Manifolds on which only tori can act

Contractible Open 3-manifolds Which Non-trivially Cover Only Non-compact 3-manifolds

Symplectic actions of two tori on four manifolds

We classify symplectic actions of 2-tori on compact, connected symplectic 4-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin–Sternberg, Delzant and Benoist. The classification is in terms of a collection of invariants, which are invariants of the topology of the manifold, of the torus action and of the symplectic form. We construct explicit models of su...

Symplectic actions of 2-tori on 4-manifolds

We classify symplectic actions of 2-tori on compact, connected symplectic 4-manifolds, up to equivariant symplectomorphisms, hence extending the theory of Atiyah, Guillemin–Sternberg, Delzant and Benoist to actions of tori which are not necessarily Hamiltonian. The classification is in terms of a collection of up to nine invariants. We construct an explicit model of such symplectic manifolds wi...

The Calabi–yau Equation on 4-manifolds over 2-tori

This paper pursues the study of the Calabi–Yau equation on certain symplectic non-Kähler 4-manifolds, building on a key example of Tosatti and Weinkove in which more general theory had proved less effective. Symplectic 4-manifolds admitting a 2-torus fibration over a 2-torus base T2 are modelled on one of three solvable Lie groups. Having assigned an invariant almost-Kähler structure and a volu...

Potential Functions and Actions of Tori on Kähler Manifolds

inherits from M a Kähler structure; and in the first part of this paper we will describe what the Kähler form and Ricci form look like locally on coordinate patches in Mλ. Then in the second part of this paper we will discuss some global implications of these results. This will include (1) A Kählerian proof of the Duistermaat-Heckman theorem. (2) A formula, due to Biquard and Gauduchon, for the...