Notebook implementations of algorithms developed in the following research preprints (in preparation).
Leonor Godinho and Silvia Sabatini,
Given the dimension of a compact almost complex manifold (M,J) admitting a J-preserving circle action with a nonempty, discrete fixed point set and the value of the Chern number c_1 c_(n - 1)[M], this program gives a lower bound for the Euler characteristic of M or, equivalently, for the number of fixed points of the circle action. It also gives this lower bound in the case where M is symplectic and the action is Hamiltonian.
Open notebookCarlos Florentino and Leonor Godinho
Given the number of steps n of a polygon, this program produces a table with the possible diffeomorphism types of smooth Fano polygon spaces with n steps, presenting the corresponding Picard Numbers and Poincar\'e polynomials, and giving an example of a monotone length vector for each diffeomorphism type. Moreover, it explains how to obtain these spaces from the complex projective space P^(n-3) through a sequence of partial blow-ups and blow-downs. The notation used is the following: If the diffeomorphism type is (k_ 1,...,k_(n-3)), then the space can be obtained from P^(n-3) through k_1 blow ups at a point, and a sequence of k_i partial blow downs of P^(i-1) followed by k_i partial blow ups of the points obtained, yielding k_i copies of "exceptional" P^(n-i-3)s, for i=2,...,n-3; this process is denoted by k_i Bl(P^(i-1),P^(n-i-3)), i=2,...,n-3 .
In a separate part, given an arbitrary vector of lengths a:=(a_1,...a_n), it determines if a is admissible and generic and, if this is the case, computes the Poincare Polynomial of the smooth polygon space M(a) (not necessarily Fano), its volume and describes how to obtain this manifold from a complex projective space through partial blow downs and blow ups.
Open notebook