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.



Lattice polarized K3 surfaces
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.


