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)



Sections of tangent bundle on hypersurface
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 $H^0(X,\Omega ^{d-1}\_X)=0$.


