Denis Auroux

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface Xk obtained by blowing up ℂℙ2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration Wk:Mk→ℂ with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of Xk, and give an explicit correspondence between the deformation parameters for Xk and the cohomology class [B+iω]∈H2(Mk,ℂ).

Singular Lefschetz pencils

We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4–manifold equipped with a “near-symplectic” structure (ie, a closed 2–form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, every near-symplectic 4–manifold (X, ω) can be decomposed into (a) two symplectic Lefschetz fibrations over discs, and (b) a fibre bundle over S 1 which relates the boundaries of the Lefschetz fibrations to each other via a sequence of fibrewise handle additions taking place in a neighbourhood of the zero set of the 2–form. Conversely, from such a decomposition one can recover a near-symplectic structure.

Homological mirror symmetry for punctured spheres

We prove that the wrapped Fukaya category of a punctured sphere (S 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.

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.

ESTIMATED TRANSVERSALITY IN SYMPLECTIC GEOMETRY AND PROJECTIVE MAPS

The main theorem of this paper is a result of estimated transversality with respect to stratifications of jet spaces in the approximately holomorphic category over an almost-complex manifold. The notion of asymptotic ampleness of complex vector bundles over an almost-complex manifold is also discussed, as well as applications to the construction of maps with generic singularities from compact symplectic manifolds to projective spaces.

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.

Regular homotopy of Hurwitz curves

We prove that any two irreducible cuspidal Hurwitz curves and (or, more generally, two curves with -type singularities) in the Hirzebruch surface with the same homology classes and sets of singularities are regular homotopic. Moreover, they are symplectically regular homotopic if and are symplectic with respect to a compatible symplectic form.

Sutured Khovanov homology, Hochschild homology, and the Ozsváth-Szabó spectral sequence

In 2001, Khovanov and Seidel constructed a faithful action of the (m+1)-strand braid group on the derived category of left modules over a quiver algebra, A_m. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.

LEFSCHETZ PENCILS, BRANCHED COVERS AND SYMPLECTIC INVARIANTS

This set of lectures aims to give an overview of Donaldsons theory of linear systems on symplectic manifolds and the algebraic and geometric invariants to which they give rise. After collecting some of the relevant background, we discuss topological, algebraic and symplectic viewpoints on Lefschetz pencils and branched covers of the projective plane. The later lectures discuss invariants obtained by combining this theory with pseudo-holomorphic curve methods.

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

We construct infinitely many families of monotone Lagrangian tori in