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For each smooth, projective, complex variety $X$ that is simply connected, John Morgan constructed a natural mixed Hodge structure on the homotopy group $\pi_k(X,x)\otimes \mathbb{Q}$. This was extended to (possibly singular) quasi-projective varieties independently by Hain, by Deligne, and by Navarro Aznar. For any the basepoint connected component of the double loop space $\Omega^2 X_0\sim [(S^2,0),(X,x)]_0$, each homotopy group $\pi_k(\Omega^2 X_0)\otimes \mathbb{Q}$ is canonically isomorphic to $\pi_{k+2}(X)\otimes \mathbb{Q}$, $k>0$. Thus these also have natural mixed Hodge structures.

Grothendieck constructed a Hom scheme $$H=\text{Hom}_{\mathbb{C}}((\mathbb{CP}^1,0),(X,x))$$ parameterizing algebraic maps (i.e., holomorphic maps in case $X$ is projective). Each quasi-compact, open and closed subset $H_\beta$ of the Hom scheme is quasi-projective. Assuming that $H_\beta$ is connected, there is a well-defined homotopy class of maps $$\Phi_{\beta}:H_\beta \to \Omega^2 X_0,$$ that associates to each holomorphic map from $\mathbb{CP}^1 = S^2$ the underlying continuous map. There are associated pushforward maps on homotopy groups, $$\Phi_{k,\beta}: \pi_k(H_\beta)\otimes \mathbb{Q} \to \pi_{k+2}(X,x)\otimes \mathbb{Q}.$$ Assume now that $H_\beta$ is simply connected so that the groups $\pi_k(H_\beta)\otimes \mathbb{Q}$ have mixed Hodge structures. (There are many cases of $X$ where the associated spaces $H_\beta$ are connected and simply connected. Also, with no hypothesis on $H_\beta$, one can also ask about compatibility with mixed Hodge structures of the induced homomorphism $H_1(H_\beta;\mathbb{Q}) \to \pi_3(X,x)\otimes \mathbb{Q}.$) Also, as Dan Petersen points out, it is necessary to form the Tate twist of the target, i.e., $\pi_{k+2}(X,x)\otimes \mathbb{Q}(1)$.

Question. Is the pushforward map $\Phi_{k,\beta}$ a morphism of mixed Hodge structures?

Of course there are many complex varieties $X$ for which $H$ is empty, and then the answer is trivially yes. Yet there are many other complex projective manifolds, e.g., Fano manifolds, where the maps $\Phi_{k,\beta}$ are nontrivial.

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  • $\begingroup$ What does the subscript 0 mean in $\Omega^2 X_0\sim [(S^2,0),(X,x)]_0$? If it means "basepoint component", then why do you say "for any connected component"? If not, why is the isomorphism between $\pi_k(\Omega^2 X_0)\otimes \mathbb{Q}$ and $\pi_{k+2}(X)\otimes \mathbb{Q}$ canonical? $\endgroup$
    – Tom Church
    Commented Mar 18, 2016 at 15:30
  • $\begingroup$ @TomChurch. You are correct. I am eliding the issue that the function $\Phi_\beta$ is only well-defined up to homotopy. There is a canonical function $\Phi_\beta$ to the connected component of $\Omega^2 X$ corresponding to the homotopy class $\beta\in \pi_2(X)$. The homotopy equivalence of this component with the basepoint component is only well-defined up to homotopy. $\endgroup$
    – Jason Starr
    Commented Mar 18, 2016 at 15:40
  • $\begingroup$ Is it easily seen that there is a continuous $\Phi_\beta$? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Mar 18, 2016 at 16:48
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    $\begingroup$ "Is it easy to see?" Yes; this is part of adjointness of the space of continuous functions with the compact-open topology. Let $C$, $X$, and $H$ be topological spaces with $H$ Hausdorff. Let $F:H\times C \to X$ be continuous. The claim is that the induced function $\Phi:H\to [C,X]$ is continuous, where $[C,X]$ has the compact-open topology. For an open subset $U$ of $X$, for a compact subset $K$ of $C$, the subset $(H\times K)\cap \Phi^{-1}(X\setminus U)$ is a closed subset of $H\times K$, hence it is proper over $H$. So it has closed image in $H$, and the complement of the image is open. $\endgroup$
    – Jason Starr
    Commented Mar 18, 2016 at 23:29
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    $\begingroup$ I'd expect there to be a Tate twist at least. If $k=0$ you get a map from $H_0(H_\beta)=\mathbf Q(0)$ to $H_2(X)$, which has weight $-2$. $\endgroup$
    – Dan Petersen
    Commented Mar 19, 2016 at 6:10

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