I am looking for an example (if such exist) of a smooth projective variety $X$ whose $\mathbb{Q}$-homology $H_*(X,\mathbb{Q})$ is generated by algebraic cycles, and yet does not have a second homotopy group, $\pi_2(X)=0.$ Thus, algebraic cycles that span $H_2(X,\mathbb{Q})$ are coming from some non-rational curves.
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$\begingroup$ Do you only require that the even-dimensional homology is generated by cycles or do you mean that there is no odd-dimensional rational homology? $\endgroup$– Dustin CartwrightCommented Mar 26, 2021 at 16:26
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$\begingroup$ The whole cohomology, thus there is no odd homology indeed. $\endgroup$– FilipCommented Mar 26, 2021 at 17:02
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1 Answer
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Fake projective planes have $H_2(X,\mathbb{Z}) \cong \mathbb{Z}$. They have metrics of pinched negative curvature, so they have $\pi_2(X) \cong \{0\}$. Thus $H_2(X,\mathbb{Q}) \cong \mathbb{Q}$ is generated by a hyperplane section, and this is not a rational curve.
A Picard maximal fake quadric (a surface of general type with the same rational cohomology as $\mathbb{P}^1 \times \mathbb{P}^1$) with universal cover the product of two hyperbolic planes will also have the desired property with $H_2(X,\mathbb{Q}) \cong \mathbb{Q}^2$. They have metrics of nonpositive curvature, so again $\pi_2$ is trivial. For example, there are "product quotient" examples; see The classification of surfaces with pg=q=0 isogenous to a product of curves, Pure Appl. Math. Q. 4 (2008), no. 2, Special Issue: In honor of Fedor Bogomolov. Part 1, 547–586 by Bauer, Catanese, and Grunewald.
You should also look into other surfaces with $p_g = q = 0$. You certainly want $p_g = 0$, and so $q = 0$ if the surface is minimal of general type (you certainly want minimal for $\pi_2(X)$ to be $\{0\}$). For example, see Bauer, Catanese, and Pignatelli's Surfaces of general type with geometric genus zero: a survey, Complex and differential geometry, 1-48, Springer Proc. Math., 8, Springer, Heidelberg, 2011.
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1$\begingroup$ Many thanks! Strictly speaking, later examples that you mention can have odd-degree homology so are non-examples for my question, but fake projective planes and fake quadrics are awesome. $\endgroup$– FilipCommented Mar 26, 2021 at 17:42
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$\begingroup$ Oh, good point. I forgot about generating all $H_*(X,\mathbb{Q})$ when I threw the bielliptics in there; removed. $\endgroup$– ToffeeCommented Mar 26, 2021 at 17:48
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1$\begingroup$ The minimal smooth projective surfaces with $p_g = q = 0$ are still good though. The Hodge diamond gives you that only $H^{1,1}(X)$ is nonzero away from degree $0$ and $4$, so then your question is reduced to Picard maximality and triviality of $\pi_2$, so there are probably more examples out there. $\endgroup$– ToffeeCommented Mar 26, 2021 at 17:54
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2$\begingroup$ Oh, and for a higher-dimensional analogue, Prasad and Yeung built fake $\mathbb{P}^4$s in Arithmetic fake projective spaces and arithmetic fake Grassmannians, Amer. J. Math. 131 (2009), no. 2, 379–407. These again have metrics of negative curvature, so all higher homotopy groups vanish. $\endgroup$– ToffeeCommented Mar 26, 2021 at 17:58