Abstract
Let \(g\) be an involution of a compact closed manifold \(X\) such that the fixed-point set \(X^{g}\) is middle dimensional. Under the assumption that the normal bundle of the fixed-point set is either the tangent or co-tangent bundle conditions on the equivariant invariants of \(X\) arise. In particular if \(X\) is a holomorphic-symplectic manifold and \(g\) an anti holomorphic-symplectic involution one arrives at a generalisation of Beauville’s result that for \(X\) a hyper-Kähler manifold the \(\hat{A}\) genus of \(X^{g}\) is one.
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Notes
The orientation convention used here is the natural one that arises if \(M\) is a complex manifold so that \(T^{*}M\) is the complex dual of \(TM\). In such a case we have that \(c_{n}(T^{*}M)= e(T^{*}M)= (-1)^{n}e(TM)\).
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Acknowledgments
It is a pleasure to thank Lothar Göttsche for asking a related question, his interest in this work and numerous discussions. Many thanks are due to Arnaud Beauville who read an initial draft and suggested some improvements. I would also like to thank Alberto Verjovsky for pointing out that some of the Introduction is standard lore in symplectic geometry.
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Thompson, G. On involutions with middle-dimensional fixed-point locus and holomorphic-symplectic manifolds. Ann Glob Anal Geom 45, 239–250 (2014). https://doi.org/10.1007/s10455-013-9398-5
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DOI: https://doi.org/10.1007/s10455-013-9398-5