Equivariant cohomology of K-contact manifolds

We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen–Macaulay, the natural substitute of equivariant formality for torus actions without fixed points. As a consequence, generic components of the contact moment map are perfect Morse-Bott functions for the basic cohomology of the orbit foliation ${{\mathcal F}}$ of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of ${{\mathcal F}}$ vanishes in odd degrees, and that its dimension equals the number of closed Reeb orbits. We characterize K-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM-type theorem for K-contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated GKM condition.     

Lagrangian Curves in a 4-Dimensional Affine Symplectic Space

Lagrangian curves in \(\mathbb {R}^{4}\) entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangian curves with constant symplectic curvatures, to construct a class of Lagrangian tori in \(\mathbb {R}^{4}\) and determine Lagrangian geodesics.     

Arc Spaces and Equivariant Cohomology

We present a new geometric interpretation of equivariant cohomology in which one replaces a smooth, complex G-variety X by its associated arc space J _∞ X, with its induced G-action. This not only allows us to obtain geometric classes in equivariant cohomology of arbitrarily high degree, but also provides more flexibility for equivariantly deforming classes and geometrically interpreting multiplication in the equivariant cohomology ring. Under appropriate hypotheses, we obtain explicit bijections between $ \mathbb{Z} $ -bases for the equivariant cohomology rings of smooth varieties related by an equivariant, proper birational map. We also show that self-intersection classes can be represented as classes of contact loci, under certain restrictions on singularities of subvarieties. We give several applications. Motivated by the relation between self-intersection and contact loci, we define higher-order equivariant multiplicities, generalizing the equivariant multiplicities of Brion and Rossmann; these are shown to be local singularity invariants, and computed in some cases. We also present geometric $ \mathbb{Z} $ -bases for the equivariant cohomology rings of a smooth toric variety (with respect to the dense torus) and a partial flag variety (with respect to the general linear group).     

Spherical Lagrangians via ball packings and symplectic cutting

In this paper, we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, $S^{2}$ or $\mathbb{RP }^{2}$ , in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction, this is a natural extension of McDuff’s connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of the symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floer-theoretically essential Lagrangian tori in the del Pezzo surfaces.     

A note on Sen��s theory in the imperfect residue field case

In Sen??s theory in the imperfect residue field case, Brinon defined a functor from the category of ${{\mathbb C}_p}$ -representations to the category of linear representations of a certain Lie algebra. We give a comparison theorem between the continuous Galois cohomology of ${{\mathbb C}_p}$ -representations and the Lie algebra cohomology of the associated representations. The key ingredients of the proof are Hyodo??s calculation of Galois cohomology and the effaceability of Lie algebra cohomology for solvable Lie algebras.     

Euler classes and Bredon cohomology for groups with restricted families of finite subgroups

We explore some of the special features with respect to Bredon cohomology for groups whose finite subgroups are all either nilpotent or $p$ -groups or cyclic $p$ -groups. We get some results on dimensions and also a formula for the equivariant Euler class for certain groups. We consider the generalization for Bredon cohomology of the properties of being duality or Poincaré duality and study their behavior under $p$ -power index extensions with coefficients in a field of characteristic $p$ .     

Self-shrinkers for the mean curvature flow in arbitrary codimension

In this paper, we generalize Colding–Minicozzi’s recent results about codimension-1 self-shrinkers for the mean curvature flow to higher codimension. In particular, we prove that the sphere ${bf S}^{n}(\sqrt{2n})$ is the only complete embedded connected $F$ -stable self-shrinker in $\mathbf{R}^{n+k}$ with $\mathbf{H}\ne 0$ , polynomial volume growth, flat normal bundle and bounded geometry. We also discuss some properties of symplectic self-shrinkers, proving that any complete symplectic self-shrinker in $\mathbf{R}^4$ with polynomial volume growth and bounded second fundamental form is a plane. As a corollary, we show that there is no finite time Type I singularity for symplectic mean curvature flow, which has been proved by Chen–Li using different method. We also study Lagrangian self-shrinkers and prove that for Lagrangian mean curvature flow, the blow-up limit of the singularity may be not $F$ -stable.     

Constructing exact Lagrangian immersions with few double points

We establish, as an application of the results from Eliashberg and Murphy (Lagrangian caps, 2013), 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 ${\mathbb{R}^6_{\rm st}}$ with exactly one transverse double point. Our construction also yields a Lagrangian embedding ${S^1 \times S^2 \to \mathbb{R}^6_{\rm st}}$ with vanishing Maslov class.     

New Tools for Classifying Hamiltonian Circle Actions with Isolated Fixed Points

For every compact almost complex manifold \((\mathsf {M},\mathsf {J})\) equipped with a \(\mathsf {J}\) -preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to obtain linear relations among the isotropy weights at the fixed points. Suppose that \(\mathsf {M}\) is symplectic and the action is Hamiltonian. If the manifold satisfies an extra so-called positivity condition, then this algorithm determines a family of vector spaces that contain the admissible lattices of weights. When the number of fixed points is minimal, this positivity condition is necessarily satisfied whenever \(\dim (\mathsf {M})\le 6\) and, when \(\dim (\mathsf {M})=8\) , whenever the \(S^1\) -action extends to an effective Hamiltonian \(T^2\) -action, or none of the isotropy weights is \(1\) . Moreover, there are no known examples with a minimal number of fixed points contradicting this condition, and their existence is related to interesting questions regarding fake projective spaces. We run the algorithm for \(\dim (\mathsf {M})\le 8\) , quickly obtaining all the possible families of isotropy weights. In particular, we simplify the proofs of Ahara and Tolman for \(\dim (\mathsf {M})=6\) and, when \(\dim (\mathsf {M})=8\) , we prove that the equivariant cohomology ring, Chern classes, and isotropy weights agree with those of \({\mathbb {C}}P^4\) with the standard \(S^1\) -action (thereby proving the symplectic Petrie conjecture in this setting).     

Uniformly Quasiregular Maps on the Compactified Heisenberg Group

We show the existence of a non-injective uniformly quasiregular mapping acting on the one-point compactification $\bar{ {\mathbb{H}}}^{1}={\mathbb{H}}^{1}\cup\{\infty\}$ of the Heisenberg group ?¹ equipped with a sub-Riemannian metric. The corresponding statement for arbitrary quasiregular mappings acting on sphere ${\mathbb{S}}^{n} $ was proven by Martin (Conform. Geom. Dyn. 1:24?C27, 1997). Moreover, we construct uniformly quasiregular mappings on $\bar{ {\mathbb{H}}}^{1}$ with large-dimensional branch sets. We prove that for any uniformly quasiregular map g on $\bar{ {\mathbb{H}}}^{1}$ there exists a measurable CR structure ?? which is equivariant under the semigroup ?? generated by g. This is equivalent to the existence of an equivariant horizontal conformal structure.     

Equivariant constrained Willmore tori in the 3-sphere

In this paper we study equivariant constrained Willmore tori in the 3-sphere. These tori admit a 1-parameter group of Möbius symmetries and are critical points of the Willmore energy under conformal variations. We show that the spectral curve associated to an equivariant torus is given by a double covering of \(\mathbb {C}\) and classify equivariant constrained Willmore tori by the genus \(g\) of their spectral curve. In this case the spectral genus satisfies \(g \le 3\) .     

Self-adjoint curl operators

We study the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on a 3D bounded domain D. We aim to identify boundary conditions that render this operator self-adjoint. By the symplectic version of the Glazman-Krein-Naimark theorem, this amounts to identifying complete Lagrangian subspaces of the trace space of H(curl, D) equipped with a symplectic pairing arising from the ${\wedge}$ -product of 1-forms on ${\partial D}$ . Substantially generalizing earlier results, we characterize Lagrangian subspaces associated with closed and co-closed traces. In the case of non-trivial topology of the domain, different contributions from co-homology spaces also distinguish different self-adjoint extensions. Finally, all self-adjoint extensions discussed in the paper are shown to possess a discrete point spectrum, and their relationship with curl curl-operators is discussed.     

The Lagrangian Radon Transform and the Weil Representation

We consider the operator $\mathcal {R}$ , which sends a function on ${\mathbb {R}}^{2n}$ to its integrals over all affine Lagrangian subspaces in ${\mathbb {R}}^{2n}$ . We discuss properties of the operator $\mathcal {R}$ and of the representation of the affine symplectic group in several function spaces on ${\mathbb {R}}^{2n}$ .     

Lagrangian caps

We establish an h-principle for exact Lagrangian embeddings with concave Legendrian boundary. We prove, in particular, that in the complement of the unit ball B in the standard symplectic ${\mathbb{R}^{2n}, 2n \geq 6}$ R 2 n , 2 n ≥ 6 , there exists an embedded Lagrangian n-disc transversely attached to B along its Legendrian boundary, which is loose in the sense of Murphy (Loose Legendrian embeddings in high dimensional contact manifolds, arXiv:1201.2245, 2013).     

The Calabi homomorphism, Lagrangian paths and special Lagrangians

Let $\mathcal{O }$ be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $(X,\omega ).$ We define a functional $\mathcal{C }:\mathcal{O } \rightarrow \mathbb{R }$ for each differential form $\beta $ of middle degree satisfying $\beta \wedge \omega = 0$ and an exactness condition. If the exactness condition does not hold, $\mathcal{C }$ is defined on the universal cover of $\mathcal{O }.$ A particular instance of $\mathcal{C }$ recovers the Calabi homomorphism. If $\beta $ is the imaginary part of a holomorphic volume form, the critical points of $\mathcal{C }$ are special Lagrangian submanifolds. We present evidence that $\mathcal{C }$ is related by mirror symmetry to a functional introduced by Donaldson to study Einstein–Hermitian metrics on holomorphic vector bundles. In particular, we show that $\mathcal{C }$ is convex on an open subspace $\mathcal{O }^+ \subset \mathcal{O }.$ As a prerequisite, we define a Riemannian metric on $\mathcal{O }^+$ and analyze its geodesics. Finally, we discuss a generalization of the flux homomorphism to the space of Lagrangian submanifolds, and a Lagrangian analog of the flux conjecture.     

Stability inequalities and universal Schubert calculus of rank 2

The goal of the paper is to introduce a version of Schubert calculus for each dihedral reflection group W. That is, to each ??sufficiently rich?? spherical building Y of type W we associate a certain cohomology theory $ H_{BK}^*(Y) $ and verify that, first, it depends only on W (i.e., all such buildings are ??homotopy equivalent??), and second, $ H_{BK}^*(Y) $ is the associated graded of the coinvariant algebra of W under certain filtration. We also construct the dual homology ??pre-ring?? on Y. The convex ??stability?? cones in $ {\left( {{\mathbb{R}^2}} \right)^m} $ defined via these (co)homology theories of Y are then shown to solve the problem of classifying weighted semistable m-tuples on Y in the sense of [KLM1]; equivalently, they are cut out by the generalized triangle inequalities for thick Euclidean buildings with the Tits boundary Y. The independence of the (co)homology theory of Y refines the result of [KLM2], which asserted that the Stability Cone depends on W rather than on Y. Quite remarkably, the cohomology ring $ H_{BK}^*(Y) $ is obtained from a certain universal algebra A _t by a kind of ??crystal limit?? that has been previously introduced by Belkale?CKumar for the cohomology of ag varieties and Grassmannians. Another degeneration of A _t leads to the homology theory H _*(Y).     

Symplectic homology of displaceable Liouville domains and leafwise intersection points

In this note we prove that the symplectic homology of a Liouville domain $W$ displaceable in the symplectic completion vanishes. Nevertheless if the Euler characteristic of $(W,\partial W)$ is odd, the filtered symplectic homologies of $W$ do not vanish and give rise to leafwise intersection points on the symplectic completion of $W$ for a perturbation displacing $W$ from itself. In contrast to the existing results we can find a leafwise intersection point for a given period but its energy varies by period instead.     

Symplectic Twistor Operator on \mathbb R ^{2n} and the Segal–Shale–Weil Representation

The aim of our article is the study of solution space of the symplectic twistor operator $T_s$ in symplectic spin geometry on standard symplectic space $(\mathbb R ^{2n},\omega )$ , which is the symplectic analogue of the twistor operator in (pseudo) Riemannian spin geometry. In particular, we observe a substantial difference between the case $n=1$ of real dimension $2$ and the case of $\mathbb R ^{2n}, n>1$ . For $n>1$ , the solution space of $T_s$ is isomorphic to the Segal–Shale–Weil representation.     

K-Orbit Closures on G/B as Universal Degeneracy Loci for Flagged Vector Bundles with Symmetric or Skew-Symmetric Bilinear Form

We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of K-orbit closures on the flag variety G/B, where G = GL(n, $ \mathbb{C} $ ), and where K is one of the symmetric subgroups O(n, $ \mathbb{C} $ ) or Sp(n, $ \mathbb{C} $ ). We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formulas for the classes of such loci in terms of the Chern classes of the various bundles.