2025/07/25 by Mounda, Badre
#14C30 (Secondary) #14D07 (Primary) #14J28 #32G20 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.18999
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 α∈ H1,1(X,ℚ), we algorithmically build a one-parameter family of quartic K3's acquiring at most ten A1-nodes. On the central fibre \widetildeX0, the class α specializes to a 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 \GrW2 H2lim≅ H2(\widetildeX0) 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.