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.
$\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$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 $IJK = -1$.