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Ivan Smith

Publications and source records attributed to Ivan Smith.

At least 37 records · Page 2Linked to original sources
arXiv2020

Fukaya-Seidel categories of Hilbert schemes and parabolic category $\mathcal{O}$

We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein-Gelfan'd-Gelfan'd category $\mathcal{O}$. As an application, we give a new geometric construction of the spectral sequence from annular to ordinary Khovanov homology. The heart of the paper is the development of a cylindrical model to compute Fukaya categories of (affine open subsets of) Hilbert schemes of quasi-projective surfaces, which may be of independent interest.

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arXiv2020

Floer theory of higher rank quiver 3-folds

We study threefolds $Y$ fibred by $A_m$-surfaces over a curve $S$ of positive genus. An ideal triangulation of $S$ defines, for each rank $m$, a quiver $Q(Δ_m)$, hence a $CY_3$-category $(C,W)$ for any potential $W$ on $Q(Δ_m)$. We show that for $ω$ in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of $(Y,ω)$ is quasi-isomorphic to $(C,W_{[ω]})$ for a certain generic potential $W_{[ω]}$. This partially establishes a conjecture of Goncharov concerning `categorifications' of cluster varieties of framed $PGL_{m+1}$-local systems on $S$, and gives a symplectic geometric viewpoint on results of Gaiotto, Moore and Neitzke.

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arXiv2019

Symplectic topology of K3 surfaces via mirror symmetry

We study the symplectic topology of certain K3 surfaces (including the "mirror quartic" and "mirror double plane"), equipped with certain Kähler forms. In particular, we prove that the symplectic Torelli group may be infinitely generated, and derive new constraints on Lagrangian tori. The key input, via homological mirror symmetry, is a result of Bayer and Bridgeland on the autoequivalence group of the derived category of an algebraic K3 surface of Picard rank one.

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arXiv2019

Irrationality and monodromy for cubic threefolds

We show the cohomological monodromy for the universal family of smooth cubic threefolds does not factor through the genus five mapping class group. This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds.

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arXiv2019

Rational equivalence and Lagrangian tori on K3 surfaces

Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L in X to the skyscraper sheaf of a point y of Y. We show there are Lagrangian tori with vanishing Maslov class in X whose class in the Grothendieck group of the Fukaya category is not generated by Lagrangian spheres. This is mirror to a statement about the `Beauville--Voisin subring' in the Chow groups of Y, and fits into a conjectural relationship between Lagrangian cobordism and rational equivalence of algebraic cycles.

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arXiv2019

Bounds on Wahl singularities from symplectic topology

Let X be a minimal surface of general type with positive geometric genus ($b_+ > 1$) and let $K^2$ be the square of its canonical class. Building on work of Khodorovskiy and Rana, we prove that if X develops a Wahl singularity of length $\ell$ in a Q-Gorenstein degeneration, then $\ell \leq 4K^2 + 7$. This improves on the current best-known upper bound due to Lee ($\ell \leq 400(K^2)^4$). Our bound follows from a stronger theorem constraining symplectic embeddings of certain rational homology balls in surfaces of general type. In particular, we show that if the rational homology ball $B_{p,1}$ embeds symplectically in a quintic surface, then $p \leq 12$, partially answering the symplectic version of a question of Kronheimer.

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arXiv2018

Khovanov homology from Floer cohomology

This paper realises the Khovanov homology of a link in the 3-sphere as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k of characteristic zero. Here we prove the symplectic cup and cap bimodules which relate different symplectic arc algebras are themselves formal over k, and construct a long exact triangle for symplectic Khovanov cohomology. We then prove the symplectic and combinatorial arc algebras are isomorphic over the integers in a manner compatible with the cup bimodules. It follows that Khovanov homology and symplectic Khovanov cohomology co-incide in characteristic zero.

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arXiv2017

Nearby Lagrangian fibers and Whitney sphere links

Let n>3, and let L be a Lagrangian embedding of an n-disk into the cotangent bundle of n-dimensional Euclidean space that agrees with the cotangent fiber over a non-zero point x outside a compact set. Assume that L is disjoint from the cotangent fiber at the origin. The projection of L to the base extends to a map of the n-sphere into the complement of the origin in Euclidean n-space . We show that this map is homotopically trivial, answering a question of Y. Eliashberg. We give a number of generalizations of this result, including homotopical constraints on embedded Lagrangian disks in the complement of another Lagrangian submanifold, and on two-component links of immersed Lagrangian spheres with one double point in 2n-dimensional space, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from the authors' previous work, constructing bounding manifolds from moduli spaces of Floer-holomorphic disks, with symplectic field theory.

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arXiv2017

Markov numbers and Lagrangian cell complexes in the complex projective plane

We study Lagrangian embeddings of a class of two-dimensional cell complexes $L_{p,q}$ into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type $\frac{1}{p^2}(pq-1,1)$ (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into $\mathbf{CP}^2$ then $p$ is a Markov number and we completely characterise $q$. We also show that a collection of Lagrangian pinwheels $L_{p_i,q_i}$, $i=1,\ldots,N$, cannot be made disjoint unless $N\leq 3$ and the $p_i$ form part of a Markov triple. These results are the symplectic analogue of a theorem of Hacking and Prokhorov, which classifies complex surfaces with quotient singularities admitting a $\mathbf{Q}$-Gorenstein smoothing whose general fibre is $\mathbf{CP}^2$.

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arXiv2017

Symplectomorphisms of exotic discs

We construct a symplectic structure on a disc that admits a compactly supported symplectomorphism which is not smoothly isotopic to the identity. The symplectic structure has an overtwisted concave end; the construction of the symplectomorphism is based on a unitary version of the Milnor-Munkres pairing. En route, we introduce a symplectic analogue of the Gromoll filtration.

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arXiv2016

The monotone wrapped Fukaya category and the open-closed string map

We build the wrapped Fukaya category W(E) for any monotone symplectic manifold, convex at infinity. We define the open-closed and closed open-string maps. We study their algebraic properties and prove that the string maps are compatible with the eigenvalue splitting of W(E). We extend Abouzaid's generation criterion from the exact to the monotone setting. We construct an acceleration functor from the compact Fukaya category which on Hochschild (co)homology commutes with the string maps and the canonical map from quantum cohomology QH(E) to symplectic cohomology SH(E). We define the QH(E)- and SH(E)-module structure on the Hochschild (co)homology of W(E) which is compatible with the string maps. The module and unital algebra structures, and the generation criterion, also hold for the compact Fukaya category F(E), and also hold for closed monotone symplectic manifolds. As an application, we show that the wrapped category of any monotone negative line bundle over any projective space is proper (cohomologically finite). For any monotone negative line bundle E over a toric Fano variety, we show that SH(E) is non-trivial and that W(E) contains an essential non-displaceable monotone Lagrangian torus.

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arXiv2015

The symplectic arc algebra is formal

We prove a formality theorem for the Fukaya categories of the symplectic manifolds underlying symplectic Khovanov cohomology, over fields of characteristic zero. The key ingredient is the construction of a degree one Hochschild cohomology class on a Floer A-infinity algebra associated to the (k,k)-nilpotent slice Y, obtained by counting holomorphic discs which satisfy a suitable conormal condition at infinity in a partial compactification of Y. The partial compactification is obtained as the Hilbert scheme of a partial compactification of a Milnor fibre. A sequel to this paper will prove formality of the symplectic cup and cap bimodules, and infer that symplectic Khovanov cohomology and Khovanov cohomology have the same total rank over characteristic zero fields.

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arXiv2015

Lagrangian exotic spheres

Let k>2. We prove that the cotangent bundles of oriented homotopy (2k-1)-spheres S and S' are symplectomorphic only if the classes defined by S and S' agree up to sign in the quotient group of oriented homotopy spheres modulo those which bound parallelizable manifolds. We also show that if the connect sum of real projective space of dimension (4k-1) and a homotopy (4k-1)-sphere admits a Lagrangian embedding in complex projective space, then twice the homotopy sphere framed bounds. The proofs build on previous work of Abouzaid and the authors, in combination with a new cut-and-paste argument, which also gives rise to some interesting explicit exact Lagrangian embeddings into plumbings. As another application, we show that there are re-parameterizations of the zero-section in the cotangent bundle of a sphere which are not Hamiltonian isotopic (as maps, rather than as submanifolds) to the original zero-section.

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arXiv2015

Quiver algebras as Fukaya categories

We embed triangulated categories defined by quivers with potential arising from ideal triangulations of marked bordered surfaces into Fukaya categories of quasi-projective 3-folds associated to meromorphic quadratic differentials. Together with previous results, this yields non-trivial computations of spaces of stability conditions on Fukaya categories of symplectic 6-manifolds.

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arXiv2015

Conifold transitions and Mori theory

We show there is a symplectic conifold transition of a projective 3-fold which is not deformation equivalent to any Kaehler manifold. The key ingredient is Mori's classification of extremal rays on smooth projective 3-folds. It follows that there is a Lagrangian sphere in a projective variety which is not the vanishing cycle of any Kaehler degeneration, answering a question of Donaldson.

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arXiv2014

Exact Lagrangian immersions with a single double point

Let k > 2. We show that if a closed orientable 2k-manifold K, with Euler characteristic not equal to -2, admits an exact Lagrangian immersion into complex Euclidean 2k-space with one transverse double point and no other self-intersections, then K is diffeomorphic to the standard sphere. The proof combines Floer homological arguments with a detailed study of moduli spaces of holomorphic disks with boundary in a monotone Lagrangian submanifold obtained by Lagrange surgery on K.

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arXiv2014

A symplectic prolegomenon

A symplectic manifold gives rise to a triangulated A-infinity category, the derived Fukaya category, which encodes information on Lagrangian submanifolds and dynamics as probed by Floer cohomology. This survey aims to give some insight into what the Fukaya category is, where it comes from and what symplectic topologists want to do with it.

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