K3 surface
A complex K3 surface has real dimension 4, and it plays an important role in the study of smooth 4-manifolds.
[2] (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 Serre duality, As a result, the arithmetic genus (or holomorphic Euler characteristic) of X is: On the other hand, the Riemann–Roch theorem (Noether's formula) says: where
The two definitions agree for a complex algebraic K3 surface, by Jean-Pierre Serre's GAGA theorem.
The Hodge index theorem implies that the Picard lattice of an algebraic K3 surface has signature
Many properties of a K3 surface are determined by its Picard lattice, as a symmetric bilinear form over the integers.
This leads to a strong connection between the theory of K3 surfaces and the arithmetic of symmetric bilinear forms.
One clear statement, due to Viacheslav Nikulin and David Morrison, is that every even lattice of signature
"Elliptic" means that all but finitely many fibers of this morphism are smooth curves of genus 1.
[15] Another contrast to negatively curved varieties is that the Kobayashi metric on a complex analytic K3 surface X is identically zero.
The proof uses that an algebraic K3 surface X is always covered by a continuous family of images of elliptic curves.
A stronger question that remains open is whether every complex K3 surface admits a nondegenerate holomorphic map from
[18] The set of isomorphism classes of complex analytic K3 surfaces is the quotient of N by the orthogonal group
[20]) For the same reason, there is not a meaningful moduli space of compact complex tori of dimension at least 2.
When stated carefully, the Torelli theorem holds: a K3 surface is determined by its Hodge structure.
However, the global Torelli theorem for K3 surfaces says that the quotient map of sets is bijective.
In contrast, Valery Gritsenko, Klaus Hulek and Gregory Sankaran showed that
Formally, it works better to view this as a moduli space of K3 surfaces Y with du Val singularities.
By the Hodge index theorem, the intersection form on the real vector space
The orthogonal complements of the roots form a set of hyperplanes which all go through the positive cone.
Any two such components are isomorphic via the orthogonal group of the lattice Pic(X), since that contains the reflection across each root hyperplane.
[26] A related statement, due to Sándor Kovács, is that knowing one ample divisor A in Pic(X) determines the whole cone of curves of X. Namely, suppose that X has Picard number
K3 surfaces are somewhat unusual among algebraic varieties in that their automorphism groups may be infinite, discrete, and highly nonabelian.
By a version of the Torelli theorem, the Picard lattice of a complex algebraic K3 surface X determines the automorphism group of X up to commensurability.
A related statement, due to Hans Sterk, is that Aut(X) acts on the nef cone of X with a rational polyhedral fundamental domain.
[28] K3 surfaces appear almost ubiquitously in string duality and provide an important tool for the understanding of it.
String compactifications on these surfaces are not trivial, yet they are simple enough to analyze most of their properties in detail.
were studied by Ernst Kummer, Arthur Cayley, Friedrich Schur and other 19th-century geometers.
Kunihiko Kodaira completed the basic theory around 1960, in particular making the first systematic study of complex analytic K3 surfaces which are not algebraic.
An important later advance was the proof of the Torelli theorem for complex algebraic K3 surfaces by Ilya Piatetski-Shapiro and Igor Shafarevich (1971), extended to complex analytic K3 surfaces by Daniel Burns and Michael Rapoport (1975).
