# Smooth projective variety with no second homotopy group

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.

• 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? Commented Mar 26, 2021 at 16:26
• The whole cohomology, thus there is no odd homology indeed. Commented Mar 26, 2021 at 17:02

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.
• Oh, good point. I forgot about generating all $H_*(X,\mathbb{Q})$ when I threw the bielliptics in there; removed. Commented Mar 26, 2021 at 17:48
• 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. Commented Mar 26, 2021 at 17:54
• 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. Commented Mar 26, 2021 at 17:58