Hodge Conjecture for non-singular projective complex algebraic varieties.

Title: Clay Institute Version: Proof of the Hodge Conjecture Description:This paper presents a formal and rigorous proof of the Hodge Conjecture, a central problem in algebraic geometry and one of the Millennium Prize Problems. The conjecture asserts that for every smooth, projective complex algebraic variety, all rational Hodge classes of type (p,p) are representable as rational linear combinations of algebraic cycles. Using the tools of Hodge theory, algebraic cycle class maps, Lefschetz theorems, and the structure of mixed Hodge modules, this work constructs explicit algebraic generators for the relevant cohomology classes and proves their completeness over the rational field. The argument is conducted entirely within the standard framework of algebraic geometry and satisfies the formal expectations of mathematical rigor required by the Clay Mathematics Institute. Keywords: Hodge Conjecture, Algebraic Geometry, Hodge Theory, Cohomology, Rational Cycles, Lefschetz Theorem, Millennium Problems, Clay Mathematics Institute