; moment maps; geometric invariant theory; singular spaces; moduli spaces; gauge theory; Yang—Mills equations.
We introduce a notion of coisotropics on 1-shifted symplectic Lie groupoids (i.e. quasi-symplectic groupoids) using twisted Dirac structures and show that it satisfies properties analogous to the corresponding derived-algebraic notion in shifted Poisson geometry. In particular, intersections of 1-coisotropics are 0-shifted Poisson. We also show that 1-shifted coisotropic structures transfer through Morita equivalences, giving a well-defined notion for differentiable stacks. Most results are formulated with clean-intersection conditions weaker than transversality while avoiding derived geometry. Examples of 1-coisotropics that are not Lagrangians include Hamiltonian actions of quasi-symplectic groupoids on Dirac manifolds. This recovers several generalizations of Marsden-Weinstein-Meyer's symplectic reduction and introduces new ones.
We develop a general procedure for reduction along strong Dirac maps, which are a broad generalization of Poisson momentum maps. We recover a large number of familiar constructions in Poisson and quasi-Poisson geometry, and we introduce new examples of Poisson, quasi-Poisson, and Dirac reduced structures. In particular, we obtain quasi-Poisson analogues of several classes of spaces that are studied in geometric representation theory.
We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden--Weinstein--Meyer reduction, Mikami--Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg--Kazhdan construction of Moore--Tachikawa varieties in TQFT. A key feature of our construction is a concrete and systematic association of a Hamiltonian G-space 𝔐G,S to each pair (G,S), where G is any Lie group and S ⊆ Lie(G)∗ is any submanifold satisfying certain non-degeneracy conditions. The spaces 𝔐G,S satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. While these Hamiltonian G-spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new.
The first part of this paper is a generalization of the Feix-Kaledin theorem on the existence of a hyperkähler metric on a neighbourhood of the zero section of the cotangent bundle of a Kähler manifold. We show that the problem of constructing a hyperkähler structure on a neighbourhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix-Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler structure on a neighbourhood of its identity section. More generally, we reduce the existence of a hyperkähler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin's unobstructedness theorem.
We prove a version of the affine Kempf-Ness theorem for non-algebraic symplectic structures and shifted moment maps, and use it to describe hyperkähler quotients of T*G, where G is a complex reductive group.
Hyperkähler quotients by non-free actions are typically highly singular, but are remarkably still partitioned into smooth hyperkähler manifolds. We show that these partitions are topological stratifications, in a strong sense. We also endow the quotients with global Poisson structures which induce the hyperkähler structures on the strata. Finally, we give a local model which shows that these quotients are locally isomorphic to linear complex-symplectic reductions in the GIT sense. These results can be thought of as the hyperkähler analogues of Sjamaar-Lerman's theorems for symplectic reduction. They are based on a local normal form for the underlying complex-Hamiltonian manifold, which may be of independent interest.
We study singular hyperkähler quotients of the cotangent bundle of a complex semisimple Lie group as stratified spaces whose strata are hyperkähler. We focus on one particular case where the stratification satisfies the frontier condition and the partial order on the set of strata can be described explicitly by Lie theoretic data.
The equations of motion of a charged particle in the field of Yang's $\mathrm{SU}(2)$ monopole in 5-dimensional Euclidean space are derived by applying the Kaluza-Klein formalism to the principal bundle $\mathbb{R}^8\setminus\{0\}\to\mathbb{R}^5\setminus\{0\}$ obtained by radially extending the Hopf fibration $S^7\to S^4$, and solved by elementary methods. The main result is that for every particle trajectory $r:I\to\mathbb{R}^5\setminus\{0\}$, there is a 4-dimensional cone with vertex at the origin on which $r$ is a geodesic. We give an explicit expression of the cone for any initial conditions.
When a compact Lie group acts freely and in a Hamiltonian way on a symplectic manifold, the Marsden-Weinstein theorem says that the reduced space is a smooth symplectic manifold. If we drop the freeness assumption, the reduced space is usually fairly singular, but Sjamaar and Lerman showed that it can still be stratified into smooth symplectic manifolds which “fit together nicely”, in a precise sense. In this thesis, we prove analogues of Sjamaar-Lerman's results in hyperkähler geometry, yielding to the notion of stratified hyperkähler spaces.
We also study examples of stratified hyperkähler spaces coming from hyperkähler quotients of certain moduli spaces of solutions to the so-called Nahm equations. In particular, we prove a Kempf-Ness type theorem which realises these spaces as quasi-projective algebraic varieties. We then focus on an interesting family of examples whose stratification structure can be described explicitly by combinatorial data associated with the root system of a complex semisimple Lie algebra.
Leaving stratified spaces aside, we investigate how Nahm's equations, which are non-linear systems of ODEs, can generate groupoid structures by concatenation of paths, in a manner analogous to the fundamental groupoid of a topological space.
Finally, we study another moduli space of solutions to Nahm's equations, which is a smooth hyperkähler manifold diffeomorphic to a variety studied in geometric representation theory called the universal centraliser. We draw analogies between this space and the moduli space of Higgs bundles, and explain how mirror symmetry naturally enters into the picture.