scholarly journals On the Definition of Irreducible Holomorphic Symplectic Manifolds and Their Singular Analogs

2021 ◽  
Author(s):  
Martin Schwald
Keyword(s):  
Symplectic Manifolds ◽  
Complex Tori ◽  
Finite Quotients ◽  
Definition Of ◽  
Simply Connected

Abstract In the definition of irreducible holomorphic symplectic manifolds the condition of being simply connected can be replaced by vanishing irregularity. We discuss holomorphic symplectic, finite quotients of complex tori with ${\operatorname{h}}^0(X,\,\Omega ^{[2]}_X)=1$ and their Lagrangian fibrations. Neither $X$ nor the base can be smooth unless $X$ is a $2$-torus.

scholarly journals Affine varieties, singularities and the growth rate of wrapped Floer cohomology

2018 ◽  
Vol 10 (03) ◽  
pp. 493-530
Author(s):  
Mark McLean
Keyword(s):  
Growth Rate ◽  
Cotangent Bundle ◽  
Betti Numbers ◽  
Symplectic Manifolds ◽  
Isolated Singularity ◽  
Contact Manifolds ◽  
Floer Cohomology ◽  
Complex Singularities ◽  
Smooth Affine ◽  
Simply Connected

In this paper, we give partial answers to the following questions: Which contact manifolds are contactomorphic to links of isolated complex singularities? Which symplectic manifolds are symplectomorphic to smooth affine varieties? The invariant that we will use to distinguish such manifolds is called the growth rate of wrapped Floer cohomology. Using this invariant we show that if [Formula: see text] is a simply connected manifold whose unit cotangent bundle is contactomorphic to the link of an isolated singularity or whose cotangent bundle is symplectomorphic to a smooth affine variety then M must be rationally elliptic and so it must have certain bounds on its Betti numbers.

l′-Isolated Maps and Localizations

1983 ◽  
Vol 35 (2) ◽  
pp. 193-217
Author(s):  
Sara Hurvitz
Keyword(s):  
Finite Number ◽  
Free Algebra ◽  
Nilpotent Groups ◽  
Homotopy Groups ◽  
Homology Groups ◽  
Rational Cohomology ◽  
Locally Nilpotent Groups ◽  
Definition Of ◽  
Simply Connected ◽  
Path Connected

Let P be the set of primes, l ⊆ P a subset and l′ = P – l Recall that an H0-space is a space the rational cohomology of which is a free algebra.Cassidy and Hilton defined and investigated l′-isolated homomorphisms between locally nilpotent groups. Zabrodsky [8] showed that if X and Y are simply connected H0-spaces either with a finite number of homotopy groups or with a finite number of homology groups, then every rational equivalence f : X → Y can be decomposed into an l-equivalence and an l′-equivalence.In this paper we define and investigate l′-isolated maps between pointed spaces, which are of the homotopy type of path-connected nilpotent CW-complexes. Our definition of an l′-isolated map is analogous to the definition of an l′-isolated homomorphism. As every homomorphism can be decomposed into an l-isomorphism and an l′-isolated homomorphism, every map can be decomposed into an l-equivalence and an l′-isolated map.

Simply Connected Limits

1990 ◽  
Vol 42 (4) ◽  
pp. 731-746 ◽  
Author(s):  
Robert Paré
Keyword(s):  
Left Adjoint ◽  
Topos Theory ◽  
Essential Ingredient ◽  
Original Definition ◽  
The Subject ◽  
Definition Of ◽  
Simply Connected ◽  
Geometric Morphisms ◽  
Exact Categories

The importance of finite limits in completeness conditions has been long recognized. One has only to consider elementary toposes, pretoposes, exact categories, etc., to realize their ubiquity. However, often pullbacks suffice and in a sense are more natural. For example it is pullbacks that are the essential ingredient in composition of spans, partial morphisms and relations. In fact the original definition of elementary topos was based on the notion of partial morphism classifier which involved only pullbacks (see [6]). Many constructions in topos theory, involving left exact functors, such as coalgebras on a cotriple and the gluing construction, also work for pullback preserving functors. And pullback preserving functors occur naturally in the subject, e.g. constant functors and the Σα. These observations led Rosebrugh and Wood to introduce partial geometric morphisms; functors with a pullback preserving left adjoint [9]. Other reasons led Kennison independently to introduce the same concept under the name semi-geometric functors [5].

scholarly journals On the formality and strong Lefschetz property of symplectic manifolds – Erratum

2009 ◽  
Vol 147 (1) ◽  
pp. 255-255
Author(s):  
Taek Kyu Hwang ◽  
Jin Hong Kim
Keyword(s):  
Fundamental Group ◽  
Symplectic Manifold ◽  
Symplectic Manifolds ◽  
Finite Covering ◽  
Massey Product ◽  
Lefschetz Property ◽  
The Stability ◽  
Simply Connected ◽  
Open Question ◽  
Strong Lefschetz Property

Professor Vicente Muñoz kindly informed us that there is an inaccuracy in Lemma 3.5 of [1]. The correct statement of Lemma 3.5 is now that the fundamental group π1(X′) of the manifold X′ is Z, since the monodromy coming from φ8 does not imply that g4 = g4−1. Therefore, what we have actually constructed in Section 3 of [1] is a closed non-formal 8-dimensional symplectic manifold with π1 = Z whose triple Massey product is non-zero, so that the simply-connectedness in Theorem 1.1 should be dropped. As far as we know, the existence of a simply connected closed non-formal 8-dimensional symplectic manifold whose triple Massey product is non-zero still remains an open question. All other main results, especially Theorem 1.2 and Corollary 1.3, in [1] are not affected by this mistake. Furthermore, the stability of the non-formality under a finite covering as in Subsection 3.3 holds in general. We want to thank Professor Muñoz for his careful reading.

scholarly journals Neo-Normal Functions in Arbitrary Regions

1972 ◽  
Vol 48 ◽  
pp. 19-36
Author(s):  
Kam-Fook Tse
Keyword(s):  
Unit Disk ◽  
Meromorphic Functions ◽  
Boundary Behavior ◽  
Universal Covering ◽  
Value Distribution ◽  
Covering Surface ◽  
Normal Functions ◽  
Definition Of ◽  
Simply Connected Region ◽  
Simply Connected

It is well known that many properties possessed by functions holomorphic and bounded in a region are also possessed by functions meromorphic and omitting three values. Noshiro [14] in 1938 and Lehto and Virtanen [12] in 1957 independently defined the notion of “normal functions” ; they and many others subsequently discovered that most properties concerning boundary behavior and value distribution acquired by meromorphic functions omitting three values in the unit disk (or more general, in a simply-connected region) are also valid properties of “normal functions” defined there. In their research on the problems of value distribution of normal functions, Lange [9], Gavrilov [5] and Gauthier [4] have discovered that functions normal in the disk are exactly those which omit three values “locally,” i.e., they do not possess any “p-sequence” (see above references). However, the definition of a function being normal in a region depends on the simply-connectedness of the region or its universal covering surface. It is thus difficult to judge if a function defined in an arbitrary region is normal.

scholarly journals Bounds for tentacular Hamiltonians

2018 ◽  
Vol 12 (01) ◽  
pp. 209-265 ◽  
Author(s):  
Federica Pasquotto ◽  
Jagna Wiśniewska
Keyword(s):  
Floer Homology ◽  
Energy Levels ◽  
Symplectic Manifolds ◽  
Rabinowitz Floer Homology ◽  
The Subject ◽  
Definition Of

This paper represents a first step toward the extension of the definition of Rabinowitz Floer homology to non-compact energy hypersurfaces in exact symplectic manifolds. More concretely, we study under which conditions it is possible to establish [Formula: see text]-bounds for the Floer trajectories of a Hamiltonian with non-compact energy levels. Moreover, we introduce a class of Hamiltonians, called tentacular Hamiltonians which satisfy the conditions: how to define Rabinowitz Floer homology for these examples will be the subject of a follow-up paper.

scholarly journals Derivatives, Products, and Pullbacks in Forman's Combinatorial Differential Forms

2020 ◽  
Vol 3 (1) ◽  
pp. 7-16
Author(s):  
Vu Quang Huynh ◽  
Thach Phu Nguyen ◽  
Phuong Van Phan
Keyword(s):  
Connected Domain ◽  
Differential Forms ◽  
Sufficient Condition ◽  
Simply Connected Domain ◽  
Definition Of ◽  
Simply Connected

We study derivatives, closedness, and exactness of 0-forms and 1-forms in the theory of combinatorial differential forms constructed by Robin Forman. We give an example of a closed but not exact 1-form on a non-simply connected domain. We give a sufficient condition on the domain for a closed 1-form to be exact. We show that the product of forms proposed by Forman is not anti-commutative. We propose a definition of pullbacks of forms and show that this operation has several properties analogous to pullbacks on smooth forms.

scholarly journals On nonformal simply-connected symplectic manifolds

2000 ◽  
Vol 41 (2) ◽  
pp. 204-217 ◽  
Author(s):  
I. K. Babenko ◽  
I. A. Taîmanov
Keyword(s):  
Symplectic Manifolds ◽  
Simply Connected

scholarly journals Formality and the Lefschetz property in symplectic and cosymplectic geometry

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Giovanni Bazzoni ◽  
Marisa Fernández ◽  
Vicente Muñoz
Keyword(s):  
Betti Numbers ◽  
Symplectic Manifolds ◽  
Topological Properties ◽  
Cosymplectic Manifolds ◽  
Connected Case ◽  
Lefschetz Property ◽  
Simply Connected

AbstractWe review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the b

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