1. The transcendental block is constrained by Hodge symmetry and Mumford--Tate rigidity
Description.
The transcendental lattice T(X)T(X)T(X) (or the transcendental part of the Hodge structure) is governed by its Hodge decomposition and by its Mumford--Tate group. For many varieties VVV, the Mumford--Tate group of the transcendental Hodge structure is large (often reductive with few nontrivial algebraic invariants), which forces the space of Hodge tensors in T(V)mT(V)^{\otimes m}T(V)m to be small and highly structured.
Consequence.
Algebraic correspondences produce Hodge tensors that must be invariant, or transform in a prescribed way, under the Mumford--Tate group. If the transcendental part has generic Mumford--Tate (large), then there are very few available invariant tensors for correspondences to realize. Thus the mere existence of a transcendental class in Hp,pH^{p,p}Hp,p is not enough; one must produce a correspondence whose induced action aligns with the symmetries of TTT. This is a severe representation-theoretic restriction.
2. Variation of Hodge structure and non-algebraicity in families
Description.
Transcendental classes typically vary nontrivially in moduli: the Hodge decomposition of T(Y)T(Y)T(Y) moves with parameters and only special loci (Hodge loci) support extra Hodge tensors. Algebraic cycles, conversely, typically lie on Hodge loci---subvarieties of moduli with special Hodge properties.
Consequence.
To exhibit algebraicity one must either (a) work on special members of a family for which the transcendental part lies in a Hodge locus (often a countable union of algebraic subvarieties), or (b) find correspondences that are defined uniformly in families and whose cohomology classes track the needed Hodge loci. For a generic variety, transcendental Hodge classes are expected not to be algebraic. This separates generic from arithmetic/special cases and complicates any attempt at a uniform proof.
3. Rationality versus integrality subtleties
Description.
The Hodge Conjecture is a statement about Hodge classes with rational coefficients. However, many geometric techniques and counterexamples concern integral classes or subtle torsion phenomena. Voisin-type results show that integral Hodge classes need not be algebraic; rational Hodge classes are more subtle because rational equivalence and -structures intervene.
Consequence.
One cannot naively lift rational conclusions from integral computations, nor assume that constructions that yield cohomology classes integrally will reflect the rational span in the required way. This complicates attempts to manufacture algebraic cycles by elementary integer-lattice arguments unless careful control of denominators and rational equivalence is maintained.
4. Absence or scarcity of algebraic correspondences acting nontrivially on TTT
Description.
Practical realization of TTT\otimes TTT (or related transcendental tensors) requires correspondences ZVVZ\subset V\times VZVV whose induced maps Z\Phi_ZZ restrict nontrivially to TTT. For many varieties such correspondences are uncommon: automorphisms that act nontrivially on TTT may be absent; derived equivalences that induce the desired maps may not exist or be difficult to describe.
