2025/06/21 by Zheng, Dongzhe
Mathematics · #14C30 (Primary) #22E46 #32Q25 #53C55 (Secondary) #58H10 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #General Mathematics (math.GM)
paper · pdf · doi:10.48550/arxiv.2506.17754
openalex publication_date 2025/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Classical variational Hodge structure theory characterizes the algebraicity of Hodge classes by studying the transversality of period mappings under geometric deformations. However, when algebraic varieties lack appropriate deformation families, this method faces applicability limitations. This paper develops a non-constructive method based on exceptional Lie group constraints to handle such cases. Our main technical contribution is establishing a dimension control mechanism for Spencer cohomology theory under Lie group constraints. Specifically, we prove that when a compact Kähler manifold X is equipped with E7 group constraints, the corresponding Spencer kernel Kλ1,1 has complex dimension simultaneously constrained by two bounds: representation theory gives the lower bound dimℂKλ1,1 ≥ 56, while the degenerate Spencer-de Rham mapping gives the upper bound dimℂKλ1,1 ≤ h1,1(X). For 5-dimensional Calabi-Yau manifolds satisfying h1,1(X) = 56, this dimension constraint becomes the equality dimℂKλ1,1 = 56 = h1,1(X). Combined with our established Spencer-calibration equivalence principle, this dimension matching result is sufficient to verify the (1,1)-type Hodge conjecture. Our proof completely avoids explicit algebraic cycle construction, instead achieving the goal through abstract dimensional arguments. This method demonstrates the application potential of Lie group representation theory in algebraic geometry, providing new theoretical tools for handling geometric objects where traditional deformation methods are not applicable.