fiber ∞-bundle, principal ∞-bundle, associated ∞-bundle, group cohomology, nonabelian group cohomology, Lie group cohomology, groupoid cohomology, nonabelian groupoid cohomology, generalized (Eilenberg-Steenrod) cohomology, universal principal ∞-bundle, groupal model for universal principal ∞-bundles, (∞,1)-vector bundle / (∞,n)-vector bundle, cohomology with constant coefficients / with a local system of coefficients, differential generalized (Eilenberg-Steenrod) cohomology, differential cohomology in a cohesive topos, connecting homomorphism, Bockstein homomorphism, de Rham theorem, Poincare lemma, Stokes theorem, nonabelian Hodge theory, noncommutative Hodge theory, dim=1dim = 1: Riemann surface, super Riemann surface. The Hodge conjecture has also been proved for the following classes of varieties: 1) $ X $ is a smooth four-dimensional uniruled variety, that is, a variety such that there exists a rational mapping of finite degree $ P ^ {1} \times Y \rightarrow X $, where $ Y $ is a smooth variety (see [2] ). Steenbrink, "Some remarks about the Hodge conjecture" E. Cattani (ed.) https://encyclopedia2.thefreedictionary.com/Hodge+conjecture, For very general Z, it is known that [N.sup.2][H.sup.4](X) (which is the subspace of Hodge classes, as the, Specifically, they are the Riemann hypothesis, which lingers from Hilbert's list, Yang-Mills theory and the mass gap hypothesis, the P Versus NP problem, the Navier-Stokes equations, the Poincaire conjecture, the Birch and Swinnerton-Dyer conjecture, and the, Professor Miles Reid, of Warwick University, has spent more than 30 years mulling over the, Dictionary, Encyclopedia and Thesaurus - The Free Dictionary, the webmaster's page for free fun content, A remark on the Chow ring of some hyperkahler fourfolds, The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, Professor who figures maths prize poser doesn't add up. is an odd number. This is too optimistic, because there are not enough subvarieties to make this work. 149-198, pdf. A motivic reformulation is as follows. This article was adapted from an original article by S.G. Tankeev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. https://encyclopediaofmath.org/index.php?title=Hodge_conjecture&oldid=47239. 3) $ X $

over the field $ \mathbf C $ is a smooth variety (see [2]). The European Mathematical Society, The statement that for any smooth projective variety $ X $ Then it should be algebraic in nearby fibres. is an element of $ H ^ {2p} ( X , \mathbf Q ) \cap F ^ { p } H ^ {2p} ( X , \mathbf C ) $ Deligne’s Clay formulation. fiber sequence/long exact sequence in cohomology. W.V.D. This page was last edited on 5 June 2020, at 22:10. where $ d/2l $ Lecture notes from a summer school on Hodge theory: Claire Voisin, Lectures on the Hodge and Grothendieck-Hodge conjectures, Rend. groupal model for universal principal ∞-bundles, Hodge theory and Complex algebraic geometry, Equivalent descriptions of Hodge conjecture, Generalized Hodge conjecture for triangulated motives, Equivalence between statements of Hodge conjecture, Interesting implications on the theory of Motives if the hodge conjecture holds. for some $ p $, Hodge in [1]. All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. On an Abelian variety, every Hodge class is a Betti component of an absolute Hodge class [a3]. . In the case $ p = 1 $, Let X be a complex Kähler manifold. for $ i \leq p $. The Hodge conjecture regards the algebraicity of the Hodge classes. Let SmProj C corSmProj^{cor}_\mathbf{C} denote the category of algebraic correspondences of smooth projective algebraic varieties over the complex numbers. The Hodge Conjecture The sixth Millennium Problem pretty much sums up why a lot of people don’t like maths. This conjecture was put forth by W.V.D.

Absolute Hodge classes are used to define a weak notion of motif for algebraic varieties. A weaker form is the variational Hodge conjecture. Univ. \oplus _ {r = 0 } ^ { 2p } H ^ {r, 2p - r } , the Hodge conjecture is equivalent to the Lefschetz theorem on cohomology of type $ ( 1, 1) $. is generated by the cohomology classes of algebraic cycles of codimension $ p $ This is an introduction to the Hodge conjecture, which, although intended to a general mathematical audience, assumes some knowledge of topology and complex … Claire Voisin, Hodge theory and Complex algebraic geometry. On a smooth projective complex manifold, every rational Hodge cycle is a linear combination of classes cl(Z)cl(Z) of algebraic cycles. P. Deligne (ed.) (The Tate conjecture is the analogue where C \mathbf{C} is replaced by a finite field or number field, and the target category of pure Hodge structures is replaced by the l-adic representations? The Hodge Conjecture actually uses what are known as complex number coordinates and complex spatial dimensions. Hodge structure). H ^ {2p} ( X, \mathbf Q ) \otimes _ {\mathbf Q } \mathbf C = \ where $ d/l $ is a Hodge substructure such that $ M ^ {i,m-} i = 0 $ Hodge conjecture for Kähler varieties, naive version. It is by far the hardest to explain in any terms, never mind simple ones, it is incredibly far out of reality and everyday experiences and mathematicians can’t agree on what the actual problem is – never mind how to go about trying to find a solution. Uniruled varieties are, for example, the unirational varieties and the four-dimensional complete intersections with an ample anti-canonical class (see [3]). such that $ M \subseteq \mathop{\rm Ker} ( H ^ {m} ( X , \mathbf C ) \rightarrow H ^ {m} ( X \setminus Z , \mathbf C )) $. of the absolute Galois group.). 2) $ X $

Torino, Vol. Specifically, they are the Riemann hypothesis, which lingers from Hilbert's list, Yang-Mills theory and the mass gap hypothesis, the P Versus NP problem, the Navier-Stokes equations, the Poincaire conjecture, the Birch and Swinnerton-Dyer conjecture, and the Hodge conjecture.Devlin profiles each problem and offers insight into how it came about and its significance. The Hodge conjecture has also been proved for the following classes of varieties: 1) $ X $ 69, 2 (2011), pp. is a simple $ d $- where $ Y $ dimensional Abelian variety, and $ \mathop{\rm End} ( X) \otimes _ {\mathbf Z } \mathbf R = \mathbf R ^ {l} $, A Hodge class on a smooth complex projective variety $ X $ This has been verified in certain cases [a1], [a2]. The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. is the component of type $ ( p, p) $ of the absolute Galois group.

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F. Guillán (ed.) over $ X $. So as much as I would love to describe the whole conjecture … A. Conte, J.P. Murre, "The Hodge conjecture for Fano complete intersections of dimension four" , Z. in the Hodge decomposition, $$ space $ H ^ {2p} ( X, \mathbf Q ) \cap H ^ {p,p} $,

to the category of rational pure Hodge structures, given by taking rational Betti cohomology, is fully faithful. arXiv:0907.2503 The Hodge conjecture for self-products of certain K3 surfaces from arXiv Front: math.AG by Ulrich Schlickewei We use a result of van Geemen to determine the endomorphism algebra of the Kuga–Satake variety of a K3 surface with real multiplication. Suppose one has a smooth family of complex projective varieties and a locally constant cohomology class in the fibres which is everywhere a Hodge class and is algebraic at one fibre. Sem. Milne (ed.) 4) $ X $ is a smooth Fermat hypersurface of prime order (see [4], [5]). This is applied to prove the Hodge conjecture for self-products of double covers of PP 2\PP^2 which are ramified along six lines.

is a simple five-dimensional Abelian variety (see [6]). A. Kaplan (ed.) of complex numbers and for any integer $ p \geq 0 $ J.S. (ed.) 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. Add info also on Hodge conjecture: Lewis: A Survey of the Hodge conjecture. See the history of this page for a list of all contributions to it.

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