# An application of Lefschetz Hyperplane Theorem

Lefschetz Hyperplane Theorem says: Let $$X\subset\mathbb{C}P^n$$ be a smooth projective variety of complex dimension $$m$$ and let $$Y = X \cap H$$ be a generic hyperplane section. Then the natural morphism $$\pi_i(Y)\to \pi_i(X)$$ is an isomophism for $$i\le m-2$$ and is onto for $$i=m-1$$.

Now $$X\subset\mathbb{C}P^n$$ be a simply connected smooth projective variety of complex dimension $$m$$ and $$Y$$ is the intersection of $$X$$ with hyperplanes s.t. $$\dim Y=m-d$$.

Then why does Lefschetz Hyperplane Theorem show that $$Y$$ is connected and simply connected if $$\dim(Y) = \dim(X)-d = m-d\ge 2$$?

I think it used the fundamental group of $$X$$ is trivial, however, how is that related to the dimension of $$Y$$ in Lefschetz Hyperplane Theorem?

• Well, this certainly require an assumption on $X$. If $d=0$, then $Y=X$, so we need $X$ to be simply connected. – Roland Dec 6 '18 at 8:35
• @Roland Thanks, I have fixed – 6666 Dec 6 '18 at 8:41
• @Roland now how to apply Lefschetz Hyperplane Theorem to get the result? – 6666 Dec 6 '18 at 8:47
• Well then this is easy : let $X=Y_0\supset Y_1\supset...\supset Y_d=Y$ where $Y_i$ is $X$ intersected by $i$-hyperplane, so that $\dim Y_i=m-i$. Then $\pi_1(Y_d)\to \pi_1(Y_{d-1})\to...\to \pi_1(Y_0)$ are isomorphisms : indeed for $1\leq i\leq d$, we have $1\leq m-(i-1)-2$ by assumption. So $\pi_1(Y_i)\to\pi_1(Y_{i-1})$ is an isomorphism. – Roland Dec 6 '18 at 8:52
• @Roland can you explain why simply connected rules out the case of $d=0$? – 6666 Dec 6 '18 at 18:43