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The fixed point set of \(\mathbb{C}\) actions on a compact complex space. (English) Zbl 0876.14032
Let \(G\times X\to X\) be a holomorphic action of a unipotent complex linear algebraic group \(G\) on a connected compact complex space \(X\) which can be extended to a meromorphic map \(G^*\times X\to X\) for an (arbitrary) algebraic compactification \(G^*\) of \(G\). Then the fixed point set \(X^G\) of the \(G\)-action is connected and non-empty [see G. Horrocks, Topology 8, 233-242 (1969; Zbl 0159.22401) and J. B. Carrell and A. J. Sommese in: Group actions and vector fields, Proc. Pol.-North. Am. Semin., Vancouver 1981, Lect. Notes 956, 23-28 (1982; Zbl 0493.32026)]. Topologically the relation between \(X^G\) and \(X\) is in fact much closer. The author shows that the canonical injection \(X^G \subset X\) induces an isomorphism between \(\pi_1(X^G)\) and \(\pi_1(X)\). A main ingredient for the proof is the author’s construction of a flat family of closures of \(G\)-orbits, birationally equivalent to \(X\) [see A. Fujiki, Invent. Math. 44, 225-258 (1978; Zbl 0367.32004)].
As a consequence of the above results the Chow variety of \(q\)-cycles of degree \(d\) in \(\mathbb{P}_n\) is connected and simply connected.

14L30 Group actions on varieties or schemes (quotients)
32M05 Complex Lie groups, group actions on complex spaces
32J18 Compact complex \(n\)-folds
14F35 Homotopy theory and fundamental groups in algebraic geometry