Existence of Line Bundles on K3-surfaces

Jörg Winkelmann

In fact, this happens for a generic K3-surface of algebraic dimension zero, but not for every one. There are K3-surfaces of algebraic dimension zero with some non-trivial line bundles.

(1) A K3-surface X has algebraic dimension zero if and only if the intersection product on Pic(X) is negative definit.
Proof: K3-Surfaces have trivial canonical bundle. Hence Riemann-Roch yields X(D)=(1/2)D.D + 2 and X(D) h⁰(D)+h²(D) = h⁰(D) + h⁰(-D) for every line bundle D. If L is a non-trivial line bundle with L.L 0, this implies that dim H⁰(L) 2 or dim H⁰(L^-1) 2. Thus there is a line bundle with two linearly independent sections and therefore a non-constant meromorphic function.
Conversely, if the algebraic dimension is two or one, then there exists a curve which is ample resp. the fiber of the algebraic reduction. Now C.C >0 for ample curves and C.C=0 for fibers of surjective holomorphic maps from a surface to a curve.

(2) All K3-surfaces are diffeomorphic, hence H²(X,Z) with cup-product is isomorphic to a fixed Z-module of rank 22 with fixed bilinear product (This Z-module is called L=L_Z in [BPV], Ch.VIII). This bilinear form is non degenerate and of signature (19,3) (19 negative, 3 positive). For every K3-surface, there is a non-zero holomorphic two-form which yields an element w in L_C ( L_C= tensor product of L_Z with C over Z) such that

(w,w)=0 and (w,\bar w)>0. (*)

Conversely, for every w in L_C fulfilling (*) there exists a corresponding K3-surface (Torelli-Theorem, [BPV], Ch.VIII, (14.2) ). Now, the wedge-product of any (1,1)-form with w or \bar w is zero. Hence H^1,1(X) is the orthogonal complement of w,\bar w in H²(X,C)=L_C. Let w=u+iv denote the decomposition of w into real and imaginary part. Then (*) is equivalent to:

(u,u)=(v,v)>0 and (u,v)=0 (**)

Thus u,v is a real two dimensional subspace of L_R such that the intersection product is positive definit. Conversely, if H is a real two-dimensional subvectorspace of L_R on which the product is positive definit, one can choose u,v fulfilling (**). Together with the Torelli theorem quoted above this implies: Let V be a subvectorspace of L_R of real codimension two and let H denote its orthogonal complement. Then there exists a K3-surface X such that H^1,1(X)=V if and only if the product on H is positive definit.

(3) We also need (see [BPV], Ch, IV, thm. (2.12) ): The image of Pic(X) in H²(X) is precisely the intersection of H²(X,Z) and H^1,1(X).

(4) Let x be an element in L_Z such that (x,x)<0. Since the signature is (19,3), there is a subvectorspace V of L_R of codimension two with a positive definit orthogonal complement. Let W denote the set of all subvectorspaces V of L_R of codimension two with positiv definit orthogonal complement such that x is in V. This set W is an open set in some algebraic subvariety of some Grassmannian. Now, for every y in L_Z, the set of all V in W with W containing y defines a subset S_y of W. It is clear that this set S_y is nowhere dense in W. Everything here is at least real analytic (or sub/semi-algebraic if you prefer), hence it follows that a countable union of such S_y can not be the whole W. For this reason there exists a subvectorspace V of L_R of codimension two, containing x such that the orthogonal complement is positive definit and V does not contain any element y in L_Z with (y,y) 0. Thus the intersection of V with L_Z is non-trivial and the product on this intersection is negativ definit.
It follows that the corresponding K3-surface has algebraic dimension zero, but does admit non-trivial line bundles.

(5) Note that x in L_Z was arbitrary except for the condition x,x<0, and that, by similar arguments, we may require that x _Z is exactly Pic(X) if x is primitive, i.e., if there does not exist any y in L_Z with ny=x for some natural number n>1. The structure of L_Z is described in detail in [BPV]. In particular, it contains as a direct summand a hyperbolic Z-plane, i.e. a Z² with the product given by the matrix

.

(This direct summand is called H in [BPV]). Now let p be a natural number. Then

Thus there is a primitive element x in L_Z with (x,x)= -2p. It follows, by the arguments used above, that there exists a K3-surface X ( of algebraic dimension zero) such that Pic(X) is generated by a single element x with (x,x)=-2p. Clearly, Pic(X) is not generated by (-2)-curves. Actually, [BPV],Ch,VIII, Prop.3.7 now implies that this surface contains no curves. Thus there exists an example of a K3-surface of algebraic dimension zero which does not contain any curve, but which nevertheless admits a non-trivial line bundle.

[BPV] : Barth, Peters, van de Ven: Complex Surfaces.

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