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Let $X$ be a smooth projective connected variety over $\mathbb{C}$ and let $T_X$ be its tangent bundle.

If $T_X$ is ample, then $X$ is isomorphic to a projective space by Mori's theorem.

If $T_X$ is trivial, then $X$ is isomorphic to an abelian variety.

If $T_X$ is nef, then the Albanese map $X\to A_X$ is smooth, and the fibres are Fano varieties with nef tangent bundle by work of Demailly-Peternell-Schneider. Conjecturally, these Fano fibres are even rational homogeneous spaces.

I wonder whether there are other questions one might ask.

For example, what if $T_X$ is big? What do we expect then?

What if $T_X$ is $m$-ample for some integer $m\geq 1$?

What about $\Lambda^pT_X$ being trivial (resp. ample) for some $p=1,\ldots, \dim X$. (In the ample case, this interpolates between between projective space where $p=1$ and Fano varieties where $p=\dim X$.)

Are there any other properties of $T_X$ which force (even just conjecturally) properties of $X$?

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    $\begingroup$ Are all of these properties defined as the line bundle $\mathcal O_{\mathbb P T_X}(1)$ having that property, as with ampleness? $\endgroup$
    – Tabes Bridges
    Commented Sep 16, 2020 at 11:23
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    $\begingroup$ @TabesBridges Yes, although there are indeed other ways of defining bigness, due to Viehweg I believe. $\endgroup$
    – Pat
    Commented Sep 16, 2020 at 11:29
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    $\begingroup$ You may add a weak version of ampleness like m-ample, take m=1, see google.com/url?sa=t&source=web&rct=j&url=http://… $\endgroup$
    – user160903
    Commented Sep 16, 2020 at 14:05
  • $\begingroup$ But when $K_X$ is1-ample then finding a type of canonical metrics like Kähler-Einstein metric is related to new conditions. $\endgroup$
    – user160903
    Commented Sep 16, 2020 at 20:54
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    $\begingroup$ On bigness: recent work of arxiv.org/abs/2003.09476 has completely settled the bigness of $T_X$ for del Pezzo surfaces (also considered in arxiv.org/abs/2002.11010, if it's not poor form to refer to one's own papers)- basically, hard to know what bigness of $T_X$ means, as del Pezzos of degrees 5 or more have big tangent bundle and those of degrees 4 or less do not- I don't know why this should be true more systematically, but maybe someone does! $\endgroup$
    – Devlin Mallory
    Commented Sep 17, 2020 at 4:34

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