Algebraic torus actions on contact manifolds

In this paper, for any Milnor hypersurface, we find the largest dimension of effective algebraic torus actions on it. The proof of the corresponding theorem is based on the computation of the automorphism group for any Milnor hypersurface. We find all generalized Buchstaber–Ray and Ray hypersurfaces that are toric varieties. We compute the Betti numbers of these hypersurfaces and describe their integral singular cohomology rings in terms of the cohomology of the corresponding ambient varieties.

Complete quotients by algebraic torus actions

Given an algebraic torus action on a normal projective variety with finitely generated total coordinate ring, we study the GIT-equivalence for not necessarily ample linearized divisors, and we provide a combinatorial description of the partially ordered set of GIT-equivalence classes. As an application, we extend in the $\QQ$-factorial case a basic feature of the collection of ample GIT-classes to the partially ordered collection of maximal subsets with a quasiprojective quotient: for any two members there is at most one minimal member comprising both of them. Moreover, we demonstrate in an example, how our theory can be applied for a systematic treatment of ``exotic projective orbit spaces'', i.e., projective geometric quotients that do not arise from any linearized ample divisor.

We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our approach extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.

A contact structure in a 3-dimensional manifold is a completely nonintegrable 2-dimensional distribution: if the distribution is the kernel of a (locally defined) 1-form λ then λ∧dλ � 0everywhere. The standard contact structure in 3-space, arising from the identification of R 3 with the space of 1-jets of functions on the line, is given by the contact form λ = dz − yd x; here x is a point on the line, z is the value of a function and y the value of the derivative at x. The same formula defines the standard contact structure in the space of 1-jets of functions on the circle (x being the cyclic coordinate), the space which is topologically the solid torus. In this paper we will be concerned only with these two contact manifolds. In a contact 3-dimensional manifold one considers two classes of knots and links: the transverse and the Legendrian ones. The former are everywhere transverse to the contact distribution, and the latter are everywhere tangent to it. Every topological isotopy class of knots in a contact 3-fold contains a transverse and a Legendrian knot. The main problem of contact knot theory (which has made, so far, only a few first steps) is to classify, up to contact isotopies, transverse and Legendrian knots within each topological isotopy class. In particular, one would like to have specifically contact invariants of transverse and Legendrian knots and links. One such invariant is easily defined. Legendrian and transverse knots in the standard contact space have natural framings given by the vector fields ∂z and ∂y, respectively. The corresponding self-linking number is called the Bennequin number; it is denoted by β(K) where K is a Legendrian or a transverse knot 1 . Similarly one defines the Bennequin number of an oriented Legendrian or transverse link. The study of knots in contact 3-dimensional manifolds was put forward by the seminal paper by D. Bennequin [Be]. One of the main results of this paper

We study a generalization of K-contact and (k, μ)-contact manifolds, and show that if such manifolds of dimensions ≥ 5 are conformally flat, then they have constant curvature +1. We also show under certain conditions that such manifolds admitting a non-homothetic closed conformal vector field are isometric to a unit sphere. Finally, we show that such manifolds with parallel Ricci tensor are either Einstein, or of zero $${\xi}$$ -sectional curvature.

By a result of Eliashberg, every symplectic filling of a three-dimensional contact connected sum is obtained by performing a boundary connected sum on another symplectic filling. We prove a partial generalization of this result for subcritical contact surgeries in higher dimensions: given any contact manifold that arises from another contact manifold by subcritical surgery, its belt sphere is zero in the oriented bordism group Ω * SO (W) of any symplectically aspherical filling W, and in dimension five, it will even be nullhomotopic. More generally, if the filling is not aspherical but is semipositive, then the belt sphere will be trivial in H * (W). Using the same methods, we show that the contact connected sum decomposition for tight contact structures in dimension three does not extend to higher dimensions: in particular, we exhibit connected sums of manifolds of dimension at least five with Stein fillable contact structures that do not arise as contact connected sums. The proofs are based on holomorphic disk-filling techniques, with families of Legendrian open books (so-called “Lobs”) as boundary conditions.

Whenever one studies an algebraic variety X on which an algebraic group G acts, one of the natural questions is which geometric quantites are preserved by the action, i.e. what are the geometric invariants? For example, one may want to know the G-invariant open sets, the G-invariant divisors or, possibly, the G-invariant points, that is, the G-fixed points. If G is an algebraic torus, its geometric invariants are often very rich, especially if X is projective. In particular, an algebraic torus acting on a projective variety always has fixed points. Therefore, one of the most fruitful techniques for studying a general G-action is to consider the induced action of an algebraic subtorus.

An Ozsváth–Szabó Floer homology invariant of knots in a contact manifold

We prove that all flexible Weinstein fillings of a given contact manifold with vanishing first Chern class have isomorphic integral cohomology. As an application, we show that in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. We also construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic. These contact structures are distinguished by positive symplectic homology, which we prove is a contact invariant for flexibly-filled contact structures. The key step is a procedure for increasing the degrees of Reeb chords of loose Legendrians.

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 admitting minimal strong fillings of arbitrarily large $b_2$. Moreover, we introduce the notion of symplectic uniruled/adjunction caps and uniruled/adjunction contact structures to present a unified picture to the existing finiteness results on the topological invariants of exact/strong fillings of a contact 3-manifold. As a byproduct, we find new classes of contact 3-manifolds with the finiteness property and extend Wand's obstruction of planar contact 3-manifolds to uniruled/adjunction contact structures with complexity zero. structures with complexity zero.

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure ξ is tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S 3. We also describe geometric conditions in dimension three for ξ to be universally tight in the nonpositive curvature setting.

Let a contact 3-manifold (Y, \xi_0) be the link of a normal surface singularity equipped with its canonical contact structure \xi_0 . We prove a special property of such contact 3-manifolds of "algebraic" origin: the Heegaard Floer invariant c^+(\xi_0)\in \mathrm {HF}^+(-Y) cannot lie in the image of the U -action on \mathrm {HF}^+(-Y) . It follows that Karakurt's "height of U -tower" invariants are always 0 for canonical contact structures on singularity links, which contrasts the fact that the height of U -tower can be arbitrary for general fillable contact structures. Our proof uses the interplay between the Heegaard Floer homology and Némethi's lattice cohomology.

Owing to the existence of a dilatation generator of eigenvalues ± 2 , ± 1 , 0 \pm 2, \pm 1,0 the symplectic Lie algebra is considered as a | 2 | |2| -graded Lie algebra. The corresponding decomposition of the symplectic group Sp(2( n + 1 ), R ) {\text {Sp(2(}}n + 1{\text {),}}{\mathbf {R}}{\text {)}} makes the semidirect product (denoted L 0 {L^0} ) of the ( 2 n + 1 ) (2n + 1) -dimensional Weyl group by the conformal symplectic group CSp( 2 n , R ) {\text {CSp(}}2n,{\mathbf {R}}{\text {)}} appear as a privileged subgroup and permits one to construct a 2 n + 1 2n + 1 -dimensional homogeneous space possessing a natural contact form. Then Sp ( 2 ( n + 1 ) , R ) {\text {Sp}}(2(n + 1),{\mathbf {R}}) -valued Cartan connections on a L 0 {L^0} principal fibre bundle over a 2 n + 1 2n + 1 -dimensional manifold B 2 n + 1 {B_{2n + 1}} are constructed and called symplectic Cartan connections. The conditions for obtaining a unique symplectic Cartan connection are given. The existence of this unique Cartan connection is used to define the notion of contact structure over B 2 n + 1 {B_{2n + 1}} and it is shown that any L 0 {L^0} -structure of degree 2 2 over B 2 n + 1 {B_{2n + 1}} can be considered as a contact structure on it. Moreover it is shown that a contact structure can be associated in a canonical way to any contact manifold.

This survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.