Lattice polarized K3 surfaces

Question

I've recently encountered the definition of a lattice polarized K3 surface. What is the idea behind the definition? Surely, there's something deeper to it than merely being a natural generalization of a polarized K3.

Answer 1

I'm not sure how much detail you're looking for; the point is that you can think of a lattice polarized K3 surface as a K3 surface with several different line bundles. A very general algebraic K3 surface has Picard number 1 - for instance, a very general quartic surface in $\mathbf{P}^3$ has only the divisor class coming from the hyperplane section. If you want to look at more "special" K3 surfaces and their moduli spaces, then lattice polarized K3 surfaces are the natural thing to look at. Over $\mathbf{C}$, the period map identifies the moduli space with a quotient of a suitable Hermitian symmetric domain of type $IV$ by an arithmetic subgroup of an appropriate orthogonal group. For more details, see Dolgachev's paper on mirror symmetry for lattice polarized K3 surfaces, and for a more algebraic treatment, you can look at Beauville's paper on Fano threefolds and K3 surfaces.

Answer 2

As one example, one may study singular plane sextics, or singular spacial quartics, or such. Assuming all singularities simple ($ADE$), one gets a lattice polarized $K3$-surface as the principal object of study.

Answer 3

Let me add one more, slightly different perspective: the notion of lattice polarization for K3s becomes very natural from the point of view of families of higher-dimensional Calabi-Yau manifolds. If ${\mathcal X}\to T$ is a family of projective Calabi-Yau manifolds of dimension at least three, in particular with $H^{2,0}(X_t)=0$ on fibres, then the Picard groups of fibres can be identified in the family (perhaps up to monodromy), and then one can ask interesting questions like constancy of ample cone, deformations of contractions, etc (results of Wilson in dimension three). In particular, if ${\mathcal X}\to T$ arises as a family of anticanonical sections of some fixed Fano manifold $Y$, then ${\rm Pic}(Y)\to {\rm Pic}(X_t)$ is an isomorphism by Lefschetz, and again one can ask about the ample cone, for example (still equal in dimension three, by a result of Kollár). In contrast, all these statements fail for families of K3 surfaces; lattice polarized families are a useful substitute, giving families of K3s with interesting generic Picard group.