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Denis Auroux

Publications and source records attributed to Denis Auroux.

At least 19 recordsLinked to original sources
arXiv2025

Lagrangian Floer theory, from geometry to algebra and back again

We survey various aspects of Floer theory and its place in modern symplectic geometry, from its introduction to address classical conjectures of Arnold about Hamiltonian diffeomorphisms and Lagrangian submanifolds, to the rich algebraic structures captured by the Fukaya category, and finally to the idea, motivated by mirror symmetry, of a "geometry of Floer theory" centered around family Floer cohomology and local-to-global principles for Fukaya categories.

math.SG↗
arXiv2023

Holomorphic discs of negative Maslov index and extended deformations in mirror symmetry

The SYZ approach to mirror symmetry for log Calabi-Yau manifolds starts from a Lagrangian torus fibration on the complement of an anticanonical divisor. A mirror space is constructed by gluing local charts (moduli spaces of local systems on generic torus fibers) via wall-crossing transformations which account for corrections to the analytic structure of moduli spaces of objects of the Fukaya category induced by bubbling of Maslov index 0 holomorphic discs, and made into a Landau-Ginzburg model by equipping it with a regular function (the superpotential) which enumerates Maslov index 2 holomorphic discs. When they occur, holomorphic discs of negative Maslov index deform this picture by introducing inconsistencies in the wall-crossing transformations, so that the mirror is no longer an analytic space; the geometric features of the corrected mirror can be understood in the language of extended deformations of Landau-Ginzburg models. We illustrate this phenomenon (and show that it actually occurs) by working through the construction for an explicit example (a log Calabi-Yau 4-fold obtained by blowing up a toric variety), and discuss a family Floer approach to the geometry of the corrected mirror in this setting. Along the way, we introduce a Morse-theoretic model for family Floer theory which may be of independent interest.

math.SG↗
arXiv2021

Homological mirror symmetry for hypersurfaces in $(\mathbb{C}^*)^n$

We prove a homological mirror symmetry result for maximally degenerating families of hypersurfaces in $(\mathbb{C}^*)^n$ (B-model) and their mirror toric Landau-Ginzburg A-models. The main technical ingredient of our construction is a "fiberwise wrapped" version of the Fukaya category of a toric Landau-Ginzburg model. With the definition in hand, we construct a fibered admissible Lagrangian submanifold whose fiberwise wrapped Floer cohomology is isomorphic to the ring of regular functions of the hypersurface. It follows that the derived category of coherent sheaves of the hypersurface quasi-embeds into the fiberwise wrapped Fukaya category of the mirror. We also discuss an extension to complete intersections.

math.SG↗
arXiv2021

Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves

Mirror symmetry for higher genus curves is usually formulated and studied in terms of Landau-Ginzburg models; however the critical locus of the superpotential is arguably of greater intrinsic relevance to mirror symmetry than the whole Landau-Ginzburg model. Accordingly, we propose a new approach to the A-model of the mirror, viewed as a trivalent configuration of rational curves together with some extra data at the nodal points. In this context, we introduce a version of Lagrangian Floer theory and the Fukaya category for trivalent graphs, and show that homological mirror symmetry holds, namely, that the Fukaya category of a trivalent configuration of rational curves is equivalent to the derived category of a non-Archimedean generalized Tate curve. To illustrate the concrete nature of this equivalence, we show how explicit formulas for theta functions and for the canonical map of the curve arise naturally under mirror symmetry.

math.SG↗
arXiv2021

Fukaya categories of surfaces, spherical objects, and mapping class groups

We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least two whose Chern character represents a non-zero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank one local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions, and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included.

math.SG↗
arXiv2020

GPP, the Generic Preprocessor

In computer science, a preprocessor (or macro processor) is a tool that programatically alters its input, typically on the basis of inline annotations, to produce data that serves as input for another program. Preprocessors are used in software development and document processing workflows to translate or extend programming or markup languages, as well as for conditional or pattern-based generation of source code and text. Early preprocessors were relatively simple string replacement tools that were tied to specific programming languages and application domains, and while these have since given rise to more powerful, general-purpose tools, these often require the user to learn and use complex macro languages with their own syntactic conventions. In this paper, we present GPP, an extensible, general-purpose preprocessor whose principal advantage is that its syntax and behaviour can be customized to suit any given preprocessing task. This makes GPP of particular benefit to research applications, where it can be easily adapted for use with novel markup, programming, and control languages.

cs.PL↗
arXiv2017

Speculations on homological mirror symmetry for hypersurfaces in $(\mathbb{C}^*)^n$

Given an algebraic hypersurface $H=f^{-1}(0)$ in $(\mathbb{C}^*)^n$, homological mirror symmetry relates the wrapped Fukaya category of $H$ to the derived category of singularities of the mirror Landau-Ginzburg model. We propose an enriched version of this picture which also features the wrapped Fukaya category of the complement $(\mathbb{C}^*)^n\setminus H$ and the Fukaya-Seidel category of the Landau-Ginzburg model $((\mathbb{C}^*)^n,f)$. We illustrate our speculations on simple examples, and sketch a proof of homological mirror symmetry for higher-dimensional pairs of pants.

math.SG↗
arXiv2015

Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces

We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface $H$ in a toric variety $V$ we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of $V\times\mathbb{C}$ along $H\times 0$, under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to $H$. The main applications concern affine hypersurfaces of general type, for which our results provide a geometric basis for various mirror symmetry statements that appear in the recent literature. We also obtain analogous results for complete intersections.

math.SG↗
arXiv2014

Infinitely many monotone Lagrangian tori in $\mathbb{R}^6$

We construct infinitely many families of monotone Lagrangian tori in $\mathbb{R}^6$, no two of which are related by Hamiltonian isotopies (or symplectomorphisms). These families are distinguished by the (arbitrarily large) numbers of families of Maslov index 2 pseudo-holomorphic discs that they bound.

math.SG↗
arXiv2014

Homological mirror symmetry for punctured spheres

We prove that the wrapped Fukaya category of a punctured sphere ($S^2$ with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the Landau-Ginzburg model.

math.AG↗
arXiv2013

On Khovanov-Seidel quiver algebras and bordered Floer homology

We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard-Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule defined by Khovanov and Seidel.

math.GT↗
arXiv2013

A beginner's introduction to Fukaya categories

The goal of these notes is to give a short introduction to Fukaya categories and some of their applications. The first half of the text is devoted to a brief review of Lagrangian Floer (co)homology and product structures. Then we introduce the Fukaya category (informally and without a lot of the necessary technical detail), and briefly discuss algebraic concepts such as exact triangles and generators. Finally, we mention wrapped Fukaya categories and outline a few applications to symplectic topology, mirror symmetry and low-dimensional topology. This text is based on a series of lectures given at a Summer School on Contact and Symplectic Topology at Université de Nantes in June 2011.

math.SG↗
arXiv2010

Fukaya categories of symmetric products and bordered Heegaard-Floer homology

The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of Fukaya categories of symmetric products and Lagrangian correspondences. More specifically, we give a description of the algebra A(F) which appears in the work of Lipshitz, Ozsvath and Thurston in terms of (partially wrapped) Floer homology for product Lagrangians in the symmetric product, and outline how bordered Heegaard-Floer homology itself can conjecturally be understood in this language.

math.GT↗
arXiv2010

Fukaya categories and bordered Heegaard-Floer homology

We outline an interpretation of Heegaard-Floer homology of 3-manifolds (closed or with boundary) in terms of the symplectic topology of symmetric products of Riemann surfaces, as suggested by recent work of Tim Perutz and Yanki Lekili. In particular we discuss the connection between the Fukaya category of the symmetric product and the bordered algebra introduced by Robert Lipshitz, Peter Ozsvath and Dylan Thurston, and recast bordered Heegaard-Floer homology in this language.

math.GT↗
arXiv2009

The canonical pencils on Horikawa surfaces

We calculate the monodromies of the canonical Lefschetz pencils on a pair of homeomorphic Horikawa surfaces. We show in particular that the (pluri)canonical pencils on these surfaces have the same monodromy groups, and are related by a "partial twisting" operation.

math.GT↗
arXiv2009

Special Lagrangian fibrations, wall-crossing, and mirror symmetry

In this survey paper, we briefly review various aspects of the SYZ approach to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on Lagrangian fibrations and wall-crossing phenomena in Floer homology. Various examples are presented, some of them new.

math.SG↗