### The Hodge conjecture: The complications of understanding the shape of geometric spaces

#### Abstract

The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. It was proposed by William Hodge in 1950 and is currently a stimulus for the development of several theories based on geometry, analysis, and mathematical physics. It proposes a natural condition for the existence of complex submanifolds within a complex manifold. Manifolds are the spaces in which geometric objects can be considered. In complex manifolds, the structure of the space is based on complex numbers, instead of the most intuitive structure of geometry, based on real numbers.

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Atiyah, M. F., & Hirzebruch, F. (1962). Analytic cycles on complex manifolds.

*Topology, 1*, 25–45. doi: 10.1016/0040-9383(62)90094-0Grothendieck, A. (1969). Hodge’s general conjecture is false for trivial reasons.

*Topology, 8*, 299–303. doi: 10.1016/0040-9383(69)90016-0Hodge, W. V. D. (1950). The topological invariants of algebraic varieties. In

*Proceedings of the International Congress of Mathematicians*(pp. 181–192). Cambridge, MA: American Mathematical Society.Poincaré, H. (1895). Analysis situs.

*Journal de l’École Polytechnique, 1*, 1–123.Voisin, C. (2002). A counterexample to the Hodge Conjecture extended to Kähler varieties.

*International Mathematics Research Notices, 20*, 1057–1075. doi: 10.1155/S1073792802111135Weil, A. (1980). Abelian varieties and the Hodge ring. In

*Oeuvres Scientifiques Collected Papers III*(pp. 421–429). New York: Springer-Verlag.

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