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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.

##### MSC:
 53D35 Global theory of symplectic and contact manifolds 57M60 Group actions on manifolds and cell complexes in low dimensions 55R10 Fiber bundles in algebraic topology
##### Keywords:
symplectic manifold; torus action; four-manifold; orbifold
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