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Let $H$ be a smooth projective hypersurface in $\mathbb{P}^n(\mathbb{C})$ where $n\geq 3$. Then by the Lefschetz hyperplane theorem we have that $H^1(H,\mathbb{C})= H^1(\mathbb{P}^n(\mathbb{C}),\mathbb{C})=0$. It thus follows that $\pi_1(H)^{ab}$ is a finite abelian group.

Q.1 Do we have an example of an $H$ such that $\pi_1(H)$ is infinite?

Q.2 Is it possible to compute explicitly $\pi_1(H)$ (or more modestly $\pi_1(H)^{ab})$ in term of the defining equation of $H$ ?

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2 Answers 2

up vote 15 down vote accepted

Non-singular projective hypersurfaces are simply connected. By the Lefschetz theorem $\pi_k(X)\to\pi_k(\mathbf{P}^n(\mathbf{C}))$ is an isomorphism for $k\leq n-2$ where $X$ is a nonsingular complex hypersurface: as shown e.g. in Griffiths-Harris (chapter 1, second proof of the Lefschetz hyperplane theorem) if $M$ is a smooth complex manifold and $V$ is the zero locus of a section of a positive line bundle, then (assuming $V$ smooth) there is a smooth function on $M$ with $V$ as the zero locus and all critical points outside $V$ non-degenerate and of index $\geq \dim_{\mathbf{C}}M$; so $M$ is homotopy equivalent to $V$ with cells of dimension $\geq \dim_{\mathbf{C}}M$ glued to it.

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Thanks algori, this is a key result to know. –  Hugo Chapdelaine Jan 6 '11 at 15:01

Let me supplement Algori's answer a bit. The statement for fundamental groups goes back to Zariski, I believe. Standard Morse theory proofs yield this and more ( Milnor's book gives a nice account). Over any field, there are similar results using the etale fundamental group. See for example SGA2, exp XII, cor. 3.5.

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