arXiv subjects

Ignasi Mundet-i-Riera

Publications and source records attributed to Ignasi Mundet-i-Riera.

3 recordsLinked to original sources

arXiv2014

Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds

A recent preprint of Csikós, Pyber and Szabó (arXiv:1411.7524) proves that the diffeomorphism group of $T^2\times S^2$ is not Jordan. The purpose of this paper is to generalize the arguments of Csikós, Pyber and Szabó in order to obtain many other examples of compact manifolds whose diffeomorphism group fails to be Jordan. In particular we prove that for any $ε>0$ there exist manifolds admitting effective actions of arbitrarily large $p$-groups $Γ$ all of whose abelian subgroups have at most $|Γ|^ε$ elements. Finally, we also recover some results on nonexistence of effective actions of compact connected semisimple Lie group on manifolds.

math.DG↗

arXiv2008

A Hilbert--Mumford criterion for polystability in Kaehler geometry

Consider a Hamiltonian action by biholomorphisms of a compact Lie group $K$ on a Kaehler manifold $X$, with moment map $μ:X\to\klie^*$. We characterize which orbits of the complexified action of $G=K^{\CC}$ in $X$ intersect $μ^{-1}(0)$ in terms of the maximal weights $\lim_{t\to\infty}\laμ(e^{\imag ts}\cdot x),s\ra$, where $s$ belongs to the Lie algebra of $K$. We do not impose any a priori restriction on the stabilizer of $x$. Assuming some mild growth conditions on the action of $K$ on $X$, we view the maximal weights as defining a maps $λ_x$ from the boundary at infinity of the symmetric space $K\backslash G$ to $\RR\cup\{\infty\}$. We prove that $G\cdot x$ meets $μ^{-1}(0)$ if: (1) $λ_x$ is everywhere nonnegative, (2) any boundary point $y$ such that $λ_x(y)=0$ can be connected with a geodesic in $K\backslash G$ to another boundary point $y'$ satisfying $λ_x(y')=0$. We also prove that $λ_{g\cdot x}(y)=λ_x(y\cdot g)$ for any $g\in G$ and $y\in \partial_{\infty}(K\backslash G)$.

math.SG↗

arXiv2006

The biinvariant diagonal class for Hamiltonian torus actions

Suppose that an algebraic torus $G$ acts algebraically on a projective manifold $X$ with generically trivial stabilizers. Then the Zariski closure of the set of pairs $\{(x,y)\in X\times X\mid y=gx \text{for some}g\in G\}$ defines a nonzero equivariant cohomology class $[Δ_G]\in H^*_{G\times G}(X\times X)$. We give an analogue of this construction in the case where $X$ is a compact symplectic manifold endowed with a hamiltonian action of a torus, whose complexification plays the role of $G$. We also prove that the Kirwan map sends the class $[Δ_G]$ to the class of the diagonal in each symplectic quotient. This allows to define a canonical right inverse of the Kirwan map.

math.SG↗