This is the exercise III 7.4 of Hartshorne. If you don't have this text in your hand, see this post.
I'm trying (b): If $X = \mathbb{P}^n$, identifying $H^p(X, \Omega^p)$ with $H^0(X, \Omega^0) = k$, show that $\eta(Y) = \deg Y$, the degree of the subvariety $Y$.
I have shown this in the case that $Y$ is a point (this is exactly (a), I think.) and the case that $Y$ is the intersection of $n-p$ hyperplanes in $\mathbb{P}^n$. ($p$ is the codimension of $Y$ in $\mathbb{P}^n$.) And I have shown that by Bertini and Bezout there exists $n-p$ hyperplanes $H_1, \cdots, H_{n-p}$ such that the number of points of $Y \cap H_1 \cap \cdots \cap H_{n-p}$ is the degree of $Y$. I think these results are sufficient to show this exercise, but I can't show.
The hint says that cut with a hyperplane, and reduce to the case that $Y$ is a set of finite points.
Thank you very much!
