10.5 Hodge theory

Hodge theory connects differential forms, cohomology, and harmonic analysis in complex geometry. It provides a framework for understanding the interplay between analytic and algebraic properties of complex manifolds, using harmonic forms as fundamental objects.

The Hodge decomposition theorem is central, stating that on compact Kähler manifolds, differential forms can be decomposed into harmonic, exact, and co-exact forms. This decomposition links de Rham cohomology to harmonic forms, revealing deep connections between topology and analysis.

Hodge theory fundamentals

• Hodge theory is a central tool in the study of complex geometry and topology that connects differential forms, cohomology, and harmonic analysis
• It provides a powerful framework for understanding the interplay between the analytic and algebraic properties of complex manifolds
• The fundamental objects in Hodge theory are harmonic forms, which are differential forms that satisfy certain differential equations

Harmonic forms and Laplacian

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• Harmonic forms are differential forms that are both closed ($d\omega = 0$) and co-closed ($d^*\omega = 0$), where $d$ is the exterior derivative and $d^*$ is its adjoint
• The Laplacian operator $\Delta = dd^* + d^*d$ plays a crucial role in defining harmonic forms
  • Forms in the kernel of the Laplacian ($\Delta \omega = 0$) are precisely the harmonic forms
• The space of harmonic $k$-forms on a compact Riemannian manifold $M$ is isomorphic to the $k$-th de Rham cohomology group $H^k(M, \mathbb{R})$
• Example: On a compact Riemann surface, harmonic 1-forms correspond to holomorphic differentials

Hodge decomposition theorem

• The Hodge decomposition theorem states that on a compact Kähler manifold, the space of differential $k$-forms $\Omega^k(M)$ can be decomposed into a direct sum of harmonic forms, exact forms, and co-exact forms
  • $\Omega^k(M) = \mathcal{H}^k(M) \oplus d\Omega^{k-1}(M) \oplus d^*\Omega^{k+1}(M)$
• This decomposition is orthogonal with respect to the $L^2$ inner product on differential forms
• The Hodge decomposition induces a canonical isomorphism between the de Rham cohomology and the space of harmonic forms
  • $H^k(M, \mathbb{C}) \cong \mathcal{H}^k(M)$
• Example: On a compact Kähler manifold, the Hodge decomposition of $\Omega^1(M)$ gives rise to the decomposition of the first cohomology group into holomorphic and antiholomorphic parts

Hodge star operator

• The Hodge star operator $*$ is a linear map $* : \Omega^k(M) \to \Omega^{n-k}(M)$ that depends on the Riemannian metric and orientation of the manifold
• It satisfies $**\omega = (-1)^{k(n-k)}\omega$ for $k$-forms on an $n$-dimensional manifold
• The Hodge star operator relates the exterior derivative $d$ and its adjoint $d^*$ via the formula $d^* = (-1)^{nk+n+1}*d*$
• The Hodge star operator is used to define the $L^2$ inner product on differential forms
  • $\langle \omega, \eta \rangle = \int_M \omega \wedge *\eta$
• Example: On a Riemannian surface, the Hodge star operator maps 1-forms to 1-forms, and its square is the negative identity

Cohomology groups and Hodge theory

• Hodge theory provides a powerful tool for studying the cohomology groups of a compact Kähler manifold
• The Hodge decomposition theorem implies that the $k$-th de Rham cohomology group $H^k(M, \mathbb{C})$ has a natural decomposition into a direct sum of complex subspaces
  • $H^k(M, \mathbb{C}) = \bigoplus_{p+q=k} H^{p,q}(M)$
• The spaces $H^{p,q}(M)$ are called the Hodge cohomology groups and consist of cohomology classes represented by harmonic $(p,q)$-forms
• The Hodge numbers $h^{p,q} = \dim H^{p,q}(M)$ are important invariants of the complex manifold $M$
• Example: For a compact Riemann surface of genus $g$, the Hodge numbers are $h^{1,0} = h^{0,1} = g$, and all other Hodge numbers are zero

Complex manifolds

• Complex manifolds are a central object of study in complex geometry and provide a natural setting for Hodge theory
• A complex manifold is a manifold equipped with an atlas of charts whose transition functions are holomorphic
• The complex structure on a manifold allows for the study of holomorphic and antiholomorphic objects, such as differential forms and vector bundles

Kähler manifolds and metrics

• A Kähler manifold is a complex manifold equipped with a Hermitian metric whose associated $(1,1)$-form (the Kähler form) is closed
• The Kähler condition imposes strong restrictions on the geometry and topology of the manifold
  • For example, the odd Betti numbers of a compact Kähler manifold are even
• Kähler metrics are a natural generalization of the flat metric on $\mathbb{C}^n$ and provide a rich class of examples for studying Hodge theory
• Example: Complex projective space $\mathbb{CP}^n$ with the Fubini-Study metric is a compact Kähler manifold

Dolbeault cohomology

• Dolbeault cohomology is a refinement of de Rham cohomology for complex manifolds that takes into account the complex structure
• The Dolbeault complex consists of $(p,q)$-forms with the $\bar{\partial}$ operator, which is the $(0,1)$-part of the exterior derivative $d$
  • $\bar{\partial} : \Omega^{p,q}(M) \to \Omega^{p,q+1}(M)$
• The $k$-th Dolbeault cohomology group $H^{p,q}_{\bar{\partial}}(M)$ is defined as the quotient of $\ker \bar{\partial} : \Omega^{p,q}(M) \to \Omega^{p,q+1}(M)$ by $\operatorname{im} \bar{\partial} : \Omega^{p,q-1}(M) \to \Omega^{p,q}(M)$
• On a compact Kähler manifold, the Dolbeault cohomology groups are isomorphic to the Hodge cohomology groups
  • $H^{p,q}_{\bar{\partial}}(M) \cong H^{p,q}(M)$
• Example: On a complex torus, the Dolbeault cohomology groups can be computed using harmonic $(p,q)$-forms with respect to the flat metric

Holomorphic vs antiholomorphic forms

• On a complex manifold, differential forms can be decomposed into $(p,q)$-forms, which are forms with $p$ holomorphic and $q$ antiholomorphic indices
• Holomorphic forms are $(p,0)$-forms that are $\bar{\partial}$-closed, i.e., they satisfy $\bar{\partial}\omega = 0$
  • Holomorphic forms are a key object of study in complex geometry and are closely related to the complex structure of the manifold
• Antiholomorphic forms are $(0,q)$-forms that are $\partial$-closed, i.e., they satisfy $\partial\omega = 0$, where $\partial$ is the $(1,0)$-part of the exterior derivative
• The spaces of holomorphic and antiholomorphic forms on a compact Kähler manifold are finite-dimensional and related by the Hodge star operator
  • $*: \Omega^{p,0}(M) \to \Omega^{0,n-p}(M)$
• Example: On a complex curve (Riemann surface), holomorphic 1-forms are the same as holomorphic differentials, which play a crucial role in the theory of algebraic curves

Hodge numbers and diamond

• The Hodge numbers $h^{p,q} = \dim H^{p,q}(M)$ are important invariants of a compact Kähler manifold $M$
• They satisfy symmetries that reflect the underlying structure of the manifold
  • $h^{p,q} = h^{q,p}$ (complex conjugation)
  • $h^{p,q} = h^{n-p,n-q}$ (Serre duality)
• The Hodge diamond is a visual representation of the Hodge numbers, arranging them in a diamond shape according to their bidegree $(p,q)$
• The Hodge diamond encodes important topological information about the manifold, such as its Betti numbers and Euler characteristic
  • $b_k = \sum_{p+q=k} h^{p,q}$ (Betti numbers)
  • $\chi(M) = \sum_{p,q} (-1)^{p+q} h^{p,q}$ (Euler characteristic)
• Example: The Hodge diamond of the complex projective plane $\mathbb{CP}^2$ has $h^{0,0} = h^{2,2} = 1$, $h^{1,1} = 1$, and all other entries zero

Hodge structures

• Hodge structures are a powerful tool for studying the cohomology of complex algebraic varieties and their variations in families
• A Hodge structure is a vector space equipped with a decomposition into a direct sum of complex subspaces satisfying certain compatibility conditions
• Hodge structures arise naturally from the Hodge decomposition of the cohomology of a compact Kähler manifold

Pure Hodge structures

• A pure Hodge structure of weight $k$ is a finite-dimensional vector space $H$ over $\mathbb{Q}$ equipped with a decomposition of its complexification $H_{\mathbb{C}} = \bigoplus_{p+q=k} H^{p,q}$ satisfying $\overline{H^{p,q}} = H^{q,p}$
• The numbers $h^{p,q} = \dim H^{p,q}$ are called the Hodge numbers of the pure Hodge structure
• Pure Hodge structures form a category, with morphisms being linear maps that preserve the Hodge decomposition
• Example: The cohomology groups $H^k(X, \mathbb{Q})$ of a compact Kähler manifold $X$ carry a pure Hodge structure of weight $k$ induced by the Hodge decomposition

Hodge filtration

• The Hodge filtration is a decreasing filtration $F^{\bullet}$ on the complexification $H_{\mathbb{C}}$ of a pure Hodge structure $H$ of weight $k$, defined by $F^p H_{\mathbb{C}} = \bigoplus_{r \geq p} H^{r,k-r}$
• The Hodge filtration determines the Hodge decomposition, as $H^{p,q} = F^p H_{\mathbb{C}} \cap \overline{F^q H_{\mathbb{C}}}$
• The Hodge filtration is a key tool in the study of variations of Hodge structures and period mappings
• Example: For the pure Hodge structure on the cohomology of a compact Kähler manifold, the Hodge filtration is given by the subspaces of cohomology classes represented by forms of type $(r,k-r)$ with $r \geq p$

Polarized Hodge structures

• A polarized Hodge structure is a pure Hodge structure $H$ of weight $k$ equipped with a bilinear form $Q: H \otimes H \to \mathbb{Q}(-k)$ satisfying certain compatibility conditions with the Hodge decomposition
  • $Q(F^p H_{\mathbb{C}}, F^{k-p+1} H_{\mathbb{C}}) = 0$ (Hodge-Riemann bilinear relations)
  • $Q(C\bar{v},\bar{w}) = (-1)^k Q(v,C\bar{w})$ for the Weil operator $C$ (polarization condition)
• Polarized Hodge structures are a refinement of pure Hodge structures that capture additional geometric information
• The primitive cohomology of a compact Kähler manifold carries a natural polarized Hodge structure
• Example: The intersection form on the middle cohomology of a compact Kähler surface defines a polarization of the Hodge structure

Variations of Hodge structures

• A variation of Hodge structures (VHS) is a family of Hodge structures parametrized by a complex manifold, satisfying certain differential equations (Griffiths transversality)
• VHS arise naturally in the study of families of complex algebraic varieties, where the cohomology of the fibers varies in a controlled way
• The period mapping associated to a VHS encodes the variation of the Hodge structure and is a key tool in the study of moduli spaces of algebraic varieties
• Example: The family of intermediate Jacobians associated to a family of smooth projective curves defines a VHS on the first cohomology of the curves

Applications of Hodge theory

• Hodge theory has numerous applications in various areas of mathematics, including algebraic geometry, complex geometry, and representation theory
• The powerful tools and techniques developed in Hodge theory have led to significant advances in understanding the topology and geometry of complex algebraic varieties

Algebraic cycles and Hodge conjecture

• The Hodge conjecture is one of the most important open problems in algebraic geometry, relating the geometry of complex algebraic varieties to their topology
• It states that for a projective complex algebraic variety $X$, every class in the $2k$-th cohomology group $H^{2k}(X, \mathbb{Q})$ that is of type $(k,k)$ (i.e., in $H^{k,k}(X) \cap H^{2k}(X, \mathbb{Q})$) is a rational linear combination of classes of algebraic cycles of codimension $k$
• The Hodge conjecture is known to hold for $k=1$ (the Lefschetz $(1,1)$-theorem) and for certain classes of varieties (e.g., abelian varieties, hypersurfaces), but remains open in general
• Example: For a smooth projective surface, the Hodge conjecture predicts that every class in $H^{1,1}(X) \cap H^2(X, \mathbb{Q})$ is a rational linear combination of classes of algebraic curves on the surface

Moduli spaces and period maps

• Moduli spaces are spaces that parametrize isomorphism classes of geometric objects, such as algebraic varieties or vector bundles
• Hodge theory provides a powerful tool for studying the geometry and topology of moduli spaces through the period mappings associated to variations of Hodge structures
• The period mapping sends a point in the moduli space to the Hodge structure on the cohomology of the corresponding geometric object
• The study of period mappings has led to significant results in the theory of moduli spaces, such as the Torelli theorem for K3 surfaces and the Griffiths transversality theorem
• Example: The moduli space of principally polarized abelian varieties (ppav) of dimension $g$ can be studied using the period mapping, which sends a ppav to its polarized Hodge structure on the first cohomology group

Hodge theory in complex geometry

• Hodge theory is a fundamental tool in the study of complex geometry, providing a bridge between the analytic and algebraic properties of complex manifolds
• The Hodge decomposition and the $\partial\bar{\partial}$-lemma are key ingredients in the proof of the Kodaira embedding theorem, which characterizes projective complex manifolds as those admitting a positive line bundle
• Hodge theory also plays a crucial role in the study of Kähler-Einstein metrics and the Calabi conjecture, which relates the existence of such metrics to the stability of the underlying manifold
• Example: The Hodge decomposition of the second cohomology group of a compact Kähler surface determines its Kodaira dimension, which measures the complexity of the surface from the perspective of birational geometry

Hodge theory and representation theory

• Hodge theory has important connections to representation theory, particularly in the study of variations of Hodge structures and their monodromy representations
• The monodromy representation associated to a variation of Hodge