If $X\subset \mathbb{P}^n$ is a smooth hypersurface (more generally a complete intersection) of dimension at least 2, and if $K_X+\mathscr{O}_X(-1)\geq 0$, why is it true that $H^0(T_X(1))=0$?
(Source: On a conjecture of Clemens on rational curves on hypersurfaces, paragraph before Proposition 1.1)
This has little to do with hypersurfaces. If $\dim(X)=d$, the only condition you need on $X$ is $H^0(X,\Omega ^{d-1}_X)=0$.
The wedge product gives a non-degenerate pairing $\Omega ^1_X\otimes \Omega ^{d-1}_X\rightarrow K_X$, hence an isomorphism $T_X\cong \Omega ^{d-1}_X\otimes K_X^{-1}$; therefore $T_X(1)\cong \Omega ^{d-1}_X\otimes (K_X(-1)) ^{-1}$. Thus if $K_X(-1)\geq 0$, $H^0(X,T_X(1))$ injects into $H^0(X,\Omega ^{d-1}_X)=0$.
