We give a constructive proof of the Hodge conjecture for complex $K3$ surfaces that does not rely on Torelli-type results. Starting with an arbitrary rational $(1,1)$-class $\alpha\in H^{ 1,1}(X,\mathbb{ Q})$, we algorithmically build a one-parameter family of quartic $K3$'s acquiring at most ten $A_1$-nodes. On the central fibre $\widetilde{ X}_0$, the class $\alpha$ specializes to a $\mathbb{ Q}$-linear combination of the hyperplane class and the exceptional $(-2)$-curves coming from the blow-ups of the nodes. Using the Clemens--Schmid sequence together with Picard--Lefschetz theory, we identify $\Gr^W_2 H^2_{ \lim}\cong H^2(\widetilde{ X}_0)$ and transport this combination back to the original smooth surface as an algebraic divisor. This yields an explicit, finite-step procedure that realizes any rational $(1,1)$-class by an algebraic cycle. We also formulate an equivariant extension for $(2,2)$-classes on Calabi--Yau threefolds, indicating how the same strategy might apply in higher dimension.
Categories: math.AG math.CV