arXiv subjects
Publications and source records attributed to Laurent Côté.
On a complex contact manifold, or complex symplectic manifold with weight-1 circle action, we construct a sheaf of stable categories carrying a t-structure which is locally equivalent to a microlocalization of the perverse t-structure.
Given a symplectic manifold, can one pack uncountably many Lagrangian submanifolds in a given Hamiltonian isotopy class of this symplectic manifold? We address $C^\infty$ and $C^0$ versions of this question.
We associate an invariant called the completed Tate cohomology to a filtered circle-equivariant spectrum and a complex oriented cohomology theory. We show that when the filtered spectrum is the spectral symplectic cohomology of a Liouville manifold, this invariant depends only on the stable homotopy type of the underlying manifold. We make explicit computations for several complex oriented cohomology theories, including Eilenberg-Maclane spectra, Morava K-theories, their integral counterparts, and complex K-theory. We show that the result for Eilenberg-Maclane spectra depends only on the rational homology, and we use the computations for Morava K-theory to recover the integral homology (as an ungraded group). In a different direction, we use the completed Tate cohomology computations for the complex K-theory to recover the complex K-theory of the underlying manifold from its equivariant filtered Floer homotopy type. A key Floer theoretic input is the computation of local equivariant Floer theory near the orbit of an autonomous Hamiltonian, which may be of independent interest.
Kashiwara showed in 1996 that the categories of microlocalized D-modules can be canonically glued to give a sheaf of categories over a complex contact manifold. Much more recently, and by rather different considerations, we constructed a canonical notion of perverse microsheaves on the same class of spaces. Here we provide a Riemann-Hilbert correspondence.
To a conical symplectic resolution with Hamiltonian torus action, Braden--Proudfoot--Licata--Webster associate a category O, defined using deformation quantization (DQ) modules. It has long been expected, though not stated precisely in the literature, that category O also admits a "Betti-type" realization as the Fukaya--Seidel category of a Lefschetz fibration. In this paper, we confirm that the category O associated to a toric hyperkähler manifold is equivalent to the partially wrapped Fukaya category of a Liouville manifold stopped by the fiber of a J-holomorphic moment map. The proof involves relating earlier DQ-module computations to a new computation of microlocal perverse sheaves. Leveraging known results on (de Rham) hypertoric category O, we deduce several Floer-theoretic consequences, including formality of simple objects and Koszul duality for the (fully) wrapped Fukaya category; conversely, by applying results about microlocal sheaves, we produce a relative Calabi-Yau structure on category O.
We generalize the Cohen-Jones-Segal construction to the Morse-Bott setting. In other words, we define framings for Morse-Bott analogues of flow categories and associate a stable homotopy type to this data. We use this to recover the stable homotopy type of a closed manifold from Morse-Bott theory, and the stable equivariant homotopy type of a closed manifold with the action of a compact Lie group from Morse theory. We use this machinery in Floer theory to construct a genuine circle equivariant model for symplectic cohomology with coefficients in the sphere spectrum. Using the formalism of relative modules, we define equivariant maps to (Thom spectra over) the free loop space of exact, compact Lagrangians. We prove that this map is an equivalence of Borel equivariant spectra when the Lagrangian is the zero section of a cotangent bundle -- an equivariant Viterbo isomorphism theorem over the sphere spectrum.
We develop a purely categorical theory of action filtrations and their associated growth invariants. When specialized to categories of geometric interest, such as the wrapped Fukaya category of a Weinstein manifold, and the bounded derived category of coherent sheaves on a smooth algebraic variety, our categorical action filtrations essentially recover previously studied filtrations of geometric origin. Our approach is built around the notion of a smooth categorical compactification. We prove that a smooth categorical compactification induces well-defined growth invariants, which are moreover preserved under zig-zags of such compactifications. The technical heart of the paper is a method for computing these growth invariants in terms of the growth of certain colimits of (bi)modules. In practice, such colimits arise in both geometric settings of interest. The main applications are: (1) A "quantitative" refinement of homological mirror symmetry, which relates the growth of the Reeb-length filtration on the symplectic geometry side with the growth of filtrations on the algebraic geometry side defined by the order of pole at infinity (often these can be expressed in terms of the dimension of the support of sheaves). (2) A proof that the Reeb-length growth of symplectic cohomology and wrapped Floer cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds for the entropy and polynomial entropy of certain natural endofunctors acting on Fukaya categories.
We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most $3$. As a result, we resolve a well-known conjecture of Orlov for a large class of new examples, including all toric $3$-folds and certain log Calabi--Yau surfaces. On the symplectic side, we study various quantitative questions such as: (1) given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly-supported Hamiltonian diffeomorphism? (2) what is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to our knowledge the first to go beyond the basic flexible/rigid dichotomy.
We study symplectic rigidity phenomena for fibers in cotangent bundles of Riemann surfaces. Our main result can be seen as a generalization to open Riemann surfaces of arbitrary genus of work of Eliashberg and Polterovich on the Nearby Lagrangian Conjecture for $T^* \mathbb{R}^2$. As a corollary, we answer a strong version in dimension $2n=4$ of a question of Eliashberg about linking of Lagrangian disks in $T^* \mathbb{R}^n$, which was previously answered by Ekholm and Smith in dimensions $2n \geq 8$.
Codimension 2 contact submanifolds are the natural generalization of transverse knots to contact manifolds of arbitrary dimension. In this paper, we construct new invariants of codimension 2 contact submanifolds. Our main invariant can be viewed as a deformation of the contact homology algebra of the ambient manifold. We describe various applications of these invariants to contact topology. In particular, we exhibit examples of codimension 2 contact embeddings into overtwisted and tight contact manifolds which are formally isotopic but fail to be isotopic through contact embeddings. We also give new obstructions to certain relative symplectic and Lagrangian cobordisms.
The goal of this note is to describe some constructions of Weinstein manifolds with chaotic Reeb dynamics, and to explain how this property can sometimes be detected directly from the skeleton.
We prove that the ($τ$-weighted, sheaf-theoretic) SL(2,C) Casson-Lin invariant introduced by Manolescu and the first author in [CM19] is generically independent of the parameter $τ$ and additive under connected sums of knots in integral homology 3-spheres. This addresses two questions asked in [CM19]. Our arguments involve a mix of topology, microlocal analysis and algebraic geometry, and rely crucially on the fact that the SL(2,C) Casson-Lin invariant admits an alternative interpretation via the theory of Behrend functions.
We classify weakly exact, rational Lagrangian tori in $T^* \mathbb{T}^2- 0_{\mathbb{T}^2}$ up to Hamiltonian isotopy. This result is related to the classification theory of closed $1$-forms on $\mathbb{T}^n$ and also has applications to symplectic topology. As a first corollary, we strengthen a result due independently to Eliashberg-Polterovich and to Giroux describing Lagrangian tori in $T^* \mathbb{T}^2-0_{\mathbb{T}^2}$ which are homologous to the zero section. As a second corollary, we exhibit pairs of disjoint totally real tori $K_1, K_2 \subset T^*\mathbb{T}^2$, each of which is isotopic through totally real tori to the zero section, but such that the union $K_1 \cup K_2$ is not even smoothly isotopic to a Lagrangian. In the second part of the paper, we study linking of Lagrangian tori in $(\mathbb{R}^4, ω)$ and in rational symplectic $4$-manifolds. We prove that the linking properties of such tori are determined by purely algebro-topological data, which can often be deduced from enumerative disk counts in the monotone case. We also use this result to describe certain Lagrangian embedding obstructions.
Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three-manifolds, similar to the three-manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat connections with fixed holonomy around the meridian of the knot. Thus, our invariant is a sheaf-theoretic SL(2,C) analogue of the singular knot instanton homology of Kronheimer and Mrowka. We prove that for two-bridge and torus knots, the SL(2,C) invariant is determined by the l-degree of the $\widehat{A}$-polynomial. However, this is not true in general, as can be shown by considering connected sums of knots.
We prove some results about linking of Lagrangian tori in the symplectic vector space $(\mathbb{R}^4, ω)$. We show that certain enumerative counts of holomophic disks give useful information about linking. This enables us to prove, for example, that any two Clifford tori are unlinked in a strong sense. We extend work of Dimitroglou Rizell and Evans on linking of monotone Lagrangian tori to a class of non-monotone tori in $\mathbb{R}^4$ and also strengthen their conclusions in the monotone case in $\mathbb{R}^4$.