Floer Cohomology Research Articles

Hamiltonian isotopies of relatively exact Lagrangians are orientation-preserving

Abstract Given a closed, orientable Lagrangian submanifold L in a symplectic manifold (X, ω), we show that if L is relatively exact then any Hamiltonian diffeomorphism preserving L setwise must preserve its orientation. In contrast to previous results in this direction, there are no spin hypotheses on L. Curiously, the proof uses only mod 2 coefficients in its singular and Floer cohomology rings.

Erratum to “Poisson geometry, monoidal Fukaya categories, and commutative Floer cohomology rings”

This erratum corrects the false submisson date displayed on the last page of [Enseign. Math. 70, 313–381 (2024)].

Functoriality in categorical symplectic geometry

Categorical symplectic geometry is the study of a rich collection of invariants of symplectic manifolds, including the Fukaya A ∞ A_\infty -category, Floer cohomology, and symplectic cohomology. Beginning with work of Wehrheim and Woodward in the late 2000s, several authors have developed techniques for functorial manipulation of these invariants. We survey these functorial structures, including Wehrheim and Woodward’s quilted Floer cohomology and functors associated to Lagrangian correspondences, Fukaya’s alternate approach to defining functors between Fukaya A ∞ A_\infty -categories, and the second author’s ongoing construction of the symplectic ( A ∞ , 2 ) (A_\infty ,2) -category. In the last section, we describe a number of direct and indirect applications of this circle of ideas, and propose a conjectural version of the Barr–Beck monadicity criterion in the context of the Fukaya A ∞ A_\infty -category.

Poisson geometry, monoidal Fukaya categories, and commutative Floer cohomology rings

We describe connections between concepts arising in Poisson geometry and the theory of Fukaya categories. The key concept is that of a symplectic groupoid, which is an integration of a Poisson manifold. The Fukaya category of a symplectic groupoid is monoidal, and it acts on the Fukaya categories of the symplectic leaves of the Poisson structure. Conversely, we consider a wide range of known monoidal structures on Fukaya categories and observe that they all arise from symplectic groupoids. We also use the picture developed to resolve a conundrum in Floer theory: why are some Lagrangian Floer cohomology rings commutative?

Homological Lagrangian monodromy for some monotone tori

Given a Lagrangian submanifold L in a symplectic manifold X , the homological Lagrangian monodromy group \mathcal{H}_{L} describes how Hamiltonian diffeomorphisms of X preserving L setwise act on H_{*}(L) . We begin a systematic study of this group when L is a monotone Lagrangian n -torus. Among other things, we describe \mathcal{H}_{L} completely when L is a monotone toric fibre, make significant progress towards classifying the groups that can occur for n=2 , and make a conjecture for general n . Our classification results rely crucially on the arithmetic properties of Floer cohomology rings.

Brieskorn spheres, cyclic group actions and the Milnor conjecture

AbstractIn this paper we further develop the theory of equivariant Seiberg–Witten–Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain a number of applications. First, we show that the knot concordance invariants defined by the first author satisfy for torus knots, whenever is a prime not dividing . Since is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture. Second, we prove that a free cyclic group action on a Brieskorn homology 3‐sphere does not extend smoothly to any homology 4‐ball bounding . In the case of a non‐free cyclic group action of prime order, we prove that if the rank of is greater than times the rank of , then the ‐action on does not extend smoothly to any homology 4‐ball bounding . Third, we prove that for all but finitely many primes a similar non‐extension result holds in the case that the bounding 4‐manifold has positive‐definite intersection form. Finally, we also prove non‐extension results for equivariant connected sums of Brieskorn homology spheres.

The wrapped Fukaya category for semi-toric Calabi–Yau

We introduce the wrapped Donaldson–Fukaya category of a (generalized) semi-toric SYZ fibration with Lagrangian section satisfying a tameness condition at infinity. Examples include the Gross fibration on the complement of an anti-canonical divisor in a toric Calabi–Yau 3-fold. We compute the wrapped Floer cohomology of a Lagrangian section and find that it is the algebra of functions on the Hori–Vafa mirror. The latter result is the key step in proving homological mirror symmetry for this case. The techniques developed here allow the construction in general of the wrapped Fukaya category on an open Calabi–Yau manifold carrying an SYZ fibration with nice behavior at infinity. We discuss the relation of this to the algebraic vs analytic aspects of mirror symmetry.

The unbounded Lagrangian spectral norm and wrapped Floer cohomology

We investigate the question of whether the spectral metric on the orbit space of a fiber in the disk cotangent bundle of a closed manifold, under the action of the compactly supported Hamiltonian diffeomorphism group, is bounded. We utilize wrapped Floer cohomology to define the spectral invariant of an admissible Lagrangian submanifold within a Weinstein domain. We show that the pseudo-metric derived from this spectral invariant is a valid Ham-invariant metric. Furthermore, we establish that the spectral metric on the orbit space of an admissible Lagrangian is bounded if and only if the wrapped Floer cohomology vanishes. Consequently, we prove that the Lagrangian Hofer diameter of the orbit space for any fiber in the disk cotangent bundle of a closed manifold is infinite.

Knot concordance invariants from Seiberg–Witten theory and slice genus bounds in 4-manifolds

We construct a new family of knot concordance invariants [Formula: see text], where [Formula: see text] is a prime number. Our invariants are obtained from the equivariant Seiberg–Witten–Floer cohomology, constructed by the author and Hekmati, applied to the degree [Formula: see text] cyclic cover of [Formula: see text] branched over [Formula: see text]. In the case [Formula: see text], our invariant [Formula: see text] shares many similarities with the knot Floer homology invariant [Formula: see text] defined by Hom and Wu. Our invariants [Formula: see text] give lower bounds on the genus of any smooth, properly embedded, homologically trivial surface bounding [Formula: see text] in a definite [Formula: see text]-manifold with boundary [Formula: see text].

Sections and Unirulings of Families over $\mathbb{P}^{1}$

AbstractWe consider morphisms $\pi : X \to \mathbb{P}^{1}$ of smooth projective varieties over $\mathbb{C}$. We show that if π has at most one singular fibre, then X is uniruled and π admits sections. We reach the same conclusions, but with genus zero multisections instead of sections, if π has at most two singular fibres, and the first Chern class of X is supported in a single fibre of π.To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon’s virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections.

SL(2,C) Floer Cohomology for Surgeries on Some Knots

We establish a relationship between HP(Y), the Abouzaid–Manolescu sheaf-theoretic SL(2,C) Floer cohomology for a surgery Y on a small knot in S3 and Curtis’ SL(2,C) Casson invariant. We use this to compute HP for most surgeries on two-bridge knots. We also compute HP for surgeries on two nonsmall knots, the granny and square knots. We provide a partial calculation of the framed sheaf-theoretic SL(2,C) Floer cohomology HP#(Y) for surgeries on two-bridge knots and apply the data we obtain to show the nonexistence of a surgery exact triangle for HP#.

The symplectic cohomology of magnetic cotangent bundles

We construct a family version of symplectic Floer cohomology for magnetic cotangent bundles, without any restrictions on the magnetic form, using the dissipative method for compactness introduced by Groman (2023). As an application, we deduce that if N is a closed orientable manifold and is a magnetic form that is not weakly exact, then the \pi_1 1-sensitive Hofer–Zehnder capacity of any compact set in the magnetic cotangent bundle determined by \sigma is finite.

Categorical action filtrations via localization and the growth as a symplectic invariant

Abstract We develop a purely categorical theory of action filtrations and their associated growth invariants. When specialized to categories of geometric interest, such as the wrapped Fukaya category of a Weinstein manifold, and the bounded derived category of coherent sheaves on a smooth algebraic variety, our categorical action filtrations essentially recover previously studied filtrations of geometric origin. Our approach is built around the notion of a smooth categorical compactification. We prove that a smooth categorical compactification induces well-defined growth invariants, which are moreover preserved under zig-zags of such compactifications. The technical heart of the paper is a method for computing these growth invariants in terms of the growth of certain colimits of (bi)modules. In practice, such colimits arise in both geometric settings of interest. The main applications are: (1) A “quantitative” refinement of homological mirror symmetry, which relates the growth of the Reeb-length filtration on the symplectic geometry side with the growth of filtrations on the algebraic geometry side defined by the order of pole at infinity (often these can be expressed in terms of the dimension of the support of sheaves). (2) A proof that the Reeb-length growth of symplectic cohomology and wrapped Floer cohomology on a Weinstein manifold are at most exponential. (3) Lower bounds for the entropy and polynomial entropy of certain natural endofunctors acting on Fukaya categories.

Monotone Lagrangians in cotangent bundles of spheres

We study the compact monotone Fukaya category of T⁎Sn, for n≥2, and show that it is split-generated by two classes of objects: the zero-section Sn (equipped with suitable bounding cochains) and a 1-parameter family of monotone Lagrangian tori (S1×Sn−1)τ, with monotonicity constants τ>0 (equipped with rank 1 unitary local systems). As a consequence, any closed orientable spin monotone Lagrangian (possibly equipped with auxiliary data) with non-trivial Floer cohomology is non-displaceable from either Sn or one of the (S1×Sn−1)τ. In the case of T⁎S3, the monotone Lagrangians (S1×S2)τ can be replaced by a family of monotone tori Tτ3.

A Leray–Serre spectral sequence for Lagrangian Floer theory

We consider symplectic fibrations as in Guillemin-Lerman-Sternberg, and derive a spectral sequence to compute the Floer cohomology of certain fibered Lagrangians sitting inside a compact symplectic fibration with small monotone fibers and a rational base. We show if the Floer cohomology with field coefficients of the fiber Lagrangian vanishes, then the Floer cohomology with field coefficients of the total Lagrangian also vanishes. We give an application to certain non-torus fibers of the Gelfand-Cetlin system in Flag manifolds, and show that their Floer cohomology vanishes.

Construction of a linear K-system in Hamiltonian Floer theory

The notion of linear K-system was introduced by the present authors as an abstract model arising from the structure of compactified moduli spaces of solutions to Floer’s equation in the book (Fukaya et al. in Springer monographs in mathematics, Springer, Berlin, 2020). The purpose of the present article is to provide a geometric realization of the linear K-system associated with solutions to Floer’s equation in the Morse–Bott setting. Immediate consequences [when combined with the abstract theory from Fukaya et al. (Springer monographs in mathematics, Springer, Berlin, 2020)] are the construction of Floer cohomology for periodic Hamiltonian systems on general compact symplectic manifolds without any restriction, and the construction of an isomorphism over the Novikov ring between the Floer cohomology and the singular cohomology of the underlying symplectic manifold. The present article utilizes various analytical results on pseudoholomorphic curves established in our earlier papers and books. However, the paper itself is geometric in nature, and does not presume much prior knowledge of Kuranishi structures and their construction but assumes only the elementary part thereof, and results from Fukaya et al. (Surv Differ Geom 22:133–190, 2018) and Fukaya et al. (Exponential decay estimate and smoothness of the moduli space of pseudoholomorphic curves) on their construction, and the standard knowledge on Hamiltonian Floer theory. We explain the general procedure of the construction of a linear K-system by explaining in detail the inductive steps of ensuring the compatibility conditions for the system of Kuranishi structures leading to a linear K-system for the case of Hamiltonian Floer theory.

Cohomology of contact loci

We construct a spectral sequence converging to the cohomology with compact support of the m-th contact locus of a complex polynomial. The first page is explicitly described in terms of a log resolution and coincides with the first page of McLean’s spectral sequence converging to the Floer cohomology of the $m$‑th iterate of the monodromy, when the polynomial has an isolated singularity. Inspired by this connection, we conjecture that if two germs of holomorphic functions are embedded topologically equivalent, then the Milnor fibers of their tangent cones are homotopy equivalent.

Chekanov-Eliashberg $\mathrm{dg}$-algebras for singular Legendrians

The Chekanov-Eliashberg dg-algebra is a holomorphic curve invariant associated to Legendrian submanifolds of a contact manifold. We extend the definition to Legendrian embeddings of skeleta of Weinstein manifolds. Via Legendrian surgery, the new definition gives direct proofs of wrapped Floer cohomology push-out diagrams. It also leads to a proof of a conjectured isomorphism between partially wrapped Floer cohomology and Chekanov-Eliashberg dg-algebras with coefficients in chains on the based loop space.

Invariance of immersed Floer cohomology under Maslov flows

We show that immersed Lagrangian Floer cohomology in compact rational symplectic manifolds is invariant under Maslov flows such as coupled mean curvature/Kaehler-Ricci flow in the sense of Smoczyk as a pair of self-intersection points is born or dies at a self-tangency, using results of Ekholm-Etnyre-Sullivan. This proves part of a conjecture of Joyce. We give a lower bound on the time for which the Floer cohomology is invariant under the (forward or backwards) flow, if it exists.

On contact loci of hyperplane arrangements

We give an explicit expression for the contact loci of hyperplane arrangements and show that their cohomology rings are combinatorial invariants. We also give an expression for the restricted contact loci in terms of Milnor fibers of associated hyperplane arrangements. We prove the degeneracy of a spectral sequence related to the restricted contact loci of a hyperplane arrangement and which conjecturally computes algebraically the Floer cohomology of iterates of the Milnor monodromy. We give formulas for the Betti numbers of contact loci and restricted contact loci in generic cases.