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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.

##### MSC:
 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