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Torsion in the full orbifold \(K\)-theory of abelian symplectic quotients. (English) Zbl 1244.19005
In this paper, the full orbifold \(K\)-theory of abelian symplectic quotients is discussed. Let \((M, \omega)\) be a Hamilton \(T\)-space with moment map \(\Phi : M \to \mathfrak{t}^*\), where \(T\) is a torus. Let \(\beta : H \hookrightarrow T\) be a subtorus, then the moment map \(\Phi_H : M \to \mathfrak{h}^*\) for the action of \(H\) is given by the composition of \(\Phi\) and the linear projection \(\beta^* : \mathfrak{t}^* \to \mathfrak{h}^*\). Suppose \(\eta \in \mathfrak{h}^*\) is a regular value of \(\Phi_H\) and put \(Z=\Phi_H^{-1}(\eta)\). Then \(Z\) becomes a smooth submanifold of \(M\) with \(H\) acting locally freely. This allows us to define the quotient orbifold stack \(\mathfrak{X}=[Z/H]\). The authors prove that if there exists \(\xi \in \mathfrak{t}\) satisfying certain conditions (omitted here for simplicity), then \(\mathbb{K}_{\mathrm{orb}}(\mathfrak{X})\) contains no additive torsion, where \(\mathbb{K}_{\mathrm{orb}}(\mathfrak{X})\) is identified with \(\bigoplus_{t \in H}K_H(Z^t)\). However the hypothesis proposed perhaps may be already satisfied in many cases of interest. In fact, as an example of an important subclass of abelian symplectic quotients the authors mention the class of orbifold toric varieties, and check that they satisfy this hypothesis. In the last two sections, further examples are provided, which are non-toric.
The proof is done using equivariant Morse theory of the moment map. For the \(\xi\) given above, define a function \(\Phi^\xi : M \to \mathbb{R}\) by \(\Phi^\xi(x)=\langle \Phi(x), \xi \rangle\). Then it can be shown that the restriction \(\Phi^\xi|Z^t\) becomes a Morse-Bott function for each \(Z^t\). From analyzing them, the authors conclude that \(K_H(Z^t)\) is torsion-free for each \(Z^t\).

19L47 Equivariant \(K\)-theory
53D20 Momentum maps; symplectic reduction
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