arXiv subjects

Stéphane Guillermou

Publications and source records attributed to Stéphane Guillermou.

12 recordsLinked to original sources

arXiv2026

Density of fibers for the filtered Fukaya category of $T^*N$

We answer a question of Biran and Cornea about the density of iterated cones of fibers in the Fukaya category of a cotangent bundle. We prove that indeed if we take a dense set of basepoints, the iterated cones of the cotangent fibres are dense in the Filtered Fukaya category. In an appendix we prove that the space of exact Lagrangians in a symplectic manifold is never totally bounded for the spectral distance (unless it is empty). This was implicit in \cite{MCA-VH-CV} for $n=1$ and proved for cotangent bundles of negatively curved manifolds in \cite{A-B-C}.

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arXiv2025

Regular Lagrangians are smooth Lagrangians

We prove that for any element in the $γ$-completion of the space of smooth compact exact Lagrangian submanifolds of a cotangent bundle, if its $γ$-support is a smooth Lagrangian submanifold, then the element itself is a smooth Lagrangian. We also prove that if the $γ$-support of an element in the completion is compact, then it is connected.

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arXiv2024

Twisted generating functions and the nearby Lagrangian conjecture

We prove that, for closed exact embedded Lagrangian submanifolds of cotangent bundles, the homomorphism of homotopy groups induced by the stable Lagrangian Gauss map vanishes. In particular, we prove that this map is null-homotopic for all spheres. The key tool that we introduce in order to prove this is the notion of twisted generating function and we show that every closed exact Lagrangian can be described using such an object, by extending a doubling argument developed in the setting of sheaf theory. Floer theory and sheaf theory constrain the type of twisted generating functions that can appear to a class which is closely related to Waldhausen's tube space, and our main result follows by a theorem of Bökstedt which computes the rational homotopy type of the tube space.

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arXiv2023

The $γ$-support as a micro-support

We prove that for any element $L$ in the completion of the space of smooth compact exact Lagrangian submanifolds of a cotangent bundle equipped with the spectral distance, the $γ$-support of $L$ coincides with the reduced micro-support of its sheaf quantization. As an application, we give a characterization of the Vichery subdifferential in terms of $γ$-support.

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arXiv2023

The singular support of sheaves is $γ$-coisotropic

We prove that the singular support of an element in the derived category of sheaves is $γ$-coisotropic, a notion defined in [Vit22]. We prove that this implies that it is involutive in the sense of Kashiwara-Schapira, but being $γ$-coisotropic has the advantage to be invariant by symplectic homeomorphisms (while involutivity is only invariant by $C^1$ diffeomorphisms) and we give an example of an involutive set that is not $γ$-coisotropic. Along the way we prove a number of results relating the singular support and the spectral norm $γ$ and raise a number of new questions.

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arXiv2022

Sheaves and symplectic geometry of cotangent bundles

This paper is essentially made of the three preprints arXiv:1212.5818, arXiv:1311.0187, arXiv:1603.07876 gathered in a single text, with simplified proofs. We recall several results of the microlocal theory of sheaves of Kashiwara-Schapira and apply them to study the symplectic geometry of cotangent bundles. We explain how we can recover the Gromov nonsqueezing theorem, the Gromov-Eliashberg rigidity theorem, the existence of graph selectors, we prove a three cusps conjecture about curves on the sphere and we recover more recent results on the topology of exact Lagrangian submanifolds of cotangent bundles.

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arXiv2022

Viterbo's spectral bound conjecture for homogeneous spaces

We prove a conjecture of Viterbo about the spectral distance on the space of compact exact Lagrangian submanifolds of a cotangent bundle $T^*M$ in the case where $M$ is a compact homogeneous space: if such a Lagrangian submanifold is contained in the unit ball bundle of $T^*M$, its spectral distance to the zero section is uniformly bounded. This also holds for some immersed Lagrangian submanifolds if we take into account the length of the maximal Reeb chord.

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arXiv2016

The three cusps conjecture

We prove Arnol'd's three cusps conjecture about the front of Legendrian curves in the projectivized cotangent bundle of the $2$-sphere. We use the microlocal theory of sheaves of Kashiwara and Schapira and study the derived category of sheaves on the $2$-sphere with a given smooth Lagrangian microsupport.

math.SG↗

arXiv2015

Construction of sheaves on the subanalytic site

On a real analytic manifold M, we construct the linear subanalytic Grothendieck topology Msal together with the natural morphism of sites $ρ$ from Msa to Msal, where Msa is the usual subanalytic site. Our first result is that the derived direct image functor by $ρ$ admits a right adjoint, allowing us to associate functorially a sheaf (in the derived sense) on Msa to a presheaf on Msa satisfying suitable properties, this sheaf having the same sections that the presheaf on any open set with Lipschitz boundary. We apply this construction to various presheaves on real manifolds, such as the presheaves of functions with temperate growth of a given order at the boundary or with Gevrey growth at the boundary. On a complex manifold endowed with the subanalytic topology, the Dolbeault complexes associated with these new sheaves allow us to obtain various sheaves of holomorphic functions with growth. As an application, we can endow functorially regular holonomic D-modules with a filtration, in the derived sense.

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arXiv2015

Quantization of conic Lagrangian submanifolds of cotangent bundles

Let $M$ be a manifold and $Λ$ a compact exact connected Lagrangian submanifold of $T^*M$. We can associate with $Λ$ a conic Lagrangian submanifold $Λ'$ of $T^*(M\times R)$. We prove that there exists a canonical sheaf $F$ on $M\times R$ whose microsupport is $Λ'$ outside the zero section. We deduce the already known results that the Maslov class of $Λ$ is $0$ and that the projection from $Λ$ to $M$ induces isomorphisms between the homotopy groups.

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arXiv2013

The Gromov-Eliashberg theorem by microlocal sheaf theory

The Gromov-Eliashberg theorem says that the group of symplectomorphisms of a symplectic manifold is C^0-closed in the group of diffeomorphisms. This can be translated into a statement about the Lagrangian submanifolds which are graphs of symplectomorphisms. It is also known that such Lagrangian submanifolds are locally microsupports of sheaves. We explain how we can deduce the Gromov-Eliashberg theorem from the involutivity theorem of Kashiwara and Schapira which says that the microsupport of a sheaf is coisotropic.

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arXiv2004

Equivariant derived category of a complete symmetric variety

Let G be a complex algebraic semi-simple adjoint group and X a smooth complete symmetric G-variety. Let L_i be the irreducible G-equivariant intersection cohomology complexes on X, and L the direct sum of the L_i. Let E= Ext(L,L) be the extension algebra of L, computed in the G-equivariant derived category of X. We considered E as a dg-algebra with differential d=0, and the E_i = Ext(L,L_i) as E-dg-modules. We show that the bounded equivariant derived category of sheaves of C-vector spaces on X is equivalent to the subcategory of the derived category of E-dg-modules generated by the E_i.

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