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 $X=V$.
Best Answer
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.
For every abelian variety $A$, every morphism $A\to X$ is constant.
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:
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.