"Compact Kähler manifolds"의 두 판 사이의 차이

@@ 1번째 줄: / 1번째 줄: @@
-==introduction==
-* {{수학노트|url=Metrics_on_Riemann_surfaces}}
-* 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 [http://www.staff.science.uu.nl/~vando101/MRIlectures.pdf Lectures on Riemannian Geometry, Part II:Complex Manifolds]
-* [http://www.math.upenn.edu/~siegelch/Notes/Cattani1.pdf Complex manifolds, Kahler metrics, differential and harmonic forms]
-* Werner Ballmann [http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/kaehler0609.pdf‎ 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.

"Compact Kähler manifolds"의 두 판 사이의 차이

2020년 11월 13일 (금) 09:29 판

둘러보기 메뉴

검색