arXiv subjects
Publications and source records attributed to Ivan Smith.
We prove that moduli spaces of meromorphic quadratic differentials with simple zeroes on compact Riemann surfaces can be identified with spaces of stability conditions on a class of CY3 triangulated categories defined using quivers with potential associated to triangulated surfaces. We relate the finite-length trajectories of such quadratic differentials to the stable objects of the corresponding stability condition.
We study exact Lagrangian immersions with one double point of a closed orientable manifold K into n-complex-dimensional Euclidean space. Our main result is that if the Maslov grading of the double point does not equal 1 then K is homotopy equivalent to the sphere, and if, in addition, the Lagrangian Gauss map of the immersion is stably homotopic to that of the Whitney immersion, then K bounds a parallelizable (n+1)-manifold. The hypothesis on the Gauss map always holds when n=2k or when n=8k-1. The argument studies a filling of K obtained from solutions to perturbed Cauchy-Riemann equations with boundary on the image f(K) of the immersion. This leads to a new and simplified proof of some of the main results of arXiv:1111.5932, which treated Lagrangian immersions in the case n=2k by applying similar techniques to a Lagrange surgery of the immersion, as well as to an extension of these results to the odd-dimensional case.
We establish an $h$-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space $\R^6_\st$ with exactly one transverse double point. Our construction also yields a Lagrangian embedding $S^1\times S^2\to\R^6_\st$ with vanishing Maslov class.
Consider a Stein manifold M obtained by plumbing cotangent bundles of manifolds of dimension greater than or equal to 3 at points. We prove that the Fukaya category of closed exact Lagrangians with vanishing Maslov class in M is generated by the compact cores of the plumbing. As applications, we classify exact Lagrangian spheres in A2-Milnor fibres of arbitrary dimension, derive constraints on exact Lagrangian fillings of Legendrian unknots in disk cotangent bundles, and prove that the categorical equivalence given by the spherical twist in a homology sphere is typically not realised by any compactly supported symplectomorphism.
There is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces. We investigate symplectic aspects of this relationship, with a view to applications in low-dimensional topology. We construct a derived equivalence between the Fukaya category of a curve and the nilpotent summand of the Fukaya category of the associated complete intersection of two quadrics. This essentially determines the instanton Floer homology of a 3-manifold fibred by genus two curves.
We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two applications to symplectic Khovanov cohomology are included.
We use the quilt formalism of Mau-Wehrheim-Woodward to give a sufficient condition for a finite collection of Lagrangian submanifolds to split-generate the Fukaya category, and deduce homological mirror symmetry for the standard 4-torus. As an application, we study Lagrangian genus two surfaces of Maslov class zero, deriving numerical restrictions on the intersections of such a surface with linear Lagrangian 2-tori in in the 4-torus.
In this note we construct infinitely many distinct simply connected Stein fillings of a certain infinite family of contact 3--manifolds.
We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow (math/0604379, math/0612399) and the authors (math/0701783), before discussing a new approach using family Floer cohomology and the ``wrapped Fukaya category''. The latter, inspired by Viterbo's symplectic homology, emphasises the connection to loop spaces, hence seems particularly suitable when trying to extend the existing theory beyond the simply-connected case.
We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel-Whitney class of the Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically indistinguishable from the zero-section. This implies strong restrictions on their topology. An essentially equivalent result was recently proved independently by Nadler, using a different approach.
Using the symplectic geometry of certain manifolds which appear naturally in Lie theory, we define an invariant which assigns a graded abelian group to an oriented link. The relevant manifolds are transverse slices to certain nilpotent orbits inside sl_{2m}, and intersections of those with regular semisimple orbits. The invariant is conjectured to be equal to Khovanov's combinatorially defined homology theory (with the bigrading of that theory collapsed in a certain way).
Ramanujam's surface M is a contractible affine algebraic surface which is not homeomorphic to the affine plane. For any m>1 the product M^m is diffeomorphic to Euclidean space R^{4m}. We show that, for every m>0, M^m cannot be symplectically embedded into a subcritical Stein manifold. This gives the first examples of exotic symplectic structures on Euclidean space which are convex at infinity. It follows that any exhausting plurisubharmonic Morse function on M^m has at least three critical points, answering a question of Eliashberg. The heart of the argument involves showing a particular Lagrangian torus L inside M cannot be displaced from itself by any Hamiltonian isotopy, via a careful study of pseudoholomorphic discs with boundary on L.
These are notes from lectures given at the Clay Institute Summer School on "Floer homology, gauge theory and low-dimensional topology" (Budapest, 2004). The first part describes as background some of the geometry of symplectic fibre bundles and their monodromy. The second part, overviewing joint work with Paul Seidel, applies these general ideas and Floer cohomology to certain fibre bundles that arise naturally in Lie theory. This constructs an invariant of oriented links in the three-sphere which is conjecturally equal to Khovanov's combinatorial homology theory.
We apply ideas from conformal field theory to study symplectic four-manifolds, by using modular functors to "linearise" Lefschetz fibrations. In Chern-Simons theory this leads to the study of parabolic vector bundles of conformal blocks. Motivated by the Hard Lefschetz theorem, we show the bundles of SU(2) conformal blocks associated to Kaehler surfaces are Brill-Noether special, although the associated flat connexions may be irreducible if the surface is simply-connected and not spin.
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.
We present a new proof of a result due to Taubes: if X is a closed symplectic four-manifold with b_+(X) > 1+b_1(X) and with some positive multiple of the symplectic form a rational class, then the Poincare dual of the canonical class of X may be represented by an embedded symplectic submanifold. The result builds on the existence of Lefschetz pencils on symplectic four-manifolds. We approach the topological problem of constructing submanifolds with locally positive intersections via almost complex geometry. The crux of the argument is that a Gromov invariant counting pseudoholomorphic sections of an associated bundle of symmetric products is non-zero.
According to Taubes, the Gromov invariants of a symplectic four-manifold X with b_+ > 1 satisfy the duality Gr(A) = +/- Gr(K-A), where K is Poincare dual to the canonical class. Extending joint work with Simon Donaldson in math.SG/0012067, we interpret this result in terms of Serre duality on the fibres of a Lefschetz pencil, by proving an analogous symmetry for invariants counting sections of associated bundles of symmetric products. Using similar methods we give a new proof of an existence theorem for symplectic surfaces in four-manifolds with b_+ = 1 and b_1 = 0. This reproves another theorem due to Taubes: two symplectic homology projective planes with negative canonical class and equal volume are symplectomorphic.
We study Lefschetz pencils on symplectic four-manifolds via the associated spheres in the moduli spaces of curves, and in particular their intersections with certain natural divisors. An invariant defined from such intersection numbers can distinguish manifolds with torsion first Chern class. We prove that pencils of large degree always give spheres which behave `homologically' like rational curves; contrastingly, we give the first constructive example of a symplectic non-holomorphic Lefschetz pencil. We also prove that only finitely many values of signature or Euler characteristic are realised by manifolds admitting Lefschetz pencils of genus two curves.