Is it true that every quasi-projective rational irreducible algebraic complex variety is simply connected for the Euclidean topology?
Of course, this is false if we replace "complex" with "real" or if we forget "rational".
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asked Nov 8, 2015 at 22:11
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9$\begingroup$ Actually you did forget "rational". $\endgroup$– R.P.Nov 8, 2015 at 22:16
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12$\begingroup$ If I understand the question correctly, isn't $\mathbb{P}^1\backslash\{0,\infty\}$ a counterexample? $\endgroup$– Julian RosenNov 8, 2015 at 22:19
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9$\begingroup$ It's also false for singular rational varieties (e.g. for a nodal cubic). In the positive direction: a nonsingular rational projective variety is simply connected. $\endgroup$– Donu ArapuraNov 8, 2015 at 22:34
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3$\begingroup$ Donu's nice comment generalizes to rationally connected varieties, Prop. 6.14 in personal.psu.edu/jwh6013/Research/RationalConnectivity.pdf and Julian Rosen's neat example can be crossed with projective space to yield higher dimensional examples. such as a quadric surface minus two disjoint lines. This suggests many examples may exist by removing a disconnected divisor from a projective rational variety. $\endgroup$– roy smithNov 8, 2015 at 23:08
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4$\begingroup$ Pet peeve: "rational connectivity" should be "rational connectedness". For topological spaces, the nouns are "connectedness", "path connectedness", "arc connectedness", etc. $\endgroup$– Jason StarrNov 9, 2015 at 1:02
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1 Answer
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I want to mention the positive direction. Let $X$ be a smooth, projective variety over $\mathbb{C}$, resp. over an algebraically closed field of arbitrary characteristic. Let $Z\subset X$ be a proper closed subset. If $X$ is (separably) rationally connected and if $Z$ has codimension $\geq 2$ in $X$, then $X\setminus Z$ is simply connected for the Euclidean topology, resp. the algebraic fundamental group is trivial. This was first proved by Campana over $\mathbb{C}$: there is an excellent explanation in Debarre's book, "Higher dimensional algebraic geometry". There is another proof by Kollár that extends to positive characteristic: there is an excellent explanation in one of Debarre's Bourbaki seminars.
One might hope to drop the hypothesis that $Z$ has codimension $2$ if we assume that every general pair of points of $X\setminus Z$ is connected by a rational curve completely contained in $X\setminus Z$. Unfortunately there are many counterexamples, such as the smooth locus of the singular cubic surface with equation $xyz-w^3=0$. What is true is that the fundamental group is finite. However, it is quite open to understand the fundamental group of the smooth locus of $\mathbb{Q}$-log-Fano varieties, cf. work of Chenyang Xu, Zhiyu Tian, etc.