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Finite group actions on symplectic Calabi-Yau 4-manifolds with \(b_1>0\). (English) Zbl 1475.57025

Let \((M,\omega)\) be a symplectic \(4\)-manifold endowed with a symplectic action of a finite group \(G\). The present author, in [J. Gökova Geom. Topol. GGT 12, 1–39 (2018; Zbl 1479.57049)] proposed the following strategy for studying these: The singular set of the action, that is the set of points with nontrivial isotropy subgroup, will, in general, consist of \(2\)-dimensional components as well as isolated points. Although the underlying space \(|G/M|\) of the quotient orbifold \(G/M\) is smooth except for isolated singularities, the induced symplectic structure on \(G/M\) will be singular on the images of the \(2\)-dimensional components. Chen showed that one may de-singularize the symplectic form along these \(2\)-dimensional strata to obtain a symplectic \(4\)-orbifold \((|M/G|,\omega')\) with only isolated singularities. Let \(M_G\) be the minimal symplectic resolution of \(|M/G|\), and let \(D\) be the preimage of the singular set of the quotient orbifold \(M/G\) in \(M_G\). Chen proposed studying \(M\) via \(M_G\) and the embedding \(D\to M_G\). In particular, Chen showed that if \(M\) is Calabi-Yau, then \(M_G\) must be of torsion canonical class, be a rational \(4\)-manifold or be an irrational ruled manifold over \(T^2\).
In the present work, Chen continues this program by focusing on the case where \(M\) is Calabi-Yau, \(b_1(M)>0\) and \(G\) is cyclic. Under these conditions, Chen completely determines the fixed point set structure. A consequence of this is that for a symplectic Calabi-Yau \(4\)-manifold \(M\) with \(b_1(M)>0\) endowed with a finite symplectic \(G\)-action, if \(M_G\) is irrational ruled, or \(M_G\) is rational and \(G=\mathbb Z_2\), then \(M\) is diffeomorphic to a \(T^2\)-bundle over \(T^2\) with homologically essential fibres. To prove these results, Chen begins with a systematic fixed point analysis appealing to standard techniques from fixed point theory, namely, the Lefschetz fixed point theorem, the \(G\)-signature theorem and the \(G\)-index theorem for Dirac operators. Using these techniques, Chen obtains a complete description of the isolated points of the fixed point set of the \(G\)-action. However such techniques fall short when it comes to describing the \(2\)-dimensional components and, in particular, tori of self-intersecton zero. In order to describe these, using techniques from symplectic topology, Chen carries out a careful analysis of possible embeddings of disjoints unions of configurations of symplectic surfaces in symplectic rational \(4\)-manifolds.
As a byproduct of his investigations, Chen discovers that one cannot disjointly embed \(N\) symplectic \((-2)\)-spheres in \(\mathbb CP^2\# N\overline{\mathbb CP^2}\) for \(N=7,8\) or \(9\).

MSC:

57K40 General topology of 4-manifolds
57M60 Group actions on manifolds and cell complexes in low dimensions
57S17 Finite transformation groups
53D05 Symplectic manifolds (general theory)
57R55 Differentiable structures in differential topology

Citations:

Zbl 1479.57049
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Full Text: arXiv Link

References:

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