Compact Kähler manifolds

introduction

• 틀:수학노트
• Kahler geometry = intersection of Riemmanian, Complex, symplectic geometry
• A Hermitian metric \(h\) on a complex manifold \((M^{2m},J)\) \[h(X,Y)=h(JX,JY)\]
• fundamental 2-form (or Kähler form) \((1,1)\)-form given by \(\Omega=i\sum_{\alpha,\beta=1}^{m}h_{\alpha\overline{\beta}}dz^{\alpha}\wedge dz^{\overline{\beta}}\)
• If \(\Omega\) is closed, i.e., \(d\Omega=0\), we call \(h\) a Kahler metric
• there exists a real function \(K\) such that \(\Omega=i\partial \overline{\partial} K\), which we call the Kahler potential
• The Ricci form is one of the most important objects on a Kahler manifold

Hermitian metric on a complex manifold

• Let \(h\) be a Hermitian metric and the coefficient

\[ h_{\alpha\overline{\beta}}:=h(\frac{\partial}{\partial z_{\alpha}},\frac{\partial}{\partial \overline{z}_{\beta}}) \]

examples

flat matric

• \(h_{\alpha\overline{\beta}}=\frac{1}{2}\delta_{\alpha\beta}\)
• \(\Omega=\frac{i}{2}\sum_{\alpha=1}^m dz_{\alpha}\wedge d\bar{z}_{\alpha}\)
• potential \(u(z)=\frac{1}{2}|z|^2\)

dimension 1 case

• \(h_{\alpha\overline{\alpha}}=h_{\overline{\alpha}\alpha}:=h\)
• \(\Omega=-2ih\,dz \wedge d\overline{z}\)
• for \(\mathbb{P}^{1}\),

\[ \Omega=\frac{-i}{2\pi}\frac{dz \wedge d\bar{z}}{(1+|z|^2)^2} \] see Chern class

etc

• Fubini–Study metric
• K3 surfaces
• Calabi-Yau manifold
• Hyperkahler manifolds

cohomology theory

• Hodge theory of harmonic forms
• compact Kähler manifold of dimension n
• Dolbeault cohomology
• \(h^{p,q}=\operatorname{dim} H^{p,q}(X)\)
• \(h^{p,q}=h^{q,p}\)
• Serre duality \(h^{p,q}=h^{n-p,n-q}\)

Hodge decomposition theorem

• Let \(M\) be a compact Kähler manifold. Let \(H^{p,q}(M)\) be the space of cohomology classes represented by a closed form of type \((p,q)\). There is a direct sum decomposition

\[ H^{m}_{dR}(M;\mathbb{C})=\bigoplus_{p+q=m}H^{p,q}(M) \] Moreover, \(H^{p,q}(M)=\overline{H^{q,p}(M)}\). In other words, \(H^{m}_{dR}(M)\) carries a real Hodge structure of weight \(m\).

Delbeault

• cohomology of sheaves of holomorphic forms

theorem

Let \(\Omega\) be the space of holomorphic \(p\)-forms on \(M\) \[ H^{p,q}(M)\cong H^q(M,\Omega^p) \]

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxLXgxazdpTXRDR0E/edit

expositions

• Valentino Tosatti, KAWA lecture notes on the Kähler-Ricci flow, arXiv:1508.04823 [math.DG], August 19 2015, http://arxiv.org/abs/1508.04823
• Tosatti, Valentino. “Uniqueness of CP^n.” arXiv:1508.05641 [math], August 23, 2015. http://arxiv.org/abs/1508.05641.
• Weinkove, Ben. ‘The K"ahler-Ricci Flow on Compact K"ahler Manifolds’. arXiv:1502.06855 [math], 24 February 2015. http://arxiv.org/abs/1502.06855.
• Stefan Vandoren Lectures on Riemannian Geometry, Part II:Complex Manifolds
• Complex manifolds, Kahler metrics, differential and harmonic forms
• Werner Ballmann Lectures on Kahler Manifolds

articles

• Berczi, Gergely. “Towards the Green-Griffiths-Lang Conjecture via Equivariant Localisation.” arXiv:1509.03406 [math], September 11, 2015. http://arxiv.org/abs/1509.03406.
• Treger, Robert. ‘On Uniformization of Compact Kahler Manifolds’. arXiv:1507.01379 [math], 6 July 2015. http://arxiv.org/abs/1507.01379.
• Dinh, Tien-Cuong, Fei Hu, and De-Qi Zhang. ‘Compact K"ahler Manifolds Admitting Large Solvable Groups of Automorphisms’. arXiv:1502.07060 [math], 25 February 2015. http://arxiv.org/abs/1502.07060.

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Spacy 패턴 목록

• [{'LOWER': 'kähler'}, {'LEMMA': 'manifold'}]