Lefschetz theorem on (1,1)-classes

The Lefschetz theorem on (1,1)-classes is a fundamental result in complex geometry, particularly regarding the relationship between cohomology classes and complex submanifolds on Kähler manifolds. It provides a criterion for determining when a cohomology class of type (1,1) can be represented by a complex submanifold, specifically a divisor.

More formally, let $X$ be a compact Kähler manifold. The theorem states that a cohomology class $\alpha \in H^2(X, \mathbb{Z})$ is the first Chern class $c_1(L)$ of a holomorphic line bundle $L$ on $X$ (and therefore represented by a divisor, the zero locus of a section of $L$) if and only if its de Rham representative lies in $H^{1,1}(X, \mathbb{C})$. In other words, the class $\alpha$ has a harmonic representative that is a (1,1)-form.

Here, $H^2(X, \mathbb{Z})$ denotes the second cohomology group of $X$ with integer coefficients, and $H^{1,1}(X, \mathbb{C})$ represents the (1,1)-part of the second cohomology group with complex coefficients, obtained from the Hodge decomposition of $H^2(X, \mathbb{C})$. The first Chern class, $c_1(L)$, is a topological invariant associated with a holomorphic line bundle $L$. It is an element of $H^2(X, \mathbb{Z})$ and represents the obstruction to finding a flat connection on $L$.

The theorem can also be stated using the Néron-Severi group, denoted $NS(X)$. The Néron-Severi group consists of the isomorphism classes of holomorphic line bundles on $X$ whose first Chern class is in $H^2(X, \mathbb{Z})$. Then, the theorem implies that $NS(X) \otimes \mathbb{C} \cong H^{1,1}(X, \mathbb{Z}) \subset H^{1,1}(X, \mathbb{C}) \cap H^2(X, \mathbb{Q})$, where $H^{1,1}(X, \mathbb{Z}) = H^{1,1}(X, \mathbb{C}) \cap H^2(X, \mathbb{Z})$.

In essence, the Lefschetz theorem on (1,1)-classes provides a bridge between the topology of a Kähler manifold (via cohomology) and its geometry (via holomorphic line bundles and divisors). It gives a precise condition for when a (1,1) cohomology class arises from a complex submanifold defined by the vanishing of a section of a holomorphic line bundle.

This theorem is crucial in understanding the algebraic structure of Kähler manifolds and is a cornerstone of Hodge theory.