Symplectic Four-manifolds Research Articles

An elementary alternative to ECH capacities

The embedded contact homology (ECH) capacities are a sequence of numerical invariants of symplectic four-manifolds that give (sometimes sharp) obstructions to symplectic embeddings. These capacities are defined using embedded contact homology, and establishing their basic properties currently requires Seiberg-Witten theory. In this paper we define a sequence of symplectic capacities in four dimensions using only basic notions of holomorphic curves. The capacities satisfy the same basic properties as ECH capacities and agree with the ECH capacities for the main examples for which the latter have been computed, namely convex and concave toric domains. The capacities are also useful for obstructing symplectic embeddings into closed symplectic four-manifolds. This work is inspired by a recent preprint of McDuff and Siegel [D. McDuff, K. Siegel, arXiv [Preprint] (2021)], giving a similar elementary alternative to symplectic capacities from rational symplectic field theory (SFT).

Holomorphic Spheres and Four-Dimensional Symplectic Pairs

We classify four-dimensional manifolds endowed with symplectic pairs admitting embedded symplectic spheres with nonnegative self-intersection, following the strategy of McDuff’s classification of rational and ruled symplectic four-manifolds.

The Donaldson geometric flow for symplectic four-manifolds

This is an exposition of the Donaldson geometric flow on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. The original work appeared in [1].

An SO(3)-Monopole Cobordism Formula Relating Donaldson and Seiberg–Witten Invariants

We prove an analogue of the Kotschick-Morgan conjecture in the context of SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The main technical difficulty in the SO(3)-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible SO(3) monopoles, namely the moduli spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of SO(3) monopoles [arXiv:dg-ga/9710032]. In this monograph, we prove --- modulo a gluing theorem which is an extension of our earlier work in [arXiv:math/9907107] --- that these intersection pairings can be expressed in terms of topological data and Seiberg-Witten invariants of the four-manifold. This conclusion is analogous to the Kotschick-Morgan conjecture concerning the wall-crossing formula for Donaldson invariants of a four-manifold with $b_2^+=1$; that wall-crossing formula and the resulting structure of Donaldson invariants for four-manifolds with $b_2^+=1$ were established, assuming the Kotschick-Morgan conjecture, by Goettsche [arXiv:alg-geom/9506018] and Goettsche and Zagier [arXiv:alg-geom/9612020]. In this monograph, we reduce the proof of the Kotschick-Morgan conjecture to an extension of previously established gluing theorems for anti-self-dual SO(3) connections (see [arXiv:math/9812060] and references therein). Since the first version of our monograph was circulated, applications of our results have appeared in the proof of Property P for knots by Kronheimer and Mrowka [arXiv:math/0311489] and work of Sivek on Donaldson invariants for symplectic four-manifolds [arXiv:1301.0377].

A note on positive-definite, symplectic four-manifolds

We prove that a positive-definite smooth four-manifold with b_2^+\geq 2 and having either no 1-handles or no 3-handles cannot admit a symplectic structure.

A PERTURBATION AND GENERIC SMOOTHNESS OF THE VAFA–WITTEN MODULI SPACES ON CLOSED SYMPLECTIC FOUR-MANIFOLDS

AbstractWe prove a Freed–Uhlenbeck style generic smoothness theorem for the moduli space of solutions to the Vafa–Witten equations on a closed symplectic four-manifold by using a method developed by Feehan for the study of the PU(2)-monopole equations on smooth closed four-manifolds. We introduce a set of perturbation terms to the Vafa–Witten equations, and prove that the moduli space of solutions to the perturbed Vafa–Witten equations on a closed symplectic four-manifold for the structure group SU(2) or SO(3) is a smooth manifold of dimension zero for a generic choice of the perturbation parameters.

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space $\mathbb{R}^4$, the projective plane $\mathbb{C}P^2$, and the monotone $S^2 \times S^2$. The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for $T^*\mathbb{T}^2$, i.e.~it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

Cohomology and Hodge theory on symplectic manifolds: III

arXiv:1402.0427v2 [math.SG] 2 May 2014 Cohomology and Hodge Theory on Symplectic Manifolds: III Chung-Jun Tsai, Li-Sheng Tseng and Shing-Tung Yau February 3, 2014 Abstract We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered cohomologies give a two-sided resolution of Lefschetz maps, and thereby, they are directly related to the kernels and cokernels of the Lefschetz maps. We also introduce a novel, non-associative product operation on differential forms for symplectic manifolds. This product generates an A ∞ -algebra structure on forms that underlies the filtered cohomologies and gives them a ring structure. As an application, we demonstrate how the ring structure of the filtered cohomologies can distinguish different symplectic four-manifolds in the context of a circle times a fibered three-manifold. Contents 1 Introduction 2 Preliminaries Operations on differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Filtered forms and differential operators . . . . . . . . . . . . . . . . . . . . . . . 14 Short exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Einstein metrics, harmonic forms, and symplectic four-manifolds

If $M$ is the underlying smooth oriented $4$-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics $h$ on $M$ such that $W^+(\omega , \omega )> 0$, where $W^+$ is the self-dual Weyl curvature of $h$, and $\omega$ is a non-trivial self-dual harmonic $2$-form on $(M,h)$. While this open region in the space of Riemannian metrics contains all the known Einstein metrics on $M$, we show that it contains no others. Consequently, it contributes exactly one connected component to the moduli space of Einstein metrics on $M$.

Symplectic embeddings into four-dimensional concave toric domains

ECH (embedded contact homology) capacities give obstructions to symplectically embedding one symplectic four-manifold with boundary into another. We compute the ECH capacities of a large family of symplectic four-manifolds with boundary, called ‘concave toric domains’. Examples include the (nondisjoint) union of two ellipsoids in R 4 . We use these calculations to find sharp obstructions to certain symplectic embeddings involving concave toric domains. For example: (1) we calculate the Gromov width of every concave toric domain; (2) we show that many inclusions of an ellipsoid into the union of an ellipsoid and a cylinder are ‘optimal’; and (3) we find a sharp obstruction to ball packings into certain unions of an ellipsoid and a cylinder.

Kodaira dimension of fiber sums along spheres

In this note we discuss the effect of the symplectic sum along spheres in symplectic four-manifolds on the Kodaira dimension of the underlying symplectic manifold. We find that the Kodaira dimension is non-decreasing. Moreover, we are able to obtain precise results on the structure of the manifold obtained from the blow down of an embedded symplectic $$-4$$ -sphere.

Packing numbers of rational ruled four-manifolds

We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic four-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit values for when an ellipsoid of type $E(a, b)$, with $\frac{b}{a} \in \mathbb{N}$, embeds in a polydisc $P(s,t)$. Under this integrality assumption, we also give an alternative proof of a recent result of M. Hutchings showing that the embedded contact homology capacities give sharp inequalities for embedding ellipsoids into polydisks.

Inequivalent contact structures on Boothby–Wang five-manifolds

We consider contact structures on simply-connected five-manifolds which arise as circle bundles over simply-connected symplectic four-manifolds and show that invariants from contact homology are related to the divisibility of the canonical class of the symplectic structure. As an application we find new examples of inequivalent contact structures in the same equivalence class of almost contact structures with non-zero first Chern class.

Cohomology and Hodge Theory on Symplectic Manifolds: I

Author(s): Tsai, Chung-Jun; Tseng, Li-Sheng; Yau, Shing-Tung | Abstract: We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered cohomologies give a two-sided resolution of Lefschetz maps, and thereby, they are directly related to the kernels and cokernels of the Lefschetz maps. We also introduce a novel, non-associative product operation on differential forms for symplectic manifolds. This product generates an A-infinity algebra structure on forms that underlies the filtered cohomologies and gives them a ring structure. As an application, we demonstrate how the ring structure of the filtered cohomologies can distinguish different symplectic four-manifolds in the context of a circle times a fibered three-manifold.

Do Uniruled Six-manifolds Contain Sol Lagrangian Submanifolds?

We prove using symplectic field theory combined with some Gromov–Witten theory that if the suspension of a hyperbolic diffeomorphism of the two-torus Lagrangian embeds in a closed uniruled symplectic six-manifold, then its first rational homology is generated by the boundary of a symplectic disk. This prevents any Sol manifold from being in the real locus of an orientable real Del Pezzo fibration over a curve, confirming an expectation of Kollár. It also constrains Hamiltonian diffeomorphisms of uniruled symplectic four-manifolds.

Supersymmetric surface operators, four-manifold theory and invariants in various dimensions

We continue our program initiated in Integration over the u-plane in Donaldson theory with surface operators, J. High Energy Phys. 05 (2011), to consider supersymmetric surface operators in a topologically twisted ${\cal N}=2$ pure ${\it SU}(2)$ gauge theory, and apply them to the study of four-manifolds and related invariants. Elegant physical proofs of various seminal theorems in four-manifold theory obtained by Ozsath and Szabo and Taubes, will be furnished. In particular, we will show that Taubes’ groundbreaking and difficult result — that the ordinary SW invariants are in fact the Gromov invariants which count pseudo-holomorphic curves embedded in a symplectic four-manifold $X$ — nonetheless lends itself to a simple and concrete physical derivation in the presence of “ordinary” surface operators. As an offshoot, we will be led to several interesting and mathematically novel identities among the Gromov and “ramified” SW invariants of $X$, which in certain cases, also involve the instanton and monopole Floer homologies of its three-submanifold. Via these identities, and a physical formulation of the “ramified” Donaldson invariants of four-manifolds with boundaries, we will uncover completely new and economical ways of deriving and understanding various important mathematical results concerning (i) knot homology groups from “ramified” instantons by Kronheimer and Mrowka; and (ii) monopole Floer homology and SW theory on symplectic four-manifolds by Kutluhan–Taubes. Supersymmetry, as well as other physical concepts such as $R$-invariance, electric–magnetic duality, spontaneous gauge symmetry breaking and localization onto supersymmetric configurations in topologically twisted quantum field theories, play a pivotal role in our story.

Finite group action and Gromov-Witten invariants in symplectic four-manifolds

Let a finite group act semi-freely on a closed symplectic four-manifold with a 2-dimensional fixed point set. Then we show that the relative Gromov-Witten invariants are the same as the invariants on the quotient set-up with respect to the fixed point set.

Pseudoholomorphic maps into folded symplectic four-manifolds

Every oriented 4-manifold admits a folded symplectic structure, which in turn determines a homotopy class of compatible almost complex structures that are discontinuous across the folding hypersurface ("fold") in a controlled fashion. We define folded holomorphic maps, i.e. pseudo-holomorphic maps that are discontinuous across the fold. The boundary values on the fold are mediated by tunneling maps which are punctured H-holomorphic maps into the folding hypersurface with prescribed asymptotics on closed characteristics. Our main result is that the linearized operator of this boundary value problem is Fredholm and thus we obtain well behaved local finite dimensional moduli spaces. As examples we characterize the moduli space of maps into folded elliptic fibration and we construct examples of degree $d$ rational maps into $S^4$. Moreover we explicitly give the moduli space of degree 1 rational maps into $S^4$ and show that it possesses a natural compactification. This aims to generalize the tools of holomorphic maps to all oriented 4-manifolds by utilizing folded symplectic structures.

Invariant count of holomorphic disks in the cotangent bundles of the two-sphere and real projective plane

We first define enumerative invariants of the cotangent bundles of the two-sphere and real projective plane. These invariants are obtained in the framework of symplectic field theory by counting with respect to some sign holomorphic disks with punctures sitting on the zero section. Then, we relate these invariants with the ones of closed real symplectic four-manifolds which have been constructed earlier. This relation provides some congruences and recursive formulas for the latter as well as sharpness results for the associated lower bounds in real enumerative geometry. To cite this article: J.-Y. Welschinger, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Homologous non-isotopic symplectic surfaces of higher genus

We construct an infinite family of homologous, non-isotopic, symplectic surfaces of any genus greater than one in a certain class of closed, simply connected, symplectic four-manifolds. Our construction is the first example of this phenomenon for surfaces of genus greater than one.