Author: Eiko

Tags: algebraic geometry, lefschetz hyperplane theorem, algebraic topology, homology, cohomology, projective varieties

Time: 2024-10-03 06:27:25 - 2024-10-03 06:41:22 (UTC)

Lefschetz hyperplane theorem is a relation between the homological invariants of an algebraic variety and its subvarieties.

$X\subset \mathbb{C}{\mathbb{P}}^n$ be $d$ dimensional complex projective variety and $Y$ a hyperplanc section such that $U=X-Y$ is smooth. In short, the Lefschetz hyperplane theorem says that $Y$ contain enough homological information upto degree $d-1$, precisely speaking,

• $H_k(Y,\mathbb{Z})\to H_k(X,\mathbb{Z})$ is isomorphism for $k<d-1$ and surjective for $k=d-1$.

• Similar statement holds for ${\pi }_k$.

• $H^k(X,\mathbb{Z})\to H^k(Y,\mathbb{Z})$ is isomorphism for $k<d-1$ and injective for $k=d-1$.

• These are the same as saying the relative homology $H_k(X,Y;\mathbb{Z})$, cohomology $H^k(X,Y;\mathbb{Z})$ and homotopies ${\pi }_k(X,Y)$ vanish to degree $d-1$.