scholarly journals Remark on the Betti numbers for Hamiltonian circle actions

2021 ◽  
Vol 359 (2) ◽  
pp. 113-117
Author(s):  
Yunhyung Cho
Keyword(s):  
Betti Numbers ◽  
Circle Actions

scholarly journals HAMILTONIAN CIRCLE ACTIONS WITH MINIMAL FIXED SETS

2012 ◽  
Vol 23 (08) ◽  
pp. 1250071 ◽  
Author(s):  
HUI LI ◽  
SUSAN TOLMAN
Keyword(s):  
Cohomology Ring ◽  
Betti Numbers ◽  
Symplectic Manifolds ◽  
Dimensional Manifold ◽  
Effective Hamiltonian ◽  
Chern Classes ◽  
Circle Actions ◽  
Integral Cohomology ◽  
Fixed Set ◽  
Hamiltonian Actions

Consider an effective Hamiltonian circle action on a compact symplectic 2n-dimensional manifold (M, ω). Assume that the fixed set MS1 is minimal, in two senses: It has exactly two components, X and Y, and dim (X) + dim (Y) = dim (M) - 2. We prove that the integral cohomology ring and Chern classes of M are isomorphic to either those of ℂℙn or (if n ≠ 1 is odd) to those of [Formula: see text], the Grassmannian of oriented two-planes in ℝn+2. In particular, Hi(M;ℤ) = Hi(ℂℙn; ℤ) for all i, and the Chern classes of M are determined by the integral cohomology ring. We also prove that the fixed set data of M agrees exactly with the fixed set data for one of the standard circle actions on one of these two manifolds. In particular, we show that there are no points with stabilizer ℤk for any k > 2. The same conclusions hold when MS1 has exactly two components and the even Betti numbers of M are minimal, that is, b2i(M) = 1 for all i ∈ {0, …, ½ dim (M)}. This provides additional evidence that very few symplectic manifolds with minimal even Betti numbers admit Hamiltonian actions.

scholarly journals Unimodality of Betti numbers for Hamiltonian circle actions with index-increasing moment Maps

2016 ◽  
Vol 27 (05) ◽  
pp. 1650043 ◽  
Author(s):  
Yunhyung Cho
Keyword(s):  
Fixed Points ◽  
Morse Index ◽  
Equivariant Cohomology ◽  
Arbitrary Dimension ◽  
Betti Numbers ◽  
Circle Action ◽  
Dimensional Case ◽  
Circle Actions ◽  
Moment Maps ◽  
The Moment

The unimodality conjecture posed by Tolman in [L. Jeffrey, T. Holm, Y. Karshon, E. Lerman and E. Meinrenken, Moment maps in various geometries, http://www.birs.ca/workshops/2005/05w5072/report05w5072.pdf ] states that if [Formula: see text] is a [Formula: see text]-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points, then the sequence of Betti numbers [Formula: see text] is unimodal, i.e. [Formula: see text] for every [Formula: see text]. Recently, the author and Kim [Y. Cho and M. Kim, Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points, Math. Res. Lett. 21(4) (2014) 691–696] proved that the unimodality holds in eight-dimensional case by using equivariant cohomology theory. In this paper, we generalize the idea in [Y. Cho and M. Kim, Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points, Math. Res. Lett. 21(4) (2014) 691–696] to an arbitrary dimensional case. We prove the conjecture in arbitrary dimension under the assumption that the moment map [Formula: see text] is index-increasing, which means that [Formula: see text] implies [Formula: see text] for every pair of critical points [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is the Morse index of [Formula: see text] with respect to [Formula: see text].

Extremal Betti numbers of graded ideals

2003 ◽  
Vol 43 (3-4) ◽  
pp. 235-244 ◽  
Author(s):  
Marilena Crupi ◽  
Rosanna Utano
Keyword(s):  
Betti Numbers

scholarly journals Betti numbers and pseudoeffective cones in 2-Fano varieties

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Giosuè Emanuele Muratore
Keyword(s):  
Large Class ◽  
Betti Numbers ◽  
Fano Varieties ◽  
Higher Dimensional ◽  
Special Case ◽  
Answering Questions

Abstract The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher-dimensional analogous properties of Fano varieties. We consider (weak) k-Fano varieties and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties, in analogy with the case k = 1. Then we calculate some Betti numbers of a large class of k-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index at least n − 2, and we complete the classification of weak 2-Fano varieties answering Questions 39 and 41 in [2].

scholarly journals Topology of Subvarieties of Complex Semi-abelian Varieties

2020 ◽  
Author(s):  
Yongqiang Liu ◽  
Laurentiu Maxim ◽  
Botong Wang
Keyword(s):  
Euler Characteristic ◽  
Morse Theory ◽  
Characteristic Property ◽  
Abelian Varieties ◽  
Betti Numbers ◽  
Local Systems ◽  
Affine Manifolds ◽  
Cyclic Covers ◽  
Rank One ◽  
Generic Vanishing

Abstract We use the non-proper Morse theory of Palais–Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast June Huh’s extension of Varchenko’s conjecture to very affine manifolds and provide a generalization of this result in the context of smooth closed sub-varieties of semi-abelian varieties.

scholarly journals L2-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1169-1188 ◽  
Author(s):  
ROMAN SAUER
Keyword(s):  
Probability Space ◽  
Betti Numbers ◽  
Free Action ◽  
Countable Group ◽  
Equivalence Relations ◽  
Dimension Theory ◽  
Discrete Groups ◽  
Type Ii ◽  
Orbit Equivalent ◽  
Definition Of

There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L2-Betti numbers [Formula: see text] of a countable group G coincide with the L2-Betti numbers [Formula: see text] of the orbit equivalence relation [Formula: see text] of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.

scholarly journals Bigraded Betti numbers of certain simple polytopes

2013 ◽  
Vol 94 (3-4) ◽  
pp. 351-363 ◽  
Author(s):  
I. Yu. Limonchenko
Keyword(s):  
Betti Numbers ◽  
Simple Polytopes
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