On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action

Let $(M,J)$ be a compact, connected, almost complex manifold of dimension $2n$ endowed with a $J$-preserving circle action with isolated fixed points. In this note we analyse the `geography problem' for such manifolds, deriving equations relating the Chern numbers to the index $k_0$ of $(M,J)$. We study the symmetries and zeros of the Hilbert polynomial associated to $(M,J)$, which imply many rigidity results for the Chern numbers when $k_0\neq 1$. We apply these results to the category of compact, connected symplectic manifolds. A long-standing question posed by McDuff and Salamon, also known as the `McDuff conjecture', asked about the existence of non-Hamiltonian actions with isolated fixed points. This question was answered recently by Tolman, with an explicit construction of a six-dimensional manifold with such an action. One issue that this raises is whether one can find topological criteria that ensure the manifold can only support a Hamiltonian or only a non-Hamiltonian action. In this vein, we are able to deduce such criteria from our rigidity theorems in terms of relatively few Chern numbers, depending on the index. This improves upon results of Feldman for which one needs to know the entire Todd genus. Another consequence of our work is that, in the situation above with a Hamiltonian action, the minimal Chern number coincides with the index and is at most $n+1$, mirroring results of Michelsohn in the complex category and Hattori in the almost complex category.Comment: v2. 38 pages, 1 figure. Major revision, especially in Section 6 (in particular, Proposition 6.8 contained an incorrect statement

New tools for classifying Hamiltonian circle actions with isolated fixed points

For every compact almost complex manifold (M,J) equipped with a 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 M is symplectic and the action is Hamiltonian. If the manifold satisfies an extra "positivity condition" this algorithm determines a family of vector spaces which contain the admissible lattices of weights. When the number of fixed points is minimal, this positivity condition is necessarily satisfied whenever dim(M)< 8, and, when dim(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 [Y]. We run the algorithm for dim(M)< 10, quickly obtaining all the possible families of isotropy weights. In particular, we simplify the proofs of Ahara and Tolman for dim(M)=6 [Ah,T1] and, when dim(M)=8, we prove that the equivariant cohomology ring, Chern classes and isotropy weights agree with the ones of C P^4 with the standard S^1-action (thus proving the symplectic Petrie conjecture [T1] in this setting).Comment: 59 Pages; 16 Figures; Please find accompanying software at page http://www.math.ist.utl.pt/~lgodin/MinimalActions.htm

New Techniques for obtaining Schubert-type formulas for Hamiltonian manifolds

In [GT], Goldin and the second author extend some ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. (See also [Kn99] and [Kn08].) The main goal of this paper is to build on this work by finding more effective formulas. More explicitly, given a generic component of the moment map, they define a canonical class $\alpha_p$ in the equivariant cohomology of the manifold $M$ for each fixed point $p \in M$. When they exist, canonical classes form a natural basis of the equivariant cohomology of $M$. In particular, when $M$ is a flag variety, these classes are the equivariant Schubert classes. It is a long standing problem in combinatorics to find positive integral formulas for the equivariant structure constants associated to this basis. Since computing the restriction of the canonical classes to the fixed points determines these structure constants, it is important to find effective formulas for these restrictions. In this paper, we introduce new techniques for calculating the restrictions of a canonical class $\alpha_p$ to a fixed point $q$. Our formulas are nearly always simpler, in the sense that they count the contributions over fewer paths. Moreover, our formula is manifestly positive and integral in certain important special cases.Comment: v2; Significant revision. 52 pages, 1 figure. To appear in Journal of Symplectic Geometr

12, 24 and Beyond

We generalize the well-known "12" and "24" Theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope. Our methods apply to a wider category of objects, here called reflexive GKM graphs, that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action. As an application, we provide bounds on the Betti numbers for certain monotone Hamiltonian spaces which depend on the minimal Chern number of the manifold.Comment: 39 pages, 4 figure

Tailored Brushing Method (TBM): an innovative simple protocol to improve the oral care

Background. The objective of this study is to describe and assess the effectiveness and acceptability of a modern tailored protocol of oral hygiene, based on the concordance between professionals and patients, and based on the proper choice of best tools for oral hygiene regardless of the technique used. This new method has been called Tailored Brushing Method (TBM). Material and methods. Two groups of adult patients (n=200) were involved in this research, according to specific inclusion criteria. Test group followed the indications of the new Tailored Brushing Method, while control group was involved in a standard protocol of oral hygiene with the suggestions of a brushing technique and the typical approach based on the compliance. Plaque Index, bleeding on Probing index and patients' acceptability of the methods (Visual Analog Scale) were assessed at different time points. Descriptive and statistical analyses were performed. Results and conclusions. Test group had statistically lower Plaque Index and Bleeding on Probing after 30 days, in comparison with control group. Test group expressed a better acceptance of the new tailored method. This research suggests to use a tailored approach to oral hygiene, overcoming the need of patient's compliance, often affected by bias such as alteration of the protocols and wrong brushing maneuvers