Hodge via Slicing and Calibration Equality From Surface Amplification to Global Algebraicity

We develop a classical slicing–calibration route aimed at representing rational Hodge classes by algebraic cycles. Fix α ∈ H^(2p)(X,Q) ∩ H^(p,p)(X) and set A = PD(kα) ∈ H_(2n−2p)(X,Z). (1) In A, take an integral mass minimizer T (real dimension 2(n−p)). (2) Slice T by a very ample q–parameter complete–intersection family S_t with q := n−p−1, so that dim S_t = p+1 and T_t := T ⌞ S_t is a 2–current; a coarea replacement shows T_t is mass minimizing on S_t for a.e. t. (3) Amplify by adding a large multiple of a very ample class and use Gysin functoriality to form the sliced degree–2p class β_t(γ) := i_t^!(γ ⌣ Ω^q) ∈ H^(2p)(S_t) (here i_t : S_t → X). In middle degree n = 2p this yields, for very general t and after a tensor power, effective complete–intersection curves in the slicewise homology class. (4) Calibration on a positive–measure set of slices forces calibration almost everywhere by a measurable replacement, hence T_t is a holomorphic curve a.e. (5) A Grassmannian plane–detection lemma upgrades "many complex slice lines" to global complex p–planes: the approximate tangent 2(n−p)–plane of T is J–invariant a.e. (6) The structure theorem for positive closed (p,p)–currents and a coarea elimination of any residual positive part give T = Σ_i m_i[V_i] with m_i ∈ Z_(≥0) and V_i irreducible complex p–folds. (7) Since X is projective, V_i are algebraic; applying the argument to an amplified class and subtracting the pure amplification recovers the original class as a Q–algebraic cycle in the middle–degree case. We also record the general n,p program and isolate the remaining class–matching step as the only gap.