We give a new proof of the Hodge conjecture for abelian fourfolds of Weil type with discriminant 1 and all of their powers. The Hodge conjecture for these abelian fourfolds was proven by Markman using hyperholomorphic sheaves on hyper-Kähler varieties of generalized Kummer type, and by constructing semiregular sheaves on abelian varieties. Our proof instead relies on a direct geometric relation between abelian fourfolds of Weil type with discriminant 1 and the six-dimensional hyper-Kähler varieties K ˜ of O'Grady type arising as crepant resolutions K ˜ → K of a locally trivial deformation of a singular moduli space of sheaves on an abelian surface. As applications, we establish the Hodge conjecture and the Tate conjecture for any variety K ˜ of OG6-type as above, and all of its powers. Nous donnons une nouvelle démonstration de la conjecture de Hodge pour les variétés abéliennes de dimension 4 de type Weil de discriminant 1, ainsi que pour toutes leurs puissances. La conjecture de Hodge pour ces variétés abéliennes de dimension 4 a été démontrée par Markman à l'aide de faisceaux hyperholomorphes sur des variétés hyperkählériennes de type Kummer généralisé, ainsi que par la construction de faisceaux semiréguliers sur des variétés abéliennes. Notre démonstration repose au contraire sur une relation géométrique directe entre les variétés abéliennes de dimension 4 de type Weil de discriminant 1, et les variétés hyperkählériennes K ˜ de dimension six de type O'Grady, qui apparaissent comme des résolutions symplectiques K ˜ → K d'une déformation localement triviale d'un espace de modules singulier de faisceaux sur une surface abélienne. Comme applications, nous établissons la conjecture de Hodge et la conjecture de Tate pour toute variété K ˜ de type OG6 comme ci-dessus, ainsi que pour toutes ses puissances.

Abstract We prove cohomological vanishing criteria for the Ceresa cycle of a curve C embedded in its Jacobian J : (A) if $\mathrm{H}^3(J)^{\mathrm{Aut}(C)} = 0$ , then the Ceresa cycle is torsion modulo rational equivalence; (B) if $\mathrm{H}^0(J, \Omega_J^3)^{\mathrm{Aut}(C)} = 0$ , then the Ceresa cycle is torsion modulo algebraic equivalence, with criterion (B) conditional on the Hodge conjecture. We then use these criteria to study the simplest family of curves where (B) holds but (A) does not, namely the family of Picard curves $C \colon y^3 = x^4 + ax^2 + bx + c$ . Criterion (B) and work of Schoen combine to show that the Ceresa cycle of a Picard curve is torsion in the Griffiths group. We furthermore determine exactly when it is torsion in the Chow group. As a byproduct, we deduce that there exist one-parameter families of plane quartic curves with torsion Ceresa Chow class; that the torsion locus in $\mathcal{M}_3$ of the Ceresa Chow class contains infinitely many components; and that the order of a torsion Ceresa Chow class of a Picard curve over a number field K is bounded, with the bound depending only on $[K\colon \mathbb{Q}]$ . Finally, we determine which automorphism group strata are contained in the vanishing locus of the universal Ceresa cycle over $\mathcal{M}_3$ .

We continue our investigation of the geometry of the Albanese morphism on 0-cycles.We provide an example of a smooth projective variety with representable CH 0 -group but with no universal 0-cycle, which answers a question asked by Colliot-Thlne.Our construction relies on a counterexample to the integral Hodge conjecture provided by Benoist and Ottem.

We propose an informational reformulation of the classical Hodge Conjecture within the frame-work of Viscous Time Theory (VTT), introducing informational persistence as a principle extending classical harmonicity. In this formulation, harmonic representatives are reinterpreted as persistent informational configurations (∆C-stable structures) on compact K¨ahler manifolds. We define the informational coherence gradient ∆C on M × R, where M is a compact K¨ahler manifold and R denotes the informational axis, and establish a ∆C-inner product via a deformed Hodge star operator. Within this setting, ∆C-harmonic forms arise as the natural generalization of classical harmonic forms, capturing equilibrium informational flows under tempo-ral evolution. We further show that bounded informational flows converge toward ∆C-harmonic equilibrium, and we prove correspondence between ∆C-harmonic representatives and algebraic cycles under ∆C-preserving deformations. A worked example on the complex torus illustrates the feasibility of the framework, yielding explicit ∆C-harmonic representatives with algebraic support. This formulation embeds the classical Hodge setting as the limiting case (κ → 0) while opening broader perspectives for informational geometry, with implications for algebraic cycles, quantum coherence, and informational models of gravitation.

Abstract Let be a smooth cubic hypersurface, and let be the variety of lines on . We prove the surjectivity of the cylinder maps on the Chow groups of and if contains a one‐cycle of degree . Mongardi and Ottem previously proved the integral Hodge conjecture for curve classes on hyperkähler manifolds. Using the cylinder maps, we provide an alternative proof for the of a smooth complex cubic fourfold , which is a special hyperkähler fourfold. In addition, we confirm the integral Tate conjecture for of a smooth cubic fourfold over a finitely generated field.

Abstract Let S and T be smooth projective varieties over an algebraically closed field k. Suppose that S is a surface admitting a decomposition of the diagonal. We show that, away from the characteristic of k, if an algebraic correspondence $T \to S$ acts trivially on the unramified cohomology, then it acts trivially on any normalized, birational and motivic functor. This generalizes Kahn’s result on the torsion order of S. We also exhibit an example of S over $\mathbb {C}$ for which $S \times S$ violates the integral Hodge conjecture.

Abstract Let $X$ be a curve of genus at least 4 that is very general or very general hyperelliptic. We classify all the ways in which a power $(JX)^k$ of the Jacobian of $X$ can be isogenous to a product of Jacobians of curves. As an application, we show that if $A$ is a very general principally polarized abelian variety of dimension at least 4 or the intermediate Jacobian of a very general cubic threefold, then no power $A^k$ is isogenous to a product of Jacobians of curves. This confirms various cases of the Coleman–Oort conjecture. We further deduce from our results some progress on the question of whether the integral Hodge conjecture fails for $A$ as above.

This research consists of seven mathematical frameworks fully developed through collaborative intelligence with AI, using the DeepSeek application with the Deep Reasoning (R1) feature.The system generated new equations, mathematical functions, and hybrid solutions.Additionally, ChatGPT was used for suggestions, recommendations, and result evaluation.These frameworks aim to provide a comprehensive solution to the Hodge Conjecture by integrating advanced tools from algebraic geometry, numerical analysis, dynamical systems, and quantum computing.The frameworks are based on: enhanced mixed Hodge structure theories to generalize results to non-Khler manifolds; ultraprecise algebraic quantum algorithms (such as AQA v3.0) with topological

In early January, an article titled “How a quantum innovation may quash the idea of the multiverse” appeared in New Scientist [1] It received a prompt response from German physicist Sabine Hossenfelder in the form of a video on You Tube [2]. The multiverse part got my attention. Suppose quantum gravity one day goes far beyond unifying quantum mechanics and general relativity. It might unite everything in space and time. Assuming the universe is everything that has existed or will exist, the multiverse could be temporal or all the things that happen at different zeptoseconds in this universe (a zeptosecond is the smallest unit of time ever measured and equals 10−21s or a trillionth of a billionth of a second). That quantum gravity from the far future could unify all the times in the multiverse with the one physical universe. This makes the multiverseobservable constantly (and kind of scientific). Viewing a zeptosecond as the latest step towards discovering the quantum (smallest amount) of time, this Article proposes a connection between quantum mechanics and cosmology’s holographic principle. This suggests the microscopic is united with the macroscopic when the holographic principle is combined with the precision of unrecognized quantum certainty. The micro-macro union means the holographic principle could not only be used to achieve physical quantum entanglement of particles or atoms but could also be used to unify the temporal multiverse with the physical universe. It even opens the door on quantum wave functions and particles being used to overcome otherwise inevitable phenomena like the Sun’s future red-giant phase and the 2nd law of thermodynamics’ eventual entropic decayof the entire cosmos. Vopson writes, “In an expanding universe, the entropy will always increase because more possible micro-states are being created via the expansion of the space itself/universe”[3]. Therefore, avoiding the cosmic demise due to entropy requires the universe to be static. A possible mathematical structure of such a universe is outlined. The article finishes with Vector-Tensor-Scalar Geometry that offers support for the proposed building blocks of space-time via the Hodge Conjecture, presents a feasible alternative method for formation of astronomical bodies, and submits a new picture of the Higgs boson and field.

A Deformation-Theoretic Strategy toward the Hodge Conjecture via General Hyperplane Intersections

A Local-to-Global Strategy Toward the Hodge Conjecture via Hyperplane Sections

Deformation of Hyperplane Intersections and a Strategy Toward the Hodge Conjecture

The Hodge conjecture states that the Hodge group of Hodge classes is equal to the algebraic group generated by algebraic subvarieties on a Hodge manifold. To prove the conjecture, we introduce the Hodge structure, the Lefschetz decomposition, and polarization on the cohomology groups of the manifold, and we use mathematical induction on the degrees of the primitive cohomologies in the Lefschetz decomposition. We show that every Hodge class on a Hodge manifold is a rational linear combination of the cohomology classes of algebraic subvarieties of the manifold. The Lefschetz isomorphisms on the cohomology groups of a Hodge manifold are algebraic in the product space. As a consequence, we show that the inverses of the Lefschetz isomorphisms are also algebraic in the product space.

Exploring the Hodge Conjecture with Relational Field Theory: A Dimensional Approach

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds.

We study algebraic cycles on complex Gushel-Mukai (GM) varieties. We prove the generalised Hodge conjecture, the (motivated) Mumford-Tate conjecture, and the generalised Tate conjecture for all GM varieties. We compute all integral Chow groups of GM varieties, except for the only two infinite-dimensional cases (1-cycles on GM fourfolds and 2-cycles on GM sixfolds). We prove that if two GM varieties are generalised partners or generalised duals, their rational Chow motives in middle degree are isomorphic.Comment: 23 pages, final version, in special volume in honour of C. Voisin

Let S be a K3 surface obtained as triple cover of a quadric branched along a genus 4 curve. Using the relation with cubic fourfolds, we show that S has finite-dimensional motive, in the sense of Kimura. We also establish the Kuga-Satake Hodge conjecture for S, as well as Voisin's conjecture concerning zero-cycles. As a consequence, we obtain Kimura finitedimensionality, the Kuga-Satake Hodge conjecture, and Voisin's conjecture for 2 (9-dimensional) irreducible components of the moduli space of K3 surfaces with an order 3 non-symplectic automorphism.

The article begins by mentioning the accepted correlation between Albert Einstein’s relativistic equations, as well as James Clerk Maxwell’s electromagnetic waves, with the Prandtl-Glauert equations for fluid flow. This equation-free mention nevertheless ends with 180° ΔO = + => - (180 Degree Change in Orientation Equals Positive Becomes Negative). Of course, it also means negative can become positive: 180° ΔO = - => +. Using this article’s new equations, the Electric Dipole Moment is introduced and the charges of the quarks composing a neutron are transformed. EDM’s undetectability is explained from the perspective of a hypothetical someone who doesn’t accept the Big Bang but believes the universe is literally infinite and eternal. Then quantum spin of matter particles is discussed, and extended to astronomy’s black holes, in terms of photons and gravitons not being elementary force-carrying particles but being in possession of EDM. The photon-graviton interaction producing quantum spin is proposed as electromagnetic and gravitational vectors in a figure that might be called “Vector-Tensor-Scalar Geometry”, as well as being related to quaternions plus the Higgs boson and field. Then the article returns to black holes, showing how the inability of light to escape them leads to a 4-dimensional space-time: via binary digits or base 2 mathematics, Mobius bands, figure-8 Klein bottles, and Wick rotation. The electric potential of photons and gravitons is then interpreted not strictly as a neutron-identical Electric Dipole Moment but as a vast array of pulses sharing similarities with modern computers and electronics – the binary digits of the previous sentence. As a consequence of the electric force-carriers bringing them into existence, particles of matter and antimatter are symmetrical with bosons in the sense that every boson or fermion is, at its most fundamental level, composed of binary digits. Imaginary numbers are essential to quantum mechanics, and this article connects the imaginary numbers of Wick rotation to the nature of time. Therefore, the words here are not painting a classical picture of space-time. The 1’s and 0’s of binary digits are compatible with quantum mechanics and may be referred to as the Hidden Variables which Einstein advocated to complete quantum physics, and to give its calculations an exactness which would bring a hidden order to its chaotic randomness and superficial uncertainty. If the universe can be quantized and viewed as comprised of infinitesimal ones and zeros, how could it not obey quantum physics? And if those ones and zeros are all ultimately connected by Quantum Gravity, waves and particles could never be separated but wave-particle duality would rule. To finish, many thanks to Albert Einstein for laying the foundations of this article 105 years ago when he published a paper titled “Do gravitational fields play an essential role in the structure of elementary particles?” Immediately before submission for publication, a few topics were added to this article - dark matter, precession, the Hodge conjecture, and the Riemann hypothesis.

We validly ignore even prime number 2. Based on all arbitrarily large number of even Prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with the Prime number theorem for Arithmetic Progressions, and our derived Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. With these conditions being satisfied by all Odd Primes, we argue Polignac's and Twin prime conjectures are proven to be true when they are usefully treated as Incompletely Predictable Problems. In so doing [and with Riemann hypothesis being a special case], this action also support the generalized Riemann hypothesis formulated for Dirichlet L-function. By broadly applying Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture to Dirichlet eta function (proxy function for Riemann zeta function), Riemann hypothesis is separately proven to be true when it is usefully treated as Incompletely Predictable Problem. Crucial connections exist between these Incompletely Predictable Problems and Quantum field theory whereby eligible (sub-)functions and (sub-)algorithms are treated as infinite series.

Abstract We prove the real integral Hodge conjecture for several classes of real abelian threefolds. For instance, we prove the property for real abelian threefolds whose real locus is connected, and for real abelian threefolds A which are the product A = B × E {A=B\times E} of an abelian surface B and an elliptic curve E with connected real locus E ⁢ ( ℝ ) {E(\mathbb{R})} . Moreover, we show that every real abelian threefold satisfies the real integral Hodge conjecture modulo torsion, and reduce the principally polarized case to the Jacobian case.