I heard from string theorists thinking of the so-called "(1/2) K3 surface" or "half-K3 surface" as follows:
Let $T^2 \times S^1$ be a 3-torus with spin structure periodic in all directions. $T^2 \times S^1$, with this spin structure, is the boundary of a “1/2-K3 surface,” that is, a four-manifold $M^4$ that maps to a disc $D$ with generic fiber an elliptic curve. In particular, the map $$ M^4 \to D$$ has a section $$s : D \to M^4.$$ (possibly missing contexts) ...
Are these mathematically clear? Or do we require more clarification?
What are some other Mathematical way"s" to think about or to define "(1/2) K3 surface"?