TEBAC Hodge Program II: Variation, Hodge Loci, and Algebraicity Windows

This preprint is the second module of the TEBAC Hodge program. It develops the variation-theoretic and Hodge-locus infrastructure needed for a Clay-compatible modular attack on the rational Hodge conjecture. The module does not claim a completed proof of the Hodge conjecture. Its purpose is to organize the passage from the fixed residual obstruction package of HODGE-I to the later detector no-loss theorem target. Starting from the HODGE-I carrier and obstruction data\[K_X^p := H²ᵖ(X,Q)∩ Hp,p(X),^p(X):=spanQ\{\,cl(Z): Z⊂ X algebraic of codimension p\,\},\]and\[R_X^p:=K_X^p/A^p(X),\]the paper introduces variation of Hodge structure, the Gauss--Manin local system, Hodge loci, and algebraicity windows as the natural geometric environment in which residual Hodge classes and algebraic cycle classes can be compared. The Cattani--Deligne--Kaplan theorem is used only as an admissible infrastructure theorem asserting the algebraicity of Hodge loci in the base of an algebraic family. The module explicitly separates this base-algebraicity statement from the still-unproved cycle-algebraicity assertion required by the Hodge conjecture. The main output of HODGE-II is a precise reduction of the next decisive front to detector completeness:\[Ann(K_X^p)^A^p(X)=0.\]Equivalently, the later HODGE-III module must prove that every nonzero rational Hodge detector\[0≠ λ∈ (K_X^p)^\]is detected by some algebraic cycle class:\[λ(cl(Z))≠ 0.\] Thus HODGE-II closes the variation/Hodge-locus/algebraicity-window module of the TEBAC Hodge program and prepares the first genuine battlefront toward the full Clay-level Hodge conjecture: detector no-loss and algebraic cycle separation.