# What is K3-surface and Calabi-Yau metric?

Maybe ,this is not a good question .

I am reading some paper about Ricci flow. K3-surface with Calabi-Yau metric are refered as example of Einstein manifold. But I don't know what they are . Then I google it , and find that there are many algebraic geometry's concepts. But I know nothing about algebraic topology , for understand what they are , what I should read ?

I just have some knowledge of Riemannian geometry , algebraic topology and basic group ring field theory.

• They are compact Kahler manifolds that admit a Ricci-flat Kahler metric. There's a nice article by Beauville you can read about them for a quick intro (Variétés Kahleriennes dont la première classe de Chern est nulle). See also the book "Einstein manifolds" by Besse. – Gunnar Þór Magnússon Oct 19 '16 at 3:58
• You don't need much algebraic geometry to understand what is a Kahler manifold. You definitely need to know a lot of PDE's to know why a compact Kahler manifold with zero first chern class admits a Ricci flat metric. – user99914 Oct 19 '16 at 4:19
• I consider Moroianu's lecturer notes (moroianu.perso.math.cnrs.fr/tex/kg.pdf) a very good reference. – User3773 Oct 19 '16 at 8:17
• @GunnarÞórMagnússon What is the name of Beauville's book ? – lanse7pty Oct 23 '16 at 13:31
• @Cla I am reading it , I have many question ,do you have read off it ? – lanse7pty Oct 23 '16 at 13:32

$\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$Here's a quick summary of "basic cultural facts", see also nLab and Wikipedia. For details, perhaps consult

• Principles of Algebraic Geometry by Griffiths and Harris,

• Complex Manifolds and Deformation of Complex Structures by Kodaira,

• Einstein Manifolds by Besse,

• Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics by Siu,

• Canonical Metrics in Kähler Geometry by Tian.

A K3 surface (for Kummer, Kähler, Kodaira) is a compact, simply-connected holomorphic surface with trivial canonical bundle (i.e., admitting a non-vanishing holomorphic $2$-form).

Examples of K3 surfaces include:

• A smooth quartic surface in $\Cpx\Proj^{3}$.

• A smooth surface of degree $(2, 2, 2)$ in $\Cpx\Proj^{1} \times \Cpx\Proj^{1} \times \Cpx\Proj^{1}$, i.e., the locus of a homogeneous sextic that is quadratic in the homogeneous coordinates of each projective line.

• A smooth complex surface obtained from a complex $2$-torus by quotienting out the involution $z \mapsto -z$, then blowing up the sixteen fixed points.

Every K3 surface is Kählerian, i.e., admits a Kähler metric (Siu). Any two smooth K3 surfaces are deformation equivalent, hence diffeomorphic (Kodaira).

The moduli space of holomorphic structures on a K3 surface is $20$-dimensional, and admits a branched covering by a ball in $\Cpx^{20}$ (Siu). A $19$-dimensional subfamily consists of algebraic varieties. (Wikipedia has a sketch of the calculation.)

If $(M, J, \Omega)$ is a "polarized" holomorphic manifold, i.e., a Kählerian manifold with a fixed Dolbeault $(1, 1)$-class containing a Kähler form, then for every sufficiently smooth $(1, 1)$-form $\rho$ representing the first Chern class $c_{1}(M)$, there exists a Kähler form $\omega_{\rho}$ representing $\Omega$ whose Ricci form is $2\pi\rho$ (part of Yau's solution of the Calabi conjecture).

Particularly, since the first Chern class of a K3 surface is zero, every Kähler class on a K3 surface admits a Ricci-flat Kähler metric.

Because the holonomy of a Ricci-flat Kähler surface is contained in $SU(2) \simeq Sp(1)$, a K3 surface admits a hyper-Kähler structure, an ordered triple $I$, $J$, $K$ of holomorphic structures satisfying the quaternion identity $IJK = -1$.