Is there a reasonably simple example of a smooth variety $X$ (over complex numbers) with the properties that $H^0(T_X)\neq 0$ and $H^1(T_X)\neq 0$?
1 Answer
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In dimension $1$, elliptic curves are examples. In dimension 2, every Hirzebruch surface except $\mathbb{P}^1\times \mathbb{P}^1$ and $\text{Bl}_{p}(\mathbb{P}^2)$ are examples.
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2$\begingroup$ In any dimension, abelian varieties $\endgroup$ Nov 9, 2013 at 8:29