We prove that the group of compactly supported symplectomorphisms of the standard symplectic ball admits a continuum of linearly independent real-valued homogeneous quasimorphisms. In addition these quasimorphisms are Lipschitz in the Hofer metric and have the following property: the value of each such quasimorphism on any symplectomorphism supported in any "sufficiently small" open subset of the ball equals the Calabi invariant of the symplectomorphism. By a "sufficiently small" open subset we mean that it can be displaced from itself by a symplectomorphism of the ball. As a byproduct we show that the (Lagrangian) Clifford torus in the complex projective space cannot be displaced from itself by a Hamiltonian isotopy.
We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small open subset of the sphere equals to the Calabi invariant of the diffeomorphism. This result extends to more general symplectic manifolds: If the symplectic manifold is monotone and its quantum homology algebra is semi-simple we construct a similar quasimorphism on the universal cover of the group of Hamiltonian diffeomorphisms.
We show the existence of Calabi quasimorphisms on the universal covering of the group of Hamiltonian diffeomorphisms of a monotone coadjoint orbit of a compact Lie group. We show that this result follows from positivity results of Gromov-Witten invariants and the fact that the quantum product of Schubert classes can never be zero.
Let $\hat{G}$ be a group and $G$ its normal subgroup. In this paper, we study $\hat{G}$-invariant quasimorphisms on $G$ which appear in symplectic geometry and low dimensional topology. As its application, we prove the non-existence of a section of the flux homomorphism on closed surfaces of higher genus. We also prove that Py's Calabi quasimorphism and Entov-Polterovich's partial Calabi quasimorphism are non-extendable to the group of symplectomorphisms. We show that Py's Calabi quasimorphism is the unique non-extendable quasimorphism to some group.
We develop a family of deformations of the differential and of the pair-of-pants product on the Hamiltonian Floer complex of a symplectic manifold (M,ω) which upon passing to homology yields ring isomorphisms with the big quantum homology of M. By studying the properties of the resulting deformed version of the Oh-Schwarz spectral invariants, we obtain a Floer-theoretic interpretation of a result of Lu which bounds the Hofer-Zehnder capacity of M when M has a nonzero Gromov-Witten invariant with two point constraints, and we produce a new algebraic criterion for (M,ω) to admit a Calabi quasimo…
We extend the definition of Weinstein's Action homomorphism to Hamiltonian actions with equivariant moment maps of (possibly infinite-dimensional) Lie groups on symplectic manifolds, and show that under conditions including a uniform bound on the symplectic areas of geodesic triangles the resulting homomorphism extends to a quasimorphism on the universal cover of the group. We apply these principles to finite dimensional Hermitian Lie groups like Sp(2n,R), reinterpreting the Guichardet-Wigner quasimorphisms, and to the infinite dimensional groups of Hamiltonian diffeomorphisms Ham(M,\om) of cl…
In this paper we first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entovi-Polterovich theory of spectral symplectic quasi-states and quasimorphisms by incorporating \emph{bulk deformations}, i.e., deformations by ambient cycles of symplectic manifolds, of the Floer homology and quantum cohomology. Essentially the same kind of construction is independently carried out by Usher [Us4] in a slightly less general context. Then we explore various applications of these enhancements to the symplectic topology, especially new construction of sy…
In the setting of geometric quantization, we associate to any prequantum bundle automorphism a unitary map of the corresponding quantum space. These maps are controlled in the semiclassical limit by two invariants of symplectic topology: the Calabi morphism and a quasimorphism of the universal cover of the Hamiltonian diffeomorphism group introduced by Entov, Py, Shelukhin.
We study the Calabi-Yau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature condition, we show that the conjecture holds.
We apply a local differential geometric framework from Kähler toric geometry to (re)construct Calabi's extremal Kähler metrics on $\bbC\bbP^n$ blown-up at a point from data on the moment polytope.
In this work we construct Calabi quasi-morphisms on the universal cover of the group Ham(M) of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast to their work, we show that these quasi-morphisms descend to non-trivial homomorphisms on the fundamental group of Ham(M).
Given an integer b and a finitely presented group G we produce a compact symplectic six-manifold with c_1 = 0, b_2 > b, b_3 > b and fundamental group G. In the simply-connected case we can also arrange for b_3 = 0; in particular these examples are not diffeomorphic to Kähler manifolds with c_1 = 0. The construction begins with a certain orientable four-dimensional hyperbolic orbifold assembled from right-angled 120-cells. The twistor space of the hyperbolic orbifold is a symplectic Calabi-Yau orbifold; a crepant resolution of this last orbifold produces a smooth symplectic manifold with the re…
Bavard proved a duality theorem between commutator length and quasimorphisms. Burago, Ivanov and Polterovich introduced the notion of a conjugation-invariant norm which is a generalization of commutator length. Entov and Polterovich proved that Oh-Schwarz spectral invariants are subset-controlled quasimorphisms which are geralizations of quasimorphisms. In the present paper, we prove a Bavard-type duality theorem between conjugation-invariant (pseudo-)norms and subset-controlled quasimorphisms on stable groups. %We also give an application of its generalization to symplectic geometry related t…
We discuss a conjecture of Donaldson on a version of Yau's Theorem for symplectic forms with compatible almost complex structures and survey some recent progress on this problem. We also speculate on some future possible directions, and use a monotonicity formula for harmonic maps to obtain a new local estimate in the setting of Donaldson's conjecture.
In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along $T^4$. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group $\Z$. In this paper, we also produce the first examples of simply connected symplectic Calabi-Yau and non-Calabi-Yau 6-manifolds via coisotropic Luttinger surgery.
In this paper we summarize our recent work in the construction of Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties and the symplectic Strominger-Yau-Zaslow conjecture, together with some new development. It is submittded to the Proceedings of the Conference in Symplectic Geometry and Mirror Symmetry held in KIAS, Seoul, Korea. The paper was finished in Jan. 2001, with revisions and added reference.
Let (M, ω) be a compact symplectic 4-manifold with a compatible almost complex structure J. The problem of finding a J-compatible symplectic form with prescribed volume form is an almost-Kähler analogue of Yau's theorem and is connected to a programme in symplectic topology proposed by Donaldson. We call the corresponding equation for the symplectic form the Calabi-Yau equation. Solutions are unique in their cohomology class. It is shown in this paper that a solution to this equation exists if the Nijenhuis tensor is small in a certain sense. Without this assumption, it is shown that the probl…
We study the symplectic analogue of log Calabi-Yau surfaces and show that the symplectic deformation classes of these surfaces are completely determined by the homological information.
We explicitly produce symplectic genus-3 Lefschetz pencils (with base points), whose total spaces are homeomorphic but not diffeomorphic to rational surfaces CP^2 # p (-CP^2) for p= 7, 8, 9. We then give a new construction of an infinite family of symplectic Calabi-Yau surfaces with first Betti number b_1=2,3, along with a surface with b_1=4 homeomorphic to the 4-torus. These are presented as the total spaces of symplectic genus-3 Lefschetz pencils we construct via new positive factorizations in the mapping class group of a genus-3 surface. Our techniques in addition allow us to answer in the …
Let $(X,σ,J)$ be a compact Kähler Calabi-Yau manifold equipped with a symplectic circle action. By Frankel's theorem \cite{F}, the action on $X$ is non-Hamiltonian and $X$ does not have any fixed point. In this paper, we will show that a symplectic circle action on a compact non-Kähler symplectic Calabi-Yau manifold may have a fixed point. More precisely, we will show that the symplectic $S^1$-manifold constructed by D. McDuff \cite{McD} has the vanishing first Chern class. This manifold has the Betti numbers $b_1 = 3$, $b_2 = 8$, and $b_3 = 12$. In particular, it does not admit any Kähler str…
Using dynamical stability of symplectic curvature flow, we show that on a compact Calabi-Yau manifold, any small symplectic deformation of a Kähler form remains Kähler.
Given an SO(3)-bundle with connection, the associated two-sphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to construct symplectic six-manifolds with c_1=0 which are never Kahler; e.g., we produce such manifolds with b_1=0=b_3 and also with c_2.omega <0, answering questions posed by Smith-Thomas-Yau. Examples come from Riemannian geometry, via the Levi-Civita connection on Lambda^+. The…
We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.
In this paper we are interested in the isotopy classes of symplectic log Calabi-Yau divisors in a fixed symplectic rational surface. We give several equivalent definitions and prove the stability, finiteness and rigidity results. Motivated by the problem of counting toric actions, we obtain a general counting formula of symplectic log Calabi-Yau divisors in a restrictive region of $c_1$-nef cone. A detailed count in the case of 2- and 3-point blow-ups of complex projective space for all symplectic forms is also given. In our framework the complexity of the combinatorics of analyzing Delzant po…
This is the first of a series of papers devoted to the topology of symplectic Calabi-Yau $4$-manifolds endowed with certain symplectic finite group actions. We completely determine the fixed-point set structure of a finite cyclic action on a symplectic Calabi-Yau $4$-manifold with $b_1>0$. As an outcome of this fixed-point set analysis, the $4$-manifold is shown to be a $T^2$-bundle over $T^2$ in some circumstances, e.g., in the case where the group action is an involution which fixes a $2$-dimensional surface in the $4$-manifold. Our project on symplectic Calabi-Yau $4$-manifolds is based on …
A Lagrangian field on a symplectic manifold $M$ is a family $Λ=\{Λ_x|x \in M\}$ of pointed Lagrangian submanifolds of $M$. This notion is a generalization of a real Lagrangian polarization for which each $Λ_x$ is the leaf containing $x$. Two Lagrangian fields $Λ$ and $\tilde Λ$ are called transversal if $Λ_x$ intersects $\tildeΛ_x$ transversally at $x$ for every $x$. Two transversal Lagrangian fields determine an almost para-Kähler structure on $M$. We construct a local symplectic groupoid on a neighborhood of the zero section of $T^\ast M$ from two transversal Lagrangian fields on $M$. The La…
A categorical formalism is introduced for studying various features of the symplectic geometry of Lefschetz fibrations and the algebraic geometry of Tyurin degenerations. This approach is informed by homological mirror symmetry, derived noncommutative geometry, and the theory of Fukaya categories with coefficients in a perverse Schober. The main technical results include (i) a comparison between the notion of relative Calabi-Yau structures and a certain refinement of the notion of a spherical functor, (ii) a local-to-global gluing principle for constructing Calabi-Yau structures, and (iii) the…
We construct a family of $6$-dimensional compact manifolds $M(A)$, which are simultaneously diffeomorphic to complex Calabi-Yau manifolds and symplectic Calabi-Yau manifolds. They have fundamental groups $\mathbb{Z} \oplus \mathbb{Z}$, their odd-dimensional Betti numbers are even, they satisfy the hard Lefschetz property, and their real homotopy types are formal. However, $M(A) \times Y$ are not homotopy equivalent to any compact Kähler manifold for any topological space $Y$.
We compare two functionals defined on the space of continuous functions with compact support in an open neighborhood of the zero section of the cotangent bundle of a torus. One comes from Viterbo's symplectic homogenization while the other from the Calabi quasi-states due to Entov and Polterovich. In dimension 2 we are able to say when these two functionals are equal. A partial result in higher dimensions is presented. We also give a link to asymptotic Hofer geometry on T^*S^1. Proofs are based on the theory of quasi-integrals and topological measures on locally compact spaces.
Using the Fredholm setup of [12], we study genus zero (and higher) relative Gromov-Witten invariants with maximum tangency of symplectic log Calabi-Yau fourfolds. In particular, we give a short proof of [23, Conjecture 6.2] that expresses these invariants in terms of certain integral invariants by considering generic almost complex structures to obtain a geometric count. We also revisit the localization calculation of the multiple-cover contributions in [23, Proposition 6.1] and recalculate a few terms differently to provide more details and illustrate the computation of deformation/obstructio…
The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa formula holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.
This article studies the symplectic cohomology of affine algebraic surfaces that admit a compactification by a normal crossings anticanonical divisor. Using a toroidal structure near the compactification divisor, we describe the complex computing symplectic cohomology, and compute enough differentials to identify a basis for the degree-zero part of the symplectic cohomology. This basis is indexed by integral points in a certain integral affine manifold, providing a relationship to the theta functions of Gross--Hacking--Keel. Included is a discussion of wrapped Floer cohomology of Lagrangian su…
This paper derives an equality between the slab functions in Gross-Siebert program and generating functions of open Gromov-Witten invariants for toric Calabi-Yau manifolds, and thereby confirms a conjecture of Gross-Siebert on symplectic enumerative meaning of slab functions. The proof is based on the open mirror theorem of Chan-Cho-Lau-Tseng \cite{CCLT13}. It shows an instance of correspondence between tropical and symplectic geometry in the open sector.
In this paper, we give two elementary constructions of homogeneous quasi-morphisms defined on the group of Hamiltonian diffeomorphisms of certain closed connected symplectic manifolds (or on its universal cover). The first quasi-morphism, denoted by $\calabi\_{S}$, is defined on the group of Hamiltonian diffeomorphisms of a closed oriented surface $S$ of genus greater than 1. This construction is motivated by a question of M. Entov and L. Polterovich. If $U\subset S$ is a disk or an annulus, the restriction of $\calabi\_{S}$ to the subgroup of diffeomorphisms which are the time one map of a Ha…
We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of an algebra called symplectic cohomology, which is constructed using Hamiltonian Floer cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace. Part of the proof uses the fact that subvarieties of positive codimension are stably displaceable.
We introduce symplectic Calabi-Yau caps to obtain new obstructions to exact fillings. In particular, it implies that any exact filling of the standard unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the same integral homology and intersection form as its disk cotangent bundle. This gives evidence to a conjecture that all of its exact fillings are diffeomorphic to the disk cotangent bundle. As a result, we also obtain the first infinitely family of Stein fillable contact 3-manifolds with uniform bounds on the Betti numbers of its exact fillings but admittin…
We show that any symplectically aspherical/Calabi-Yau filling of $Y:=\partial(V\times \mathbb{D})$ has vanishing symplectic cohomology for any Liouville domain $V$. In particular, we make no topological requirement on the filling and $c_1(V)$ can be nonzero. Moreover, we show that for any symplectically aspherical/Calabi-Yau filling $W$ of $Y$, the interior $\mathring{W}$ is diffeomorphic to the interior of $V\times \mathbb{D}$ if $π_1(Y)$ is abelian and $\dim V\ge 4$. And $W$ is diffeomorphic to $V\times \mathbb{D}$ if moreover the Whitehead group of $π_1(Y)$ is trivial.
We give a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. The main result of this article is the proof that the topological Calabi-Yau compactifications of M. Gross' "Topological Mirror Symmetry", can be made into symplectic compactifications. As an example, we obtain a pair of compact symplectic 6-manifolds together with Lagrangian fibrations whose underlying affine structures are dual. The symplectic manifolds obtained are homeomorphic to a smooth quintic Calabi-Yau 3-fold and its mirro…
Symplectic 4-manifolds with Kodaira dimension zero can be viewed as symplectic Calabi-Yau surfaces. We are able to completely determine their Betti numbers by proving two general results on quaternionic vector bundles.
This article uses relative symplectic cohomology, recently studied by the second author, to understand rigidity phenomena for compact subsets of symplectic manifolds. As an application, we consider a symplectic crossings divisor in a Calabi-Yau symplectic manifold $M$ whose complement is a Liouville manifold. We show that, for a carefully chosen Liouville structure, the skeleton as a subset of $M$ exhibits strong rigidity properties akin to super-heavy subsets of Entov-Polterovich. Along the way, we expand the toolkit of relative symplectic cohomology by introducing products and units. We also…
In this paper, we prove that for a generic choice of tame (or compatible) almost complex structures $J$ on a symplectic manifold $(M^{2n},ω)$ with $n \geq 3$ and with its first Chern class $c_1(M,ω) = 0$, all somewhere injective $J$-holomorphic maps from any closed smooth Riemann surface into $M$ are \emph{embedded}. We derive this result as a consequence of the general optimal 1-jet evaluation transversality result of $J$-holomorphic maps in general symplectic manifolds that we also prove in this paper.
We establish a system of formal noncommutative calculus for differential forms and polyvector fields, which forms the foundations for the study of pre-Calabi-Yau categories. Using an explicit trace map, we show that any $n$-Calabi-Yau structure on a non-positively graded dg algebra $A$ induces a $(2-n)$-shifted symplectic structure on its derived moduli stack of representations; while any $n$-pre-Calabi-Yau structure on $A$ induces a $(2-n)$-shifted Poisson structure on this derived moduli stack.
