Assistant Professor
Department of Mathematics
University of Sherbrooke
We introduce a notion of reduction of Dirac realizations induced by a submanifold of the base and give an interpretation in shifted symplectic geometry. It yields, in particular, to a notion of symplectic (resp. quasi-Hamiltonian) reduction where the level can be a submanifold of the dual of the Lie algebra (resp. the group) rather than a point, and explains some disparate constructions in symplectic geometry. This is joint work with Ana Balibanu and Peter Crooks.
The standard notion of symplectic reduction takes as input a moment map for a Lie group action together with an element of the dual of the Lie algebra called the 'level' of the reduction. In joint work with Peter Crooks (arXiv:2107.03198), we introduced a generalization where the level can be a submanifold of the dual of the Lie algebra rather than a point. One of the key features of this construction is a general and systematic way of constructing symplectic manifolds as quotients by groupoid actions. It produces, in particular, interesting Lie theoretic examples, some of which are known (e.g. Ginzburg-Kazhdan's construction of the Moore-Tachikawa varieties as quotients by subgroup schemes of universal centralizers) and some which appear to be new. In this talk, I will explain work in progress with Ana Balibanu on a generalization of this construction in Dirac geometry. In particular, we obtain a notion of quasi-Hamiltonian reduction where the level can be a submanifold of the group. I will also explain how these constructions fit in the theory of shifted symplectic structures.
In 2011, Moore and Tachikawa conjectured the existence of certain complex Hamiltonian varieties which generate two-dimensional TQFTs where the target category has Lie groups as objects and holomorphic symplectic varieties as arrows. It was solved by Ginzburg and Kazhdan using a technique that can be thought of as a “symplectic reduction by a group scheme.” We generalize their construction by introducing a notion of “symplectic reduction by a groupoid along a submanifold.” It recovers many constructions in symplectic geometry as special cases, such as standard symplectic reduction, preimages of Slodowy slices, symplectic implosion, the Mikami-Weinstein reduction, and the Ginzburg-Kazhdan examples. This is joint work with Peter Crooks.
Feix and Kaledin independently showed that the cotangent bundle of any Kähler manifold has a hyperkähler metric on a neighbourhood of its zero section. I will explain a generalization of this result, which reduces the problem of constructing a hyperkähler metric near a complex Lagrangian submanifold in a holomorphic symplectic manifold to the existence of certain deformations of holomorphic symplectic structures. I will then use it to show that any holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler-type has a hyperkähler metric near its identity section. The metric is obtained by constructing a twistor space by lifting special deformations of holomorphic Poisson structures adapted from Hitchin's unobstructedness theorem. This talk is based on arXiv:2011.09282.
Feix and Kaledin independently showed that the cotangent bundle of any Kähler manifold has a hyperkähler metric on a neighbourhood of its zero section. In this talk, I will explain a generalization of this result, which reduces the problem of constructing a hyperkähler metric on a neighbourhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold to the existence of certain deformations of holomorphic symplectic structures. The proof uses twistor theory. I will then use this result to show that any holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler metric near its identity section. The twistor space is constructed by lifting special deformations of holomorphic Poisson structures adapted from Hitchin's unobstructedness theorem. This talk is based on arXiv:2011.09282.
I will discuss the existence of hyperkähler structures on symplectic realizations of holomorphic Poisson manifolds, and show that they always exist when the Poisson manifold has complex dimension two. We obtain this structure by constructing a twistor space by lifting special deformations of the Poisson surface adapted from Hitchin's unobstructedness theorem. In the case of the zero Poisson structure, we recover the Feix-Kaledin hyperkähler structure on the cotangent bundle of a Kähler manifold. This talk is based on arXiv:2011.09282.
There is a rich history of interactions between symplectic and complex geometry. One important aspect of it, Kempf-Ness type theorems, link beautifully the two natural notions “quotient spaces” in these geometries. On the one hand, symplectic reduction, which has roots in classical mechanics, can be performed when a real Lie group acts on a smooth symplectic manifold, and a choice of moment map has been made. On the other hand, Geometric Invariant Theory, which is the main tool for moduli problems in algebraic geometry, produces quotients of complex algebraic varieties by complex Lie groups upon choosing a line bundle with a lift of the action. The natural ground where both structures coexist is Kähler geometry, and in that case, there is a correspondence between these two choices so that the quotients coincide. This mini-course aims to present this result in the general context of non-compact and singular spaces, as well as in hyperkähler geometry, and to illustrate it by many examples, both finite and infinite-dimensional.
In 2011, Moore and Tachikawa conjectured the existence of certain complex Hamiltonian varieties which generate two-dimensional TQFTs where the target category has complex reductive groups as objects and holomorphic symplectic varieties as arrows. It was solved by Ginzburg and Kazhdan using an ad hoc technique which can be thought of as a kind of “symplectic reduction by a group scheme”. We clarify their construction by introducing a general notion of “symplectic reduction by a groupoid along a submanifold”, which generalizes many constructions at once, such as standard symplectic reduction, preimages of Slodowy slices, the Mikami-Weinstein reduction, and the Ginzburg-Kazhdan examples. This is joint work with Peter Crooks.
I will discuss the existence of hyperkähler structures near the identity section of a holomorphic symplectic groupoid, and show that they always exist in complex dimension four, i.e. when the base is a holomorphic Poisson surface. The hyperkähler structure is obtained by constructing the twistor space by pulling back specific deformations of the Poisson surface adapted from Hitchin's unobstructedness result. In the special case of the zero Poisson structure, we recover the Feix-Kaledin hyperkähler structure on the cotangent bundle of a Kähler manifold.
I will discuss the existence of hyperkähler structures on local symplectic groupoids integrating holomorphic Poisson manifolds, and show that they always exist when the base is a Poisson surface. The hyperkähler structure is obtained by constructing the twistor space by lifting specific deformations of the Poisson surface adapted from Hitchin's unobstructedness result. In the special case of the zero Poisson structure, we recover the Feix-Kaledin hyperkähler structure on the cotangent bundle of a Kähler manifold.
From one point of view, Kempf-Ness type theorems give explicit complex-algebraic descriptions of some real symplectic quotients. This is particularly interesting and not obvious when the symplectic form is transcendental and the manifold is non-compact, as is often the case when studying certain gauge-theoretical moduli spaces. In this talk, I will present a general result in this context (for complex affine varieties with non-standard Kähler structures and shifted moment maps) and explain how Nahm’s equations, a system of ODEs extracted from the self-dual Yang-Mills equations, provide non-trivial examples.
Symplectic reductions by non-free group actions are, in general, fairly singular topological spaces. But Sjamaar and Lerman (1991) showed that they can nevertheless be decomposed into smooth symplectic manifolds which “fit together nicely”, and that the symplectic structures are compatible with a global Poisson bracket on some substitute for the algebra of smooth functions. In this talk, I will explain an extension of Sjamaar–Lerman’s result in hyperkähler geometry. I will also present a Kempf-Ness type theorem which gives explicit complex-algebraic expressions (as GIT quotients) of certain real symplectic reductions of complex affine varieties with non-standard Kähler structures and shifted moment maps. This theorem will then be applied to singular hyperkähler reductions constructed from moduli spaces of solutions to Nahm’s equations (a reduction of the self-dual Yang-Mills equations), giving interesting Lie theoretical examples.
I will review the close relationship between the theories of quotients in symplectic and algebraic geometries, and present new results in that direction. In particular, I will explain how Nahm’s equations, a system of non-linear ODEs extracted from the self-dual Yang-Mills equations, produce non-trivial examples of this relationship.
Symplectic reductions by non-free group actions are, in general, fairly singular topological spaces. But Sjamaar and Lerman (1991) showed that they can nevertheless be decomposed into smooth symplectic manifolds which “fit together nicely”, and that the symplectic structures are compatible with a global Poisson bracket on some substitute for the algebra of smooth functions. In this talk, I will explain an extension of Sjamaar–Lerman’s result in hyperkähler geometry. I will also present a general Kempf-Ness type theorem which gives explicit complex-algebraic expressions (as GIT quotients) of certain real symplectic reductions of complex affine varieties with non-standard Kähler structures and shifted moment maps. This theorem will then be applied to singular hyperkähler reductions constructed from moduli spaces of solutions to Nahm’s equations, giving interesting Lie theoretical examples which can be analyzed algebraically.
These two lectures will survey the theory of Nahm’s equations and their various hyperkähler moduli spaces, focusing on the case of bounded intervals. In particular, I will review Kronheimer’s result that the moduli space of solutions to Nahm’s equations on a compact interval gives a hyperkähler structure on the cotangent bundle of a complex reductive group, and also discuss Nahm’s equations with poles.
I will review the close relationship between the theories of quotients in symplectic and algebraic geometries, and present new results in that direction. In particular, I will explain how Nahm’s equations, a system of non-linear ODEs extracted from the self-dual Yang-Mills equations, produce non-trivial examples of this relationship.
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 might be singular, but Sjamaar-Lerman (1991) showed that it can still be partitioned into smooth symplectic manifolds which "fit together nicely" in the sense that they form a stratification. The goal of this talk is to discuss an analogue of this result in hyperkähler geometry. I will also explain how singular hyperkähler quotients have natural complex analytic structures with holomorphic Poisson brackets, and that those structures are locally isomorphic to complex-symplectic GIT quotients of affine spaces.
I will describe a family of 2d TQFTs, due to Moore-Tachikawa, which take values in a category whose objects are Lie groups and whose morphisms are holomorphic symplectic varieties.
Symplectic reduction is the natural quotient construction in symplectic geometry. Given a free and Hamiltonian action of a compact Lie group G on a symplectic manifold M, this produces a new symplectic manifold of dimension dim(M) - 2 dim(G). If we drop the freeness assumption, the reduced space is usually fairly singular, but Sjamaar-Lerman showed that we can still decompose it into smooth symplectic manifolds which “fit together nicely” in a precise sense. In this talk, I will explain how to get an analogue of Sjamaar-Lerman’s result in hyperkähler geometry and give interesting examples coming from the so-called Nahm equations.
I will explain a beautiful link between differential and algebraic geometry, called the Kempf-Ness theorem, which says that the natural notions of "quotient spaces" in the symplectic and algebraic categories can often be identified. I will then introduce hyperkähler manifolds and outline how the Kempf-Ness theorem can be used to endow certain algebraic varieties with hyperkähler structures.
This talk consists of two different topics related to Nahm’s equations. In the first part, I will explain how concatenation of solutions to Nahm’s equations on compact intervals give groupoid structures, in a manner analogous to the fundamental groupoid of a topological space. In the second part, I will draw analogies between the moduli space of solutions to Nahm’s equations on an open interval with fixed poles at the endpoints and the moduli space of Higgs bundles over a curve. Both spaces are hyperkähler manifolds and have fibrations onto vector spaces making them completely integrable systems. Moreover, the mirror symmetry program for moduli spaces of Higgs bundles has many features which have analogues in this Nahm moduli space, such as dual fibrations and non-trivial branes coming from involutions.
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 might be singular, but Sjamaar-Lerman (1991) showed that it can still be partitioned into smooth symplectic manifolds which “fit together nicely” in the sense that they form a stratification. In hyperkähler geometry, there is an analogue of symplectic reduction which has been a very important tool for constructing new examples of these special manifolds. In the first part of this talk, I will explain how Sjamaar-Lerman’s results can be extended to this setting, namely, hyperkähler quotients by non-free actions are stratified spaces whose strata are hyperkähler. In the second part, I will review the notion of Nahm’s equations and use them to give interesting examples of such stratified hyperkähler spaces.
Symplectic reduction is the natural quotient construction for symplectic manifolds. Given a free and proper action of a Lie group G on a symplectic manifold M, this process produces a new symplectic manifold of dimension dim(M) - 2 dim(G). For non-free actions, however, the result is usually fairly singular. But Sjamaar-Lerman (1991) showed that the singularities can be understood quite precisely: symplectic reductions by non-free actions are partitioned into smooth symplectic manifolds, and these manifolds fit together nicely, in the sense that they form a stratification.
Symplectic reduction has an analogue in hyperkähler geometry, which has been a very important tool for constructing new examples of these special manifolds. In this talk, I will explain how Sjamaar-Lerman’s results can be extended to this setting, namely, hyperkähler quotients by non-free actions are stratified spaces whose strata are hyperkähler.
Symplectic reduction is the natural quotient construction for symplectic manifolds. Given a free and proper action of a Lie group G on a symplectic manifold M, this process produces a new symplectic manifold of dimension dim(M) - 2 dim(G). For non-free actions, however, the result is usually fairly singular. But Sjamaar-Lerman (1991) showed that the singularities can be understood quite precisely: symplectic reductions by non-free actions are partitioned into smooth symplectic manifolds, and these manifolds fit nicely together in the sense that they form a stratification.
Symplectic reduction has an analogue in hyperkähler geometry, which has been a very important tool for constructing new examples of these special manifolds. In this talk, I will explain how Sjamaar-Lerman’s results can be extended to this setting, namely, hyperkähler quotients by non-free actions are stratified spaces whose strata are hyperkähler.
I will explain a beautiful link between differential and algebraic geometry, called the Kempf-Ness Theorem, which says that the natural notions of "quotient spaces" in the symplectic and algebraic categories can often be identified. The result will be presented in its most general form where actions are not necessarily free and hence I will also introduce the notion of stratified spaces.
A stratified space is a topological space that can be decomposed into a union of manifolds of possibly different dimensions. After reviewing this theory I will construct a family of stratified hyperkahler spaces whose stratification can be described explicitly using combinatorial data associated to a root system. This construction uses a tool from gauge theory called Nahm equations, which I will also review in the talk.
Nahm equations are the reduction of the self-dual Yang-Mills equations to one-dimension. By considering the moduli space of solutions to such equations on different intervals and with appropriate boundary conditions, we get a plethora of examples of hyperkahler manifolds. I will review this theory and link it with an interesting conjectural TQFT with values in a category of hyperkahler manifolds. Finally, I will introduce a family of stratified hyperkahler spaces constructed with Nahm equations and whose stratification can be described explicitly using combinatorial data associated to a root system.
Moduli spaces of Nahm equations are a useful tool to obtain hyperkahler structures on some manifolds associated to a complex semisimple Lie group G. For example, the cotangent bundle and coadjoint orbits of G can be endowed with a hyperkahler structure in this way. After reviewing this theory, we introduce a family of stratified hyperkahler spaces obtained by Nahm equations whose stratifications can be described explicitly via the root system.
A hyperkähler manifold is a Riemannian manifold $(M,g)$ with three complex structures $I,J,K$ satisfying the quaternion relations, i.e. $I^2=J^2=K^2=IJK=-1$, and such that $(M,g)$ is Kähler with respect to each of them. I will describe a construction due to Kronheimer which gives such a structure on the cotangent bundle of any complex reductive group.