Complex and Kähler geometry; Hodge theory

前置知识

62 项必需的前置知识（一直追溯到底）

你将学会

• Define complex manifolds, Kähler metrics and the Dolbeault complex, with projective varieties as the main examples.
• State the Hodge decomposition and Hodge symmetry and deduce topological restrictions on compact Kähler manifolds.
• State the Calabi conjecture (Yau's theorem) and the Hodge conjecture, clearly separating theorems from open problems.

定义

• A Kähler manifold is a complex manifold with a Hermitian metric whose associated 2-form is closed.
• The Hodge numbers h^{p,q} of a compact Kähler manifold are the dimensions of the spaces of harmonic (p, q)-forms.
• A Calabi–Yau manifold is a compact Kähler manifold with trivial canonical bundle (conventions on simple connectivity vary).

定理

• Hodge decomposition: Hᵏ(X; ℂ) ≅ ⊕_{p+q=k} H^{p,q}(X) with H^{q,p} = conj(H^{p,q}) for compact Kähler X.
• The odd Betti numbers of a compact Kähler manifold are even.
• Yau (1977): a compact Kähler manifold with vanishing first Chern class admits a Ricci-flat Kähler metric in each Kähler class.
• The Hodge conjecture — that rational Hodge classes on smooth projective varieties are algebraic — is open.

常见误解

• Thinking every complex manifold is Kähler; the Hopf surface is a compact complex manifold that is not Kähler.
• Believing the Hodge conjecture concerns integral classes; the integral version is false (Atiyah–Hirzebruch), and the conjecture is stated for rational classes.

可选发现

• Arithmetic geometry and elliptic curves Hodge theory has arithmetic analogues: the Weil conjectures (proved by Deligne) control point counts of varieties over finite fields through cohomology.
• Derived categories Derived categories of coherent sheaves on Calabi–Yau manifolds are central objects in mirror symmetry and modern algebraic geometry.
• ← Riemann surfaces
• ← Symplectic geometry and topology

延伸阅读

• Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry (1978)
• Daniel Huybrechts, Complex Geometry: An Introduction (2005)
• Claire Voisin, Hodge Theory and Complex Algebraic Geometry I (2002)