K3 Surface
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In mathematics, a complex analytic K3 surface is a compact connected
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a ...
of dimension 2 with trivial
canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, ...
and
irregularity Irregular, irregulars or irregularity may refer to any of the following: Astronomy * Irregular galaxy * Irregular moon * Irregular variable, a kind of star Language * Irregular inflection, the formation of derived forms such as plurals in ...
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 In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the notation '' ...
zero. A simple example is the Fermat
quartic surface In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface: affine and projective. An ''affine'' quartic surface ...
: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 A curve is a geometrical object in mathematics. Curve(s) may also refer to: Arts, entertainment, and media Music * Curve (band), an English alternative rock music group * ''Curve'' (album), a 2012 album by Our Lady Peace * "Curve" (song), a 20 ...
or
abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
, 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 In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
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 = \Omega^2_X is trivial, and the irregularity ''q''(''X'') (the dimension h^1(X,O_X) of the coherent sheaf cohomology group H^1(X,O_X)) is zero. By Serre duality, :h^2(X,\mathcal_X)=h^0(X,K_X)=1. As a result, the arithmetic genus (or holomorphic Euler characteristic) of ''X'' is: :\chi(X,\mathcal_X):=\sum_i (-1)^i h^i(X,\mathcal_X)=1-0+1=2. On the other hand, the
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It ...
(Noether's formula) says: :\chi(X,\mathcal_X) = \frac \left(c_1(X)^2+c_2(X)\right), where c_i(X) is the ''i''-th Chern class of the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and ...
. Since K_X is trivial, its first Chern class c_1(K_X)=-c_1(X) is zero, and so c_2(X)=24. Next, the exponential sequence 0\to \Z_X\to O_X\to O_X^*\to 0 gives an
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the conte ...
of cohomology groups 0\to H^1(X,\Z) \to H^1(X,O_X), and so H^1(X,\Z)=0. Thus the Betti number b_1(X) is zero, and by
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold ( comp ...
, b_3(X) is also zero. Finally, c_2(X)=24 is equal to the topological
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
:\chi(X)=\sum_i (-1)^ib_i(X). Since b_0(X)=b_4(X)=1 and b_1(X)=b_3(X)=0, it follows that b_2(X)=22.


Properties

*Any two complex analytic K3 surfaces are diffeomorphic as smooth 4-manifolds, by
Kunihiko Kodaira was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japane ...
. *Every complex analytic K3 surface has a Kähler metric, by
Yum-Tong Siu Yum-Tong Siu (; born May 6, 1943 in Guangzhou, China) is the William Elwood Byerly Professor of Mathematics at Harvard University. Siu is a prominent figure in the study of functions of several complex variables. His research interests invo ...
. (Analogously, but much easier: every algebraic K3 surface over a field is projective.) By
Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
's solution to the Calabi conjecture, it follows that every complex analytic K3 surface has a Ricci-flat Kähler metric. *The Hodge numbers of any K3 surface are listed in the Hodge diamond: *:: *:One way to show this is to calculate the Jacobian ideal of a specific K3 surface, and then using a variation of
Hodge structure In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structure ...
on the moduli of algebraic K3 surfaces to show that all such K3 surfaces have the same Hodge numbers. A more low-brow calculation can be done using the calculation of the Betti numbers along with the parts of the
Hodge structure In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structure ...
computed on H^2(X;\Z) for an arbitrary K3 surface. In this case, Hodge symmetry forces H^0(X;\Omega_X^2)\cong \mathbb, hence H^1(X,\Omega_X) \cong \mathbb^. For K3 surfaces in characteristic ''p'' > 0, this was first shown by Alexey Rudakov and
Igor Shafarevich Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometr ...
. *For a complex analytic K3 surface ''X'', the intersection form (or
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutat ...
) on H^2(X,\Z)\cong\Z^ is a symmetric bilinear form with values in the integers, known as the K3 lattice. This is isomorphic to the even unimodular lattice \operatorname_, or equivalently E_8(-1)^\oplus U^, where ''U'' is the hyperbolic lattice of rank 2 and E_8 is the E8 lattice. *Yukio Matsumoto's 11/8 conjecture predicts that every smooth oriented 4-manifold ''X'' with even intersection form has second Betti number at least 11/8 times the absolute value of the
signature A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...
. This would be optimal if true, since equality holds for a complex K3 surface, which has signature 3−19 = −16. The conjecture would imply that every simply connected smooth 4-manifold with even intersection form is homeomorphic to a
connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classific ...
of copies of the K3 surface and of S^2\times S^2. *Every complex surface that is diffeomorphic to a K3 surface is a K3 surface, by Robert Friedman and John Morgan. On the other hand, there are smooth complex surfaces (some of them projective) that are homeomorphic but not diffeomorphic to a K3 surface, by Kodaira and Michael Freedman. These "homotopy K3 surfaces" all have Kodaira dimension 1.


Examples

*The double cover ''X'' of the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...
branched along a smooth sextic (degree 6) curve is a K3 surface of genus 2 (that is, degree 2''g''−2 = 2). (This terminology means that the inverse image in ''X'' of a general
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hype ...
in \mathbf^2 is a smooth curve of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...
2.) *A smooth quartic (degree 4) surface in \mathbf^3 is a K3 surface of genus 3 (that is, degree 4). *A Kummer surface is the quotient of a two-dimensional
abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functi ...
''A'' by the action a\mapsto -a. This results in 16 singularities, at the 2-torsion points of ''A''. The minimal resolution of this singular surface may also be called a Kummer surface; that resolution is a K3 surface. When ''A'' is the
Jacobian In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: *Jacobian matrix and determinant *Jacobian elliptic functions *Jacobian variety *Intermediate Jacobian In mathematics, the intermediate Jacobian of a compact Kähler m ...
of a curve of genus 2, Kummer showed that the quotient A/(\pm 1) can be embedded into \mathbf^3 as a quartic surface with 16 nodes. *More generally: for any quartic surface ''Y'' with du Val singularities, the minimal resolution of ''Y'' is an algebraic K3 surface. *The intersection of a
quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections ( ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is ...
and a cubic in \mathbf^4 is a K3 surface of genus 4 (that is, degree 6). *The intersection of three quadrics in \mathbf^5 is a K3 surface of genus 5 (that is, degree 8). *There are several databases of K3 surfaces with du Val singularities in
weighted projective space In algebraic geometry, a weighted projective space P(''a''0,...,''a'n'') is the projective variety Proj(''k'' 'x''0,...,''x'n'' associated to the graded ring ''k'' 'x''0,...,''x'n''where the variable ''x'k'' has degree ''a'k''. Pro ...
s.


The Picard lattice

The
Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a globa ...
Pic(''X'') of a complex analytic K3 surface ''X'' means the abelian group of complex analytic line bundles on ''X''. For an algebraic K3 surface, Pic(''X'') means the group of algebraic line bundles on ''X''. The two definitions agree for a complex algebraic K3 surface, by
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ...
's GAGA theorem. The Picard group of a K3 surface ''X'' is always a finitely generated free abelian group; its rank is called the Picard number \rho. In the complex case, Pic(''X'') is a subgroup of H^2(X,\Z)\cong\Z^. It is an important feature of K3 surfaces that many different Picard numbers can occur. For ''X'' a complex algebraic K3 surface, \rho can be any integer between 1 and 20. In the complex analytic case, \rho may also be zero. (In that case, ''X'' contains no closed complex curves at all. By contrast, an algebraic surface always contains many continuous families of curves.) Over an algebraically closed field of characteristic ''p'' > 0, there is a special class of K3 surfaces,
supersingular K3 surface In algebraic geometry, a supersingular K3 surface is a K3 surface over a field ''k'' of characteristic ''p'' > 0 such that the slopes of Frobenius on the crystalline cohomology ''H''2(''X'',''W''(''k'')) are all equal to 1. These have also been c ...
s, with Picard number 22. The Picard lattice of a K3 surface means the abelian group Pic(''X'') together with its intersection form, a symmetric bilinear form with values in the integers. (Over \Complex, the intersection form means the restriction of the intersection form on H^2(X,\Z). Over a general field, the intersection form can be defined using the
intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
of curves on a surface, by identifying the Picard group with the divisor class group.) The Picard lattice of a K3 surface is always ''even'', meaning that the integer u^2 is even for each u\in\operatorname(X). The Hodge index theorem implies that the Picard lattice of an algebraic K3 surface has signature (1,\rho-1). 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. As a first example of this connection: a complex analytic K3 surface is algebraic if and only if there is an element u\in\operatorname(X) with u^2>0. Roughly speaking, the space of all complex analytic K3 surfaces has complex dimension 20, while the space of K3 surfaces with Picard number \rho has dimension 20-\rho (excluding the supersingular case). In particular, algebraic K3 surfaces occur in 19-dimensional families. More details about
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such ...
s of K3 surfaces are given below. The precise description of which lattices can occur as Picard lattices of K3 surfaces is complicated. One clear statement, due to Viacheslav Nikulin and David Morrison, is that every even lattice of signature (1,\rho-1) with \rho\leq 11 is the Picard lattice of some complex projective K3 surface. The space of such surfaces has dimension 20-\rho.


Elliptic K3 surfaces

An important subclass of K3 surfaces, easier to analyze than the general case, consists of the K3 surfaces with an elliptic fibration X\to\mathbf^1. "Elliptic" means that all but finitely many fibers of this morphism are smooth curves of genus 1. The singular fibers are unions of rational curves, with the possible types of singular fibers classified by Kodaira. There are always some singular fibers, since the sum of the topological Euler characteristics of the singular fibers is \chi(X)=24. A general elliptic K3 surface has exactly 24 singular fibers, each of type I_1 (a nodal cubic curve). Whether a K3 surface is elliptic can be read from its Picard lattice. Namely, in characteristic not 2 or 3, a K3 surface ''X'' has an elliptic fibration if and only if there is a nonzero element u\in\operatorname(X) with u^2=0. (In characteristic 2 or 3, the latter condition may also correspond to a quasi-elliptic fibration.) It follows that having an elliptic fibration is a codimension-1 condition on a K3 surface. So there are 19-dimensional families of complex analytic K3 surfaces with an elliptic fibration, and 18-dimensional moduli spaces of projective K3 surfaces with an elliptic fibration. Example: Every smooth quartic surface ''X'' in \mathbf^3 that contains a line ''L'' has an elliptic fibration X\to \mathbf^1, given by projecting away from ''L''. The moduli space of all smooth quartic surfaces (up to isomorphism) has dimension 19, while the subspace of quartic surfaces containing a line has dimension 18.


Rational curves on K3 surfaces

In contrast to positively curved varieties such as del Pezzo surfaces, a complex algebraic K3 surface ''X'' is not uniruled; that is, it is not covered by a continuous family of rational curves. On the other hand, in contrast to negatively curved varieties such as surfaces of general type, ''X'' contains a large discrete set of rational curves (possibly singular). In particular,
Fedor Bogomolov Fedor Alekseyevich Bogomolov (born 26 September 1946) (Фёдор Алексеевич Богомолов) is a Russian and American mathematician, known for his research in algebraic geometry and number theory. Bogomolov worked at the Steklov I ...
and
David Mumford David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded ...
showed that every curve on ''X'' is linearly equivalent to a positive linear combination of rational curves. Another contrast to negatively curved varieties is that the
Kobayashi metric In mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by Shoshichi Kobayashi in 1967. Kobayashi hyperbolic manifolds are an important class of co ...
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. (These curves are singular in ''X'', unless ''X'' happens to be an elliptic K3 surface.) A stronger question that remains open is whether every complex K3 surface admits a nondegenerate holomorphic map from \C^2 (where "nondegenerate" means that the derivative of the map is an isomorphism at some point).


The period map

Define a marking of a complex analytic K3 surface ''X'' to be an isomorphism of lattices from H^2(X,\Z) to the K3 lattice \Lambda=E_8(-1)^\oplus U^. The space ''N'' of marked complex K3 surfaces is a non- Hausdorff complex manifold of dimension 20. The set of isomorphism classes of complex analytic K3 surfaces is the quotient of ''N'' by the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
O(\Lambda), but this quotient is not a geometrically meaningful moduli space, because the action of O(\Lambda) is far from being properly discontinuous. (For example, the space of smooth quartic surfaces is irreducible of dimension 19, and yet every complex analytic K3 surface in the 20-dimensional family ''N'' has arbitrarily small deformations which are isomorphic to smooth quartics.) For the same reason, there is not a meaningful moduli space of compact complex tori of dimension at least 2. The period mapping sends a K3 surface to its
Hodge structure In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structure ...
. When stated carefully, the Torelli theorem holds: a K3 surface is determined by its Hodge structure. The period domain is defined as the 20-dimensional complex manifold :D=\. The period mapping N\to D sends a marked K3 surface ''X'' to the complex line H^0(X,\Omega^2)\subset H^2(X,\Complex)\cong \Lambda\otimes\Complex. This is surjective, and a local isomorphism, but not an isomorphism (in particular because ''D'' is Hausdorff and ''N'' is not). However, the global Torelli theorem for K3 surfaces says that the quotient map of sets :N/O(\Lambda)\to D/O(\Lambda) is bijective. It follows that two complex analytic K3 surfaces ''X'' and ''Y'' are isomorphic if and only if there is a Hodge isometry from H^2(X,\Z) to H^2(Y,\Z), that is, an isomorphism of abelian groups that preserves the intersection form and sends H^0(X,\Omega^2)\subset H^2(X,\Complex) to H^0(Y,\Omega^2).


Moduli spaces of projective K3 surfaces

A polarized K3 surface ''X'' of genus ''g'' is defined to be a projective K3 surface together with an ample line bundle ''L'' such that ''L'' is primitive (that is, not 2 or more times another line bundle) and c_1(L)^2=2g-2. This is also called a polarized K3 surface of degree 2''g''−2. Under these assumptions, ''L'' is
basepoint-free In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...
. In characteristic zero,
Bertini's theorem In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broades ...
implies that there is a smooth curve ''C'' in the
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstracti ...
, ''L'', . All such curves have genus ''g'', which explains why (''X'',''L'') is said to have genus ''g''. The vector space of sections of ''L'' has dimension ''g'' + 1, and so ''L'' gives a morphism from ''X'' to projective space \mathbf^g. In most cases, this morphism is an embedding, so that ''X'' is isomorphic to a surface of degree 2''g''−2 in \mathbf^g. There is an irreducible coarse moduli space \mathcal_g of polarized complex K3 surfaces of genus ''g'' for each g\geq 2; it can be viewed as a
Zariski open In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is ...
subset of a Shimura variety for the group ''SO''(2,19). For each ''g'', \mathcal_g is a quasi-projective complex variety of dimension 19. Shigeru Mukai showed that this moduli space is unirational if g\leq 13 or g=18,20. In contrast, Valery Gritsenko,
Klaus Hulek Klaus Hulek (born 19 August 1952 in Hindelang) is a German mathematician, known for his work in algebraic geometry and in particular, his work on moduli spaces. Life Klaus Hulek studied Mathematics from 1971 at Ludwig Maximilian University ...
and Gregory Sankaran showed that \mathcal_g is of general type if g\geq 63 or g=47,51,55,58,59,61. A survey of this area was given by . The different 19-dimensional moduli spaces \mathcal_g overlap in an intricate way. Indeed, there is a countably infinite set of codimension-1 subvarieties of each \mathcal_g corresponding to K3 surfaces of Picard number at least 2. Those K3 surfaces have polarizations of infinitely many different degrees, not just 2''g''–2. So one can say that infinitely many of the other moduli spaces \mathcal_h meet \mathcal_g. This is imprecise, since there is not a well-behaved space containing all the moduli spaces \mathcal_g. However, a concrete version of this idea is the fact that any two complex algebraic K3 surfaces are deformation-equivalent through algebraic K3 surfaces. More generally, a quasi-polarized K3 surface of genus ''g'' means a projective K3 surface with a primitive
nef Nef or NEF may refer to: Businesses and organizations * National Energy Foundation, a British charity * National Enrichment Facility, an American uranium enrichment plant * New Economics Foundation, a British think-tank * Near East Foundation, ...
and
big Big or BIG may refer to: * Big, of great size or degree Film and television * ''Big'' (film), a 1988 fantasy-comedy film starring Tom Hanks * ''Big!'', a Discovery Channel television show * ''Richard Hammond's Big'', a television show presente ...
line bundle ''L'' such that c_1(L)^2=2g-2. Such a line bundle still gives a morphism to \mathbf^g, but now it may contract finitely many (−2)-curves, so that the image ''Y'' of ''X'' is singular. (A (−2)-curve on a surface means a curve isomorphic to \mathbf^1 with self-intersection −2.) The moduli space of quasi-polarized K3 surfaces of genus ''g'' is still irreducible of dimension 19 (containing the previous moduli space as an open subset). Formally, it works better to view this as a moduli space of K3 surfaces ''Y'' with du Val singularities.


The ample cone and the cone of curves

A remarkable feature of algebraic K3 surfaces is that the Picard lattice determines many geometric properties of the surface, including the convex cone of ample divisors (up to automorphisms of the Picard lattice). The ample cone is determined by the Picard lattice as follows. By the Hodge index theorem, the intersection form on the real vector space N^1(X):=\operatorname(X)\otimes\R has signature (1,\rho-1). It follows that the set of elements of N^1(X) with positive self-intersection has two connected components. Call the positive cone the component that contains any ample divisor on ''X''. Case 1: There is no element ''u'' of Pic(''X'') with u^2=-2. Then the ample cone is equal to the positive cone. Thus it is the standard round cone. Case 2: Otherwise, let \Delta=\, the set of roots of the Picard lattice. The orthogonal complements of the roots form a set of hyperplanes which all go through the positive cone. Then the ample cone is a connected component of the complement of these hyperplanes in 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. In this sense, the Picard lattice determines the ample cone up to isomorphism. A related statement, due to Sándor Kovács, is that knowing one ample divisor ''A'' in Pic(''X'') determines the whole
cone of curves In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety X is a combinatorial invariant of importance to the birational geometry of X. Definition Let X be a proper variety. By definition, a (real) ''1-cycle ...
of ''X''. Namely, suppose that ''X'' has Picard number \rho\geq 3. If the set of roots \Delta is empty, then the closed cone of curves is the closure of the positive cone. Otherwise, the closed cone of curves is the closed convex cone spanned by all elements u\in\Delta with A\cdot u>0. In the first case, ''X'' contains no (−2)-curves; in the second case, the closed cone of curves is the closed convex cone spanned by all (−2)-curves. (If \rho=2, there is one other possibility: the cone of curves may be spanned by one (−2)-curve and one curve with self-intersection 0.) So the cone of curves is either the standard round cone, or else it has "sharp corners" (because every (−2)-curve spans an ''isolated'' extremal ray of the cone of curves).


Automorphism group

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. Namely, let the Weyl group ''W'' be the subgroup of the orthogonal group ''O''(Pic(''X'')) generated by reflections in the set of roots \Delta. Then ''W'' is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of ''O''(Pic(''X'')), and the automorphism group of ''X'' is commensurable with the quotient group ''O''(Pic(''X''))/''W''. A related statement, due to Hans Sterk, is that Aut(''X'') acts on the nef cone of ''X'' with a rational polyhedral
fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
.


Relation to string duality

K3 surfaces appear almost ubiquitously in
string duality String or strings may refer to: * String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian ani ...
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. The type IIA string, the type IIB string, the E8×E8 heterotic string, the Spin(32)/Z2 heterotic string, and M-theory are related by compactification on a K3 surface. For example, the Type IIA string compactified on a K3 surface is equivalent to the heterotic string compactified on a 4-torus ().


History

Quartic surfaces in \mathbf^3 were studied by
Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of ...
, Arthur Cayley, Friedrich Schur and other 19th-century geometers. More generally, Federigo Enriques observed in 1893 that for various numbers ''g'', there are surfaces of degree 2''g''−2 in \mathbf^g with trivial canonical bundle and irregularity zero. In 1909, Enriques showed that such surfaces exist for all g\geq 3, and
Francesco Severi Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician. He was the chair of the committee on Fields Medal on 1936, at the first delivery. Severi was born in Arezzo, Italy. He is famous for his contributions to algebr ...
showed that the moduli space of such surfaces has dimension 19 for each ''g''.Enriques (1909); Severi (1909). André gave K3 surfaces their name (see the quotation above) and made several influential conjectures about their classification. Kunihiko Kodaira completed the basic theory around 1960, in particular making the first systematic study of complex analytic K3 surfaces which are not algebraic. He showed that any two complex analytic K3 surfaces are deformation-equivalent and hence diffeomorphic, which was new even for algebraic K3 surfaces. An important later advance was the proof of the Torelli theorem for complex algebraic K3 surfaces by
Ilya Piatetski-Shapiro Ilya Piatetski-Shapiro ( Hebrew: איליה פיאטצקי-שפירו; russian: Илья́ Ио́сифович Пяте́цкий-Шапи́ро; 30 March 1929 – 21 February 2009) was a Soviet-born Israeli mathematician. During a career that s ...
and Igor Shafarevich (1971), extended to complex analytic K3 surfaces by Daniel Burns and Michael Rapoport (1975).


See also

* Enriques surface * Tate conjecture * Mathieu moonshine, a mysterious relationship between K3 surfaces and the
Mathieu group M24 In the area of modern algebra known as group theory, the Mathieu group ''M24'' is a sporadic simple group of order :   21033571123 = 244823040 : ≈ 2. History and properties ''M24'' is one of the 26 sporadic groups and was ...
.


Notes


References

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External links


Graded Ring Database homepage
for a catalog of K3 surfaces
K3 database
for the
Magma computer algebra system Magma is a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics. It is named after the algebraic structure magma. It runs on Unix-like operating systems, as well as Windows. Introduction Magma ...

The geometry of K3 surfaces
lectures by David Morrison (1988). {{DEFAULTSORT:K3 Surface Algebraic surfaces Complex surfaces Differential geometry String theory