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



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



Compactifications of group varieties
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:



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> > 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.
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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.


