the dimension problem of complex projection

Question

Is it true that $\operatorname{dim }H^{0}(P^{n},P(T^{*}P^{n}))>0$? That is, is there a global holomorphic section? Here $P^{n}$ is $n$-dimensional complex projection space and $P(T^{*}P^{n})=T^{*}P^{n}/\mathcal{C}^{*}$.

Answer 1

since $$H^{q}(\mathbb{P}^{N},\Omega^{p})=0 \quad if\quad p\neq q$$ where $\Omega^{p}$ is holomorphic $p$-form.thus $$H^{0}(\mathbb{P}^{N},T^{*}\mathbb{P}^{N})=H^{0}(\mathbb{P}^{N},\Omega^{1})=0$$ then $H^{0}(\mathbb{P}^{N},P(T^{*}\mathbb{P}^{N}))=0$