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Let $V$ be a nonempty, irreducible, smooth projective variety over $\mathbf{C}$.

Is there a smooth projective variety $X$ over $\mathbf{C}$, a surjective map $X\to V$ of varieties over $\mathbf{C}$, such that $X$ contains as a dense open $\mathbf{C}$-subscheme some $\mathbf{C}$-group scheme $G$ of finite type?

Abelian varieties and toric varieties are of course examples with $X=V$.

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  • $\begingroup$ When you say "contains as dense open subscheme some group scheme", do you mean, "contains a dense open subscheme that admits a structure of group scheme"? $\endgroup$
    – YCor
    Commented Mar 17, 2022 at 23:02
  • $\begingroup$ (Certainly $X$ should be assumed irreducible, otherwise we obtain an uninteresting trivial yes as soon as $V$ is nonempty. Also $V$ should in any case be assumed to be nonempty.) $\endgroup$
    – YCor
    Commented Mar 17, 2022 at 23:03
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    $\begingroup$ No. Any smooth projective curve of genus $> 1$ provides a counterexample. $\endgroup$
    – naf
    Commented Mar 18, 2022 at 1:40
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    $\begingroup$ @JohnPilud Let $C$ be a smooth projective curve of genus $>1$ over $\mathbb{C}$. Suppose that $C$ has a dense open $U$ which can be endowed with the structure of a group scheme. A connected finite type group scheme over $\mathbb{C}$ is isomorphic to $\mathbb{G}_m$, $\mathbb{G}_a$ or an elliptic curve. So $U$ is isomorphic to a dense open of a curve of genus $<2$. $\endgroup$
    – Ariyan Javanpeykar
    Commented Mar 18, 2022 at 10:29
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    $\begingroup$ @AriyanJavanpeykar Thank you so much! Now all is clear. $\endgroup$
    – user335418
    Commented Mar 18, 2022 at 22:34

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Smooth projective curves of genus $>1$ are counterexamples. To see this, you can use the following lemma.

Lemma. Let $X$ be a proper integral variety over $\mathbb{C}$. Then the following are equivalent.

  1. For every abelian variety $A$, every morphism $A\to X$ is constant.

  2. For every finite type connected group scheme $G$ over $\mathbb{C}$, every morphism $G\to X$ is constant.

A proof of this lemma is given in Lemma 2.5 of arxiv.org/abs/1807.03665

To show that smooth projective curves of genus $g>1$ give counterexamples, we can argue as follows:

Let $X$ be a smooth projective curve of genus $g>1$. Then, every morphism from an abelian variety $A$ to $X$ is constant. This can be seen by using the uniformisation of $A$ by affine $\dim(A)$-space or by pulling-back differentials. Now, by the above Lemma, every morphism $G\to X$ is constant, where $G$ is any finite type connected group scheme over $\mathbb{C}$. But this implies that every morphism $Y\to X$ is constant, where $Y$ is a variety containing a dense open isomorphic to the variety underlying a finite type connected group scheme.

Note: Any hyperbolic variety gives a counterexample. For example, the moduli space of genus $q$ ($q>1$) smooth proper curves with level $N$ ($N>3$) structure.

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