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Denis Auroux

Publications and source records attributed to Denis Auroux.

35 records · Page 2Linked to original sources
arXiv2008

Special Lagrangian fibrations, mirror symmetry and Calabi-Yau double covers

The first part of this paper is a review of the Strominger-Yau-Zaslow conjecture in various settings. In particular, we summarize how, given a pair (X,D) consisting of a Kahler manifold and an anticanonical divisor, families of special Lagrangian tori in X-D and weighted counts of holomorphic discs in X can be used to build a Landau-Ginzburg model mirror to X. In the second part we turn to more speculative considerations about Calabi-Yau manifolds with holomorphic involutions and their quotients. Namely, given a hypersurface H representing twice the anticanonical class in a Kahler manifold X, we attempt to relate special Lagrangian fibrations on X-H and on the (Calabi-Yau) double cover of X branched along H; unfortunately, the implications for mirror symmetry are far from clear.

math.SG↗
arXiv2007

Mirror symmetry and T-duality in the complement of an anticanonical divisor

We study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kahler manifold. In particular, we explore the connections between T-duality and mirror symmetry in concrete examples, and show how quantum corrections arise in this context.

math.SG↗
arXiv2006

A degree doubling formula for braid monodromies and Lefschetz pencils

Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavior of the braid monodromy invariants of the branch curve upon degree doubling of the linear system (from a very ample bundle $L^k$ to $L^{2k}$). As a consequence, we derive a similar degree doubling formula for the mapping class group monodromy of Donaldson's symplectic Lefschetz pencils.

math.SG↗
arXiv2005

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 X_k obtained by blowing up CP^2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W_k:M_k\to\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 X_k, and give an explicit correspondence between the deformation parameters for X_k and the cohomology class [B+iω]\in H^2(M_k,C).

math.AG↗
arXiv2005

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,omega) 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.

math.DG↗
arXiv2005

Some open questions about symplectic 4-manifolds, singular plane curves, and braid group factorizations

The topology of symplectic 4-manifolds is related to that of singular plane curves via the concept of branched covers. Thus, various classification problems concerning symplectic 4-manifolds can be reformulated as questions about singular plane curves. Moreover, using braid monodromy, these can in turn be reformulated in the language of braid group factorizations. While the results mentioned in this paper are not new, we hope that they will stimulate interest in these questions, which remain essentially wide open.

math.GT↗
arXiv2005

A stable classification of Lefschetz fibrations

We study the classification of Lefschetz fibrations up to stabilization by fiber sum operations. We show that for each genus there is a `universal' fibration f^0_g with the property that, if two Lefschetz fibrations over S^2 have the same Euler-Poincare characteristic and signature, the same numbers of reducible singular fibers of each type, and admit sections with the same self-intersection, then after repeatedly fiber summing with f^0_g they become isomorphic. As a consequence, any two compact integral symplectic 4-manifolds with the same values of (c_1^2, c_2, c_1.[w], [w]^2) become symplectomorphic after blowups and symplectic sums with f^0_g.

math.GT↗
arXiv2004

Symplectic 4-manifolds, singular plane curves, and isotopy problems

We give an overview of various recent results concerning the topology of symplectic 4-manifolds and singular plane curves, using branched covers and isotopy problems as a unifying theme. While this paper does not contain any new results, we hope that it can serve as an introduction to the subject, and will stimulate interest in some of the open questions mentioned in the final section.

math.GT↗
arXiv2004

Lagrangian submanifolds and Lefschetz pencils

Given a Lagrangian submanifold in a symplectic manifold and a Morse function on the submanifold, we show that there is an isotopic Morse function and a symplectic Lefschetz pencil on the manifold extending the Morse function to the whole manifold. From this construction we define a sequence of symplectic invariants classifying the isotopy classes of Lagrangian spheres in a symplectic 4-manifold.

math.SG↗
arXiv2004

Mirror symmetry for weighted projective planes and their noncommutative deformations

We study the derived categories of coherent sheaves of weighted projective spaces and their noncommutative deformations, and the derived categories of Lagrangian vanishing cycles of their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves (B-branes) on the weighted projective plane $\CP^2(a,b,c)$ is equivalent to the derived category of vanishing cycles (A-branes) on the affine hypersurface $X=\{x^ay^bz^c=1\}\subset (\C^*)^3$ equipped with an exact symplectic form and the superpotential $W=x+y+z$. Hence, the homological mirror symmetry conjecture holds for weighted projective planes. Moreover, we also show that this mirror correspondence between derived categories can be extended to toric noncommutative deformations of $\CP^2(a,b,c)$ where B-branes are concerned, and their mirror counterparts, non-exact deformations of the symplectic structure of $X$ where A-branes are concerned. We also obtain similar results for other examples such as weighted projective lines or Hirzebruch surfaces.

math.AG↗
arXiv2004

Regular homotopy of Hurwitz curves

We prove that any two irreducible cuspidal Hurwitz curves $C_0$ and $C_1$ (or more generally, curves with A-type singularities) in the Hirzebruch surface $F_N$ with coinciding homology classes and sets of singularities are regular homotopic; and symplectically regular homotopic if $C_0$ and $C_1$ are symplectic with respect to a compatible symplectic form.

math.SG↗
arXiv2004

Lefschetz pencils, branched covers and symplectic invariants

This set of lectures aims to give an overview of Donaldson's 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.

math.SG↗
arXiv2003

Monodromy invariants in symplectic topology

This text is a set of lecture notes for a series of four talks given at I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides a quick overview of symplectic topology and its main tools: symplectic manifolds, almost-complex structures, pseudo-holomorphic curves, Gromov-Witten invariants and Floer homology. The second and third lectures focus on symplectic Lefschetz pencils: existence (following Donaldson), monodromy, and applications to symplectic topology, in particular the connection to Gromov-Witten invariants of symplectic 4-manifolds (following Smith) and to Fukaya categories (following Seidel). In the last lecture, we offer an alternative description of symplectic 4-manifolds by viewing them as branched covers of the complex projective plane; the corresponding monodromy invariants and their potential applications are discussed.

math.SG↗
arXiv2002

Fiber sums of genus 2 Lefschetz fibrations

Using the recent results of Siebert and Tian about the holomorphicity of genus 2 Lefschetz fibrations with irreducible singular fibers, we show that any genus 2 Lefschetz fibration becomes holomorphic after fiber sum with a holomorphic fibration.

math.GT↗
arXiv2000

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.

math.SG↗
arXiv2000

Symplectic maps to projective spaces and symplectic invariants

After reviewing recent results on symplectic Lefschetz pencils and symplectic branched covers of CP^2, we describe a new construction of maps from symplectic manifolds of any dimension to CP^2 and the associated monodromy invariants. We also show that a dimensional induction process makes it possible to describe any compact symplectic manifold by a series of words in braid groups and a word in a symmetric group.

math.GT↗