Hodge Conjecture Research Articles

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 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.

Solving the Hodge Conjecture through Collaborative Intelligence.pdf

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.

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

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.

We prove that any hyper-Kähler sixfold $K$ of generalized Kummer type has a naturally associated manifold $Y_K$ of $\mathrm {K}3^{[3]}$ type. It is obtained as crepant resolution of the quotient of $K$ by a group of symplectic involutions acting trivially on its second cohomology. When $K$ is projective, the variety $Y_K$ is birational to a moduli space of stable sheaves on a uniquely determined projective $\mathrm {K}3$ surface $S_K$. As an application of this construction we show that the Kuga–Satake correspondence is algebraic for the K3 surfaces $S_K$, producing infinitely many new families of $\mathrm {K}3$ surfaces of general Picard rank $16$ satisfying the Kuga–Satake Hodge conjecture.

We introduce in this paper the notion of Hodge similarities of transcendental lattices of hyperkähler manifolds and investigate the Hodge conjecture for these Hodge morphisms. Studying K3 surfaces with a symplectic automorphism, we prove the Hodge conjecture for the square of the general member of the first four-dimensional families of K3 surfaces with totally real multiplication of degree two. We then show the functoriality of the Kuga–Satake construction with respect to Hodge similarities. This implies that, if the Kuga–Satake Hodge conjecture holds for two hyperkähler manifolds, then every Hodge similarity between their transcendental lattices is algebraic after composing it with the Lefschetz isomorphism. In particular, we deduce that Hodge similarities of transcendental lattices of hyperkähler manifolds of generalized Kummer deformation type are algebraic.

We study the Fano variety of k-planes Ω X ( k ) of a complete intersection X and show the surjectivity of the cylinder homomorphism in this setting, proving along the way the (classical) Hodge conjecture in cases where a numerical condition is satisfied. In particular, a new case is discovered in dimension 6. This work is a generalization of the second author’s paper Motives of some Fano varieties.

The paper provides a version of the rational Hodge conjecture for \mathsf{dg} categories. The noncommutative Hodge conjecture is equivalent to the version proposed by Perry (2022) for admissible subcategories. We obtain examples of evidence of the Hodge conjecture by techniques of noncommutative geometry. Finally, we show that the noncommutative Hodge conjecture for smooth proper connective \mathsf{dg} algebras is true.

Abstract We obtain examples of smooth projective varieties over ${\mathbb C}$ that violate the integral Hodge conjecture and for which the total Chow group is of finite rank. Moreover, we show that there exist such examples defined over number fields.

We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, the Jacobian of a smooth projective curve over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a smooth projective curve over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture are dense in their moduli space.

Recent papers by Markman and O’Grady give, besides their main results on the Hodge conjecture and on hyperkähler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six.In this paper we give a pedestrian introduction to these results. The spinor map, which is defined using a half-spin representation of SO(8), is used intensively. For simplicity, we use basic representation theory and we avoid the use of triality.

We give the first examples of O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}$$\\end{document}-acyclic smooth projective geometrically connected varieties over the function field of a complex curve, whose index is not equal to one. More precisely, we construct a family of Enriques surfaces over P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^{1}$$\\end{document} such that any multi-section has even degree over the base P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^{1}$$\\end{document} and show moreover that we can find such a family defined over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document}. This answers affirmatively a question of Colliot-Thélène and Voisin. Furthermore, our construction provides counterexamples to: the failure of the Hasse principle accounted for by the reciprocity obstruction; the integral Hodge conjecture; and universality of Abel–Jacobi maps.

In this paper, we give a new and simplified proof of the variational Hodge conjecture for complete intersection cycles on a hypersurface in projective space.

Abstract We prove the integral Hodge conjecture for curve classes on smooth varieties of dimension at least three constructed as a complete intersection of ample hypersurfaces in a smooth projective toric variety, such that the anticanonical divisor is the restriction of a nef divisor. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric minimal model program, we prove that in each case, is generated by classes of rational curves.

This is loosely a continuation of the author's previous paper arXiv:1802.09496. In the first part, given a fibered variety, we pull back the Leray filtration to the Chow group, and use this to give some criteria for the Hodge and Tate conjectures to hold for such varieties. In the second part, we show that the Hodge conjecture holds for a good desingularization of a self fibre product of a non-isotrivial elliptic surface under appropriate conditions. We also show that the Hodge and Tate conjectures hold for natural families of abelian varieties parameterized by certain Shimura curves. This uses Zucker's description of the mixed Hodge structure on the cohomology of a variation of Hodge structures on a curve, along with appropriate "vanishing" theorems.