Symplectic actions of 2-tori on 4-manifolds.

*(English)*Zbl 1189.53081
Mem. Am. Math. Soc. 959, vii, 81 p. (2010).

This is a memoir in which the author describes the classification of symplectic actions of \(2\)-tori in a compact connected \(4\)-manifold. This generalizes the classical classification of Hamiltonian actions of tori over a symplectic \(4\)-manifold due to M. F. Atiyah [Bull. Lond. Math. Soc. 14, 1–15 (1982; Zbl 0482.58013)], V. Guillemin [Moment maps and combinatorial invariants of Hamiltonian \(T^n\)-spaces. Progress in Mathematics (Boston, Mass.) 122. Basel: Birkhäuser. (1994; Zbl 0828.58001)], T. Delzant [Hamiltoniens périodiques et images convexes de l’application moment (1988; Zbl 0676.58029)], etc. The complete classification result is stated as follows:

Let \((M, \sigma)\) be a compact connected \(4\)-dimensional symplectic \(4\)-manifold equipped with an effective symplectic action of a \(2\)-torus \(T\). Then one and only one of the following cases occurs:

1) \((M, \sigma)\) is a \(4\)-dimensional symplectic-toric manifold, determined by its associated Delzant polygon.

2) \((M, \sigma)\) is equivariantly symplectomorphic to a product \(T^2 \times S^2\), where \(T^2 = (\mathbb R/\mathbb Z)^2\) and the first factor of \(T^2\) acts on the left factor by translations on one component, and the second factor acts on \(S^2\) by rotations about the vertical axis of \(S^2\). The symplectic form is a positive linear combination of the standard translation invariant form on \(T^2\) and the standard rotation invariant form on \(S^2\).

3) \(T\) acts freely on \((M, \sigma)\) with all \(T\)-orbits being Lagrangian \(2\)-tori, and \((M,\sigma)\) is a principal \(T\)-bundle over a \(2\)-torus with Lagrangian fibers. In this case \((M, \sigma)\) is classified (as earlier) by a discrete cocompact subgroup \(P\) of \(\mathfrak{t}^*\), an antisymmetric bilinear mapping \(c : \mathfrak{t}^* \times \mathfrak{t}^* \to \mathfrak{t}\) which satisfies certain integrality properties, and the so called holonomy invariant of an admissible connection for \( M_{\text{reg}} \to M_{\text{reg}} / T\) .

4) \(T\) acts locally freely on \((M, \sigma)\) with all \(T\)-orbits being symplectic \(2\)-tori, and \((M, \sigma)\) is a principal \(T\)-orbibundle over an oriented orbisurface with symplectic fibers. In this case \((M, \sigma)\) is classified by an antisymmetric bilinear form \(\sigma^{\mathfrak{t}}\) on \(\mathfrak{t}\), the Fuchsian signature of \(M/T\), the total symplectic area of \(M/T\), and an element in \(T^{2n+g}/G\), where \(g\) is the genus of \(M/T\) and \(n\) is the number of singular points of \(M/T\).

The first case corresponds to the classical Hamiltonian case. The other cases are deduced from more general results due to the author, namely a classification result for effective symplectic torus actions over a manifold with a coisotropic orbit [J. J. Duistermaat, A. Pelayo, Ann. Inst. Fourier 57, No. 7, 2239–2327 (2007; Zbl 1197.53114)] and a result of classification of effective symplectic \(T^{2n-2}\) actions over a \((2n)\)-dimensional manifold, that is part of the article. This implies the classification since in dimension \(4\) the orbits of the action are either Lagrangian (so in particular coisotropic) or symplectic.

Let \((M, \sigma)\) be a compact connected \(4\)-dimensional symplectic \(4\)-manifold equipped with an effective symplectic action of a \(2\)-torus \(T\). Then one and only one of the following cases occurs:

1) \((M, \sigma)\) is a \(4\)-dimensional symplectic-toric manifold, determined by its associated Delzant polygon.

2) \((M, \sigma)\) is equivariantly symplectomorphic to a product \(T^2 \times S^2\), where \(T^2 = (\mathbb R/\mathbb Z)^2\) and the first factor of \(T^2\) acts on the left factor by translations on one component, and the second factor acts on \(S^2\) by rotations about the vertical axis of \(S^2\). The symplectic form is a positive linear combination of the standard translation invariant form on \(T^2\) and the standard rotation invariant form on \(S^2\).

3) \(T\) acts freely on \((M, \sigma)\) with all \(T\)-orbits being Lagrangian \(2\)-tori, and \((M,\sigma)\) is a principal \(T\)-bundle over a \(2\)-torus with Lagrangian fibers. In this case \((M, \sigma)\) is classified (as earlier) by a discrete cocompact subgroup \(P\) of \(\mathfrak{t}^*\), an antisymmetric bilinear mapping \(c : \mathfrak{t}^* \times \mathfrak{t}^* \to \mathfrak{t}\) which satisfies certain integrality properties, and the so called holonomy invariant of an admissible connection for \( M_{\text{reg}} \to M_{\text{reg}} / T\) .

4) \(T\) acts locally freely on \((M, \sigma)\) with all \(T\)-orbits being symplectic \(2\)-tori, and \((M, \sigma)\) is a principal \(T\)-orbibundle over an oriented orbisurface with symplectic fibers. In this case \((M, \sigma)\) is classified by an antisymmetric bilinear form \(\sigma^{\mathfrak{t}}\) on \(\mathfrak{t}\), the Fuchsian signature of \(M/T\), the total symplectic area of \(M/T\), and an element in \(T^{2n+g}/G\), where \(g\) is the genus of \(M/T\) and \(n\) is the number of singular points of \(M/T\).

The first case corresponds to the classical Hamiltonian case. The other cases are deduced from more general results due to the author, namely a classification result for effective symplectic torus actions over a manifold with a coisotropic orbit [J. J. Duistermaat, A. Pelayo, Ann. Inst. Fourier 57, No. 7, 2239–2327 (2007; Zbl 1197.53114)] and a result of classification of effective symplectic \(T^{2n-2}\) actions over a \((2n)\)-dimensional manifold, that is part of the article. This implies the classification since in dimension \(4\) the orbits of the action are either Lagrangian (so in particular coisotropic) or symplectic.

Reviewer: Francisco Presas Mata (Madrid)