In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth, proper, geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface x 4 + y 4 + z 4 + w 4 = 0 {\textstyle x^{4}+y^{4}+z^{4}+w^{4}=0} in complex projective 3-space. Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy to classify) and the negatively curved surfaces of general type (which are essentially unclassifiable). K3 surfaces can be considered the simplest algebraic varieties whose structure does not reduce to curves or abelian varieties, and yet where a substantial understanding is possible. A complex K3 surface has real dimension 4, and it plays an important role in the study of smooth 4-manifolds. K3 surfaces have been applied to Kac–Moody algebras, mirror symmetry and string theory. It can be useful to think of complex algebraic K3 surfaces as part of the broader family of complex analytic K3 surfaces. Many other types of algebraic varieties do not have such non-algebraic deformations.
Definition There are several equivalent ways to define K3 surfaces. The only compact complex surfaces with trivial canonical bundle are K3 surfaces and compact complex tori, and so one can add any condition excluding the latter to define K3 surfaces. For example, it is equivalent to define a complex analytic K3 surface as a simply connected compact complex manifold of dimension 2 with a nowhere-vanishing holomorphic 2-form. (The latter condition says exactly that the canonical bundle is trivial.) There are also some variants of the definition. Over the complex numbers, some authors consider only the algebraic K3 surfaces. (An algebraic K3 surface is automatically projective.) Or one may allow K3 surfaces to have du Val singularities (the canonical singularities of dimension 2), rather than being smooth.
Calculation of the Betti numbers The Betti numbers of a complex analytic K3 surface are computed as follows. (A similar argument gives the same answer for the Betti numbers of an algebraic K3 surface over any field, defined using l-adic cohomology.) By definition, the canonical bundle K X = Ω X 2 {\displaystyle K_{X}=\Omega _{X}^{2}} is trivial, and the irregularity q(X) (the dimension h 1 ( X , O X ) {\displaystyle h^{1}(X,O_{X})} of the coherent sheaf cohomology group H 1 ( X , O X ) {\displaystyle H^{1}(X,O_{X})} ) is zero. By Serre duality,
h 2 ( X , O X ) = h 0 ( X , K X ) = 1. {\displaystyle h^{2}(X,{\mathcal {O}}_{X})=h^{0}(X,K_{X})=1.}
As a result, the arithmetic genus (or holomorphic Euler characteristic) of X is:
χ ( X , O X ) := ∑ i ( − 1 ) i h i ( X , O X ) = 1 − 0 + 1 = 2. {\displaystyle \chi (X,{\mathcal {O}}_{X}):=\sum _{i}(-1)^{i}h^{i}(X,{\mathcal {O}}_{X})=1-0+1=2.}
On the other hand, the Riemann–Roch theorem (Noether's formula) says:
χ ( X , O X ) = 1 12 ( c 1 ( X ) 2 + c 2 ( X ) ) , {\displaystyle \chi (X,{\mathcal {O}}_{X})={\frac {1}{12}}\left(c_{1}(X)^{2}+c_{2}(X)\right),}
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