Lefschetzféle

Lefschetzféle is a concept in algebraic geometry, specifically relating to the study of algebraic varieties and their properties under deformation. It is named after the mathematician Solomon Lefschetz.

The core idea behind Lefschetzféle relates to how certain properties of an algebraic variety remain invariant or change in a controlled manner as the variety is perturbed or deformed. More precisely, it often concerns the behavior of a hyperplane section of a projective variety. If a projective variety is embedded in a higher-dimensional projective space, a generic hyperplane intersects it in a variety of one dimension lower. The Lefschetz theorems provide powerful tools for understanding the cohomology of the original variety by relating it to the cohomology of its hyperplane section.

A key result is the Lefschetz hyperplane theorem, which states that for a smooth projective variety $X$ of dimension $n \geq 2$ embedded in $\mathbb{P}^{n+1}$, the natural map on the singular cohomology groups induced by the inclusion of a generic hyperplane section $H$ is an isomorphism for the first $n-1$ cohomology groups. For the $n$-th cohomology group, it states that the map is injective, and its image is related to the fundamental class of $H$. This theorem is fundamental in bridging the topological and geometric aspects of algebraic varieties.

Variations and generalizations of the Lefschetz theorems exist, extending these ideas to other types of intersections and to different cohomology theories. These results are crucial for many areas of algebraic geometry, including the classification of varieties and the study of their moduli spaces.