11.2 Kähler manifolds and Hodge theory

Kähler manifolds are complex manifolds with a special metric that connects complex and symplectic geometry. They're key to understanding the interplay between algebraic, complex, and differential geometry in modern mathematics.

Hodge theory on Kähler manifolds reveals deep connections between topology and complex structure. It provides tools like the Hodge decomposition and hard Lefschetz theorem, which are crucial for studying cohomology and geometric properties of these spaces.

Kähler Manifolds

Definition and Basic Properties

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• A Kähler manifold is a complex manifold equipped with a Hermitian metric whose imaginary part is a closed (1,1)-form, known as the Kähler form
• The Kähler condition requires that the complex structure is parallel with respect to the Levi-Civita connection
• Kähler manifolds are symplectic manifolds, with the Kähler form serving as the symplectic form
  • This means that the Kähler form is non-degenerate and closed, providing a natural symplectic structure on the manifold
• Every Kähler manifold is orientable and has even real dimension
  • The complex structure and the Kähler form induce a natural orientation on the manifold
  • The real dimension of a Kähler manifold is always twice its complex dimension

Examples of Kähler Manifolds

• Complex projective spaces (CPn) are Kähler manifolds
  • The Fubini-Study metric on CPn is a Kähler metric, making it a fundamental example of a Kähler manifold
• Complex tori, which are quotients of Cn by a lattice, are Kähler manifolds
  • The flat metric on Cn descends to a Kähler metric on the complex torus
• Smooth projective varieties, which are subvarieties of CPn defined by homogeneous polynomials, are Kähler manifolds
  • The restriction of the Fubini-Study metric to a smooth projective variety is a Kähler metric
• Other examples include Hermitian symmetric spaces and Calabi-Yau manifolds

Kähler Manifolds and Hodge Theory

Hodge Decomposition on Kähler Manifolds

• On a Kähler manifold, the de Rham cohomology admits a decomposition into bidegrees, known as the Hodge decomposition
  • The Hodge decomposition is of the form $H^k(X, C) = \oplus_{p+q=k} H^{p,q}(X)$, where $H^{p,q}(X)$ are the Hodge components
• The Hodge decomposition is compatible with the cup product, making the cohomology ring of a Kähler manifold a bigraded algebra
  • The cup product of elements in $H^{p,q}(X)$ and $H^{r,s}(X)$ lies in $H^{p+r,q+s}(X)$
• The Dolbeault cohomology groups of a Kähler manifold can be identified with certain Hodge components of the de Rham cohomology
  • Specifically, $H^{p,q}_{\bar\partial}(X) \cong H^{p,q}(X)$, where $H^{p,q}_{\bar\partial}(X)$ are the Dolbeault cohomology groups

Hodge Theory as a Tool in Kähler Geometry

• Hodge theory on Kähler manifolds provides a powerful tool for studying their topology and complex geometry
  • The Hodge decomposition allows for a more refined analysis of the cohomology of Kähler manifolds
  • The compatibility of the Hodge decomposition with the cup product reveals additional structure in the cohomology ring
• Hodge theory can be used to derive various results in Kähler geometry, such as the Kodaira vanishing theorem and the Lefschetz hyperplane theorem
  • These theorems relate the cohomology of a Kähler manifold to its complex submanifolds and provide vanishing results for certain cohomology groups

Hodge Theory for Kähler Manifolds

Hodge Numbers and Their Properties

• The Hodge numbers $h^{p,q}$ of a Kähler manifold are the dimensions of the Hodge components $H^{p,q}$ in the Hodge decomposition
  • The Hodge numbers provide a refined description of the cohomology of the manifold
• The Hodge numbers satisfy symmetries, such as $h^{p,q} = h^{q,p}$ (Hodge symmetry) and $h^{p,q} = h^{n-p,n-q}$ (Serre duality)
  • Hodge symmetry reflects the reality of the Kähler metric, while Serre duality relates the Hodge numbers to the canonical bundle
• The Betti numbers of a Kähler manifold can be expressed in terms of its Hodge numbers
  • The kth Betti number is given by $b_k = \sum_{p+q=k} h^{p,q}$, showing how the Hodge numbers refine the topological information

Hodge Diamond and Vanishing Theorems

• The Hodge diamond is a visual representation of the Hodge numbers, displaying their symmetries and relations
  • The Hodge diamond arranges the Hodge numbers in a diamond shape, with $h^{p,q}$ placed at the position $(p,q)$
  • The symmetries of the Hodge numbers are reflected in the symmetries of the Hodge diamond
• Hodge theory can be used to prove vanishing theorems for the cohomology of certain types of Kähler manifolds, such as projective varieties
  • The Kodaira vanishing theorem states that for an ample line bundle L on a projective variety X, $H^q(X, K_X \otimes L) = 0$ for $q > 0$, where $K_X$ is the canonical bundle
  • Other vanishing theorems, such as the Nakano vanishing theorem and the Kawamata-Viehweg vanishing theorem, provide similar results for more general Kähler manifolds

Hard Lefschetz Theorem for Kähler Manifolds

Statement and Proof of the Hard Lefschetz Theorem

• The hard Lefschetz theorem states that the map $L^k: H^{n-k}(X) \to H^{n+k}(X)$ given by the cup product with the kth power of the Kähler form is an isomorphism for $0 \leq k \leq n$ on a compact Kähler manifold X of complex dimension n
  • This theorem reveals a deep connection between the cohomology of a Kähler manifold and its Kähler form
• The proof of the hard Lefschetz theorem relies on the Kähler identities, which relate the Lefschetz operator L, its adjoint Λ, and the Dolbeault operators ∂ and ∂*
  • The Kähler identities, such as $[L, ∂^*] = i∂$ and $[L, ∂] = 0$, are crucial in establishing the isomorphism

Consequences of the Hard Lefschetz Theorem

• The hard Lefschetz theorem implies the Lefschetz decomposition, which expresses the cohomology groups of a Kähler manifold as a direct sum of primitive cohomology groups
  • The primitive cohomology groups $P^k \subset H^k(X)$ are the kernels of the Lefschetz operator $L^{n-k+1}$ on $H^k(X)$
  • The Lefschetz decomposition is given by $H^k(X) = \oplus_{i \geq 0} L^i P^{k-2i}$
• The primitive cohomology groups are the building blocks of the cohomology of a Kähler manifold
  • The hard Lefschetz theorem shows that the primitive cohomology groups determine the entire cohomology of the manifold
• The Hodge-Riemann bilinear relations, which are a consequence of the hard Lefschetz theorem, provide a positive definite Hermitian form on the primitive cohomology groups
  • The Hodge-Riemann bilinear relations state that $(-1)^{(k-n)/2} Q_k(α, \bar α) > 0$ for non-zero $α \in P^k$, where $Q_k$ is a certain Hermitian form on $H^k(X)$
  • These relations are essential in the study of the positivity properties of Kähler manifolds The hard Lefschetz theorem and its consequences are fundamental results in Kähler geometry, revealing deep connections between the cohomology, complex structure, and Kähler metric of a Kähler manifold. They provide powerful tools for studying the topology and algebraic geometry of these manifolds.