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\[K_X^p := H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X),\qquadA^p(X):=\operatorname{span}_{\mathbb{Q}}\{\,\operatorname{cl}(Z): Z\subset X\text{ algebraic of codimension }p\,\},\]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:\[\operatorname{Ann}_{(K_X^p)^\vee}A^p(X)=0.\]Equivalently, the later HODGE-III module must prove that every nonzero rational Hodge detector\[0\neq \lambda\in (K_X^p)^\vee\]is detected by some algebraic cycle class:\[\lambda(\operatorname{cl}(Z))\neq 0.\] 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.