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Topic: The Hodge Conjecture

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Hodge theory

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Homological conjectures in commutative algebra

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	Hodge conjecture - Wikipedia, the free encyclopedia
		The Hodge conjecture is a major unsolved problem of algebraic geometry.
		It is a conjectural description of the link between the algebraic topology of a non-singular complex algebraic variety, and its geometry as captured by polynomial equations that define sub-varieties.
		Hodge, who between 1930 and 1940 enriched the description of De Rham cohomology to include extra structure which is present in the case of algebraic varieties (though not restricted to that case).
	en.wikipedia.org /wiki/Hodge_conjecture (649 words)

	Hodge conjecture -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		The Hodge conjecture is a major unsolved problem of (Click link for more info and facts about algebraic geometry) algebraic geometry.
		Hodge, who between 1930 and 1940 enriched the description of (Click link for more info and facts about De Rham cohomology) De Rham cohomology to include extra structure which is present in the case of algebraic varieties (though not restricted to that case).
		Suppose V is a non-singular algebraic variety of dimension n over the (A number of the form a+bi where a and b are real numbers and i is the square root of -1) complex numbers.
	www.absoluteastronomy.com /encyclopedia/h/ho/hodge_conjecture.htm (608 words)

	Abstracts (Salman Abdulali)
		In the second paper, the fields of definition of the Hodge cycles are investigated, and it is shown that the relations between the zeta functions hold over the fields of definition of the canonical models.
		We investigate the relation between the usual and general Hodge conjectures, and prove that for a large class of abelian varieties A, the usual Hodge conjecture for all powers of A implies the general Hodge conjecture for A.
		We prove the general Hodge conjecture for any complex abelian variety of CM-type such that the Hodge ring of each power of the abelian variety is generated by divisors.
	personal.ecu.edu /abdulalis/abstracts.html (504 words)

	Clay Mathematics Institute - Wikipedia, the free encyclopedia
		The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.
		A solution to this conjecture has been proposed by Grigori Perelman; while still not formally published, there does appear to be a growing consensus that the argument is largely correct.
		The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions.
	en.wikipedia.org /wiki/Millennium_Prize_Problems (662 words)

	Search Results for conjecture* (Site not responding. Last check: 2007-10-22)
		This conjecture became known as "the main conjecture on cyclotomic fields" and it remained one of the most outstanding conjectures in algebraic number theory until it was solved by Mazur and Wiles in 1984 using modular curves.
		Burnside conjectured that every finite group of odd order is soluble and it is not surprising that he failed to prove this result as it was not proved until 1962 when W Feit and J C Thompson proved the result in a 300 page paper.
		In 1973 it had been conjectured that the behaviour of the logistic equation was the same in a qualitative sense for all g(x) which have a maximum value and decrease monotonically on either side of this maximum.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=conjecture*&CONTEXT=1 (9425 words)

	Research in Algebraic Geometry
		Much of the work in the field of algebraic cycles is organized around three major conjectures: the Hodge conjecture, the Tate conjecture and the generalized Birch-Swinnerton Dyer conjecture.
		A simpler problem would be to describe the cohomology classes of all closed complex submanifolds of M. The Hodge Conjecture suggests a possible answer to this question.
		The Tate conjecture is similar to the Hodge conjecture described above, the difference being that the Tate conjecture is concerned with cohomology classes of algebraic subvarieties of a smooth projective variety M defined over an arbitrary finitely generated field.
	www.math.duke.edu /~schoen/researchalggeo.html (712 words)

	Untitled Document (Site not responding. Last check: 2007-10-22)
		The International Centre for Mathematical Sciences, Edinburgh, held a conference from 20-26 July, to commemorate the centenary of Sir William Hodge, the originator of Hodge theory – ‘one of the landmarks in the history of mathematics in the 20th century’ according to Hermann Weyl.
		Hodge was born in Edinburgh and spent most of his academic career in Cambridge.
		This is the famous Hodge conjecture, one of the Clay millennium prizes for which $1 million is offered.
	www.lms.ac.uk /newsletter/319/319_02.html (989 words)

	College of Natural Sciences - Focus on Science - Spring 2001 (Site not responding. Last check: 2007-10-22)
		Since 1963, this conjecture has remained one of number theory's most famous unsolved mysteries, Number theorists Birch and Swinnerton-Dyer proposed that whether an elliptic curve has infinitely many rational number solutions is related to the behavior of an associated (zeta) function, z(s), at the point s=1.
		Dr. Dan Freed discusses the Hodge Conjecture on April 25.
		"The Hodge Conjecture is a problem of the intersection of geometry and topology, both of which involve the study of generalized shapes, or spaces," said Freed.
	www.utexas.edu /cons/admin/publications/focus/spring01/p18.html (411 words)

	abstract math/9903146 (Site not responding. Last check: 2007-10-22)
		Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface.
		We recall the Hodge conjecture and we point out a connection between the Hodge conjecture for abelian fourfolds and Kuga-Satake varieties.
		We discuss the implications of the Hodge conjecture on the geometry of surfaces whose second cohomology group has a Kuga-Satake variety.
	dragon.ian.pv.cnr.it /~geemen/abs9903146.html (128 words)

	Millennium Math in ZhurnalWiki
		It deals with objects that are so far removed from the intuitions of even the experts that not only is there no "smart money" on whether the conjecture will turnout to be true or false, there isn't even a consensus as to what it really says.
		The Hodge Conjecture illustrates perhaps most clearly of all the Millennium Problems the point I raised in Chapter 0, that the nature of modern mathematics makes much of it all but impossible for the layperson to appreciate.
		In the case of the Hodge Conjecture, the operations of calculus play a major role (differentiation, integration, etc.).
	zhurnal.net /ww/zw?MillenniumMath (670 words)

	RESEARCH
		Abstract : It is shown that if the generalized Hodge conjecture, or some weaker form of it, holds for a Calabi-Yau variety then it holds for any Calabi-Yau variety birationally equivalent to it.
		The key idea is to construct suitable homomorphisms between the Grothendieck group of varieties and the Grothendieck group of the exact category of filtered Hodge structures.
		The generalized Hodge conjecture would imply, rather trivially, that the coniveau filtration is compatible with pushforwards, pullbacks and products.
	www.math.purdue.edu /~sjkang/research.htm (155 words)

	Kuga-Satake Varieties And The Hodge Conjecture - van Geemen (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		The Hodge conjecture is discussed in section 2.
		An excellent survey of the Hodge conjecture for abelian varieties is [G].
		2 An introduction to the Hodge conjecture for abelian varietie..
	citeseer.ist.psu.edu /vangeemen00kugasatake.html (649 words)

	Errata (Site not responding. Last check: 2007-10-22)
		Errata: Abelian varieties and the general Hodge conjecture
		The error is essentially the same as in Proposition 4.4.1 of Abelian varieties and the general Hodge conjecture (see above).
		Correct results and examples may be found in Hodge structures of CM-type, J. Ramanujan Math.
	personal.ecu.edu /abdulalis/Errata.html (198 words)

	Chronology for 1950 to 1960 (Site not responding. Last check: 2007-10-22)
		Hodge puts forward the "Hodge Conjecture" on projective algebraic varieties.
		Serre uses spectral sequences to the study of the relations between the homology groups of fibre, total space and base space in a fibration.
		Taniyama poses his conjecture on elliptic curves which will play a major role in the proof of
	www.gap-system.org /~history/Chronology/1950_1960.html (231 words)

	Homepage of M392C (Site not responding. Last check: 2007-10-22)
		"The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles" (The Clay Mathematics Institute's cryptic description of the Hodge conjecture, one of the seven one-million dollar millenium problems)
		we will study harmonic forms and the Hodge theorem, the Hodge decomposition and the Hard Lefschetz theorem and discuss the Hodge conjecture.
		If time permits we may introduce Deligne's mixed Hodge structures on any algebraic variety or study deformation theory and mirror symmetry or characteristic p methods or some combination of these.
	www.ma.utexas.edu /~hausel/complex (316 words)

	On an elliptic analogue of Zagier's conjecture. - Wildeshaus (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		On an elliptic analogue of Zagier's conjecture (1997)
		Abstract: This article is the elliptic version of "Interpr'etation motivique de la conjecture de Zagier reliant polylogarithmes et r'egulateurs" ([BD]).
		The object of this work is a conjecture which predicts that certain formal linear combinations of elements of the Mordell--Weil group are homologically meaningful, i.e., yield elements in certain K--groups of symmetric powers of elliptic curves.
	citeseer.ist.psu.edu /wildeshaus97elliptic.html (498 words)

	Totaro Abstract WSU Math (Site not responding. Last check: 2007-10-22)
		The Hodge conjecture is often considered the hardest problem in algebraic geometry.
		The conjecture is an attempt to determine which homology classes on a complex algebraic variety can be represented by algebraic subvarieties.
		I will describe some examples where the conjecture can be checked, and some topological ideas which Atiyah and Hirzebruch used to disprove an overly optimistic version of the conjecture.
	www.math.wayne.edu /~claude/abst-totaro.html (100 words)

	abstracthomotopy.html (Site not responding. Last check: 2007-10-22)
		The Hodge conjecture asks whether rational Hodge classes on a smooth projective manifolds are generated by the classes of algebraic subsets, or equivalently by Chern classes of coherent sheaves.
		On a compact Kaehler manifold, Hodge conjecture is known to be false if algebraic subsets are replaced with analytic subsets.
		Here we show that it is even false that for a Kaehler manifold, Hodge classes are generated by Chern classes of coherent sheaves.
	www.math.jussieu.fr /~voisin/abstracthodgekaehler.html (91 words)

	Physics Help and Math Help - Physics Forums - View Single Post - Poincare Conjecture - Solved?
		The Riemann hypothesis not, for sure it must the most investigated by now of the 6.
		It's a pity that I don't understand completely the statement of the Poincare conjecture.
		It surely must be a very beautiful sensation to achieve to understand the prove of the conjecture.
	www.physicsforums.com /showpost.php?p=128422&postcount=4 (157 words)

	Piotr Pragacz Home Page (Site not responding. Last check: 2007-10-22)
		On the Hodge conjecture and a theorem of Nori
		The Noether-Lefschetz locus $NL$ is the subset of $L$ parametrising smooth hypersurfaces of $Y$ which vanishing cohomology has a non-zero Hodge class.
		First, I will give an explicit asymptotic description of the components of small codimension of $NL$ for $L$ sufficiently ample and will show that for these components the Hodge class is in the image of the cycle map, as predicted by Hodge Conjecture.
	www.impan.gov.pl /~pragacz/nori.htm (129 words)

	Publication list (Site not responding. Last check: 2007-10-22)
		Hodge classes on self-products of a variety with an automorphism, Comp.
		Cyclic covers of branched along v+2 hyperplanes and the generalized Hodge conjecture for certain abelian varieties, in Arithmetic of Complex Manifolds, Erlangen 1988, Springer Lecture Notes in Math.
		Addendum to: Hodge classes on self-products of a variety with an automorphism, Compositio Math.
	www.math.duke.edu /~schoen/publicationlist.html (333 words)

	Slice filtration on motives and the Hodge conjecture, by Annette Huber (Site not responding. Last check: 2007-10-22)
		Slice filtration on motives and the Hodge conjecture, by Annette Huber
		We study the slice filtration on the triangulated category of motives via the Hodge realization functor.
		Systematic use of standard conjectures (including the generalized Hodge conjecture) allows to get a conjectural picture of the expected properties of the slice filtration.
	www.math.uiuc.edu /K-theory/0725 (62 words)

	U of R Number Theory seminar (Site not responding. Last check: 2007-10-22)
		Tate's conjecture from arithmetic algebraic geometry is a p-adic analog of the Hodge conjecture.
		I will describe the conjecture for elliptic curves and abelian varieties and mention how modular forms were used by Ribet to solve the conjecture in some special cases, before Faltings gave a general proof.
		Next I will describe the general form of Tate's conjecture for an arbitrary variety, and discuss how modular forms (or automorphic representations) can be used to approach the problem for modular varieties (like modular curves, Hilbert modular surfaces and Picard modular surfaces), and products of modular varieties.
	www.math.rochester.edu /research/algebra_and_number_theory/10.24.03.html (123 words)

	Clay Mathematics Institute
		In some sense it was necessary to add pieces that did not have any geometric interpretation.
		The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles.
		There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.
	www.claymath.org /millennium/Hodge_Conjecture (162 words)

	John Baez's Description of the Millenium Prize Problemsl
		Poincare posed this puzzle in 1904 shortly after he knocked down an easier conjecture of his by finding 3-manifolds with the same homology groups as 3-spheres that weren't really 3-spheres.
		The higher-dimensional analogues of Poincare's question have all been settled in the affirmative - Smale, Stallings and Wallace solved it in dimensions 5 and higher, and Freedman later solved the subtler 4-dimensional case - but the 3-dimensional case is still unsolved.
		The Hodge conjecture, posed in 1950 states: every Hodge form is a rational linear linear combination of algebraic cycles.
	www.math.usf.edu /~eclark/baez2000problems.html (1152 words)

	Transcendental Aspects of Algebraic Cycles - Cambridge University Press (Site not responding. Last check: 2007-10-22)
		The advanced lectures are grouped under three headings: Lawson (co)homology, motives and motivic cohomology and Hodge theoretic invariants of cycles.
		Among the topics treated are: cycle spaces, Chow topology, morphic cohomology, Grothendieck motives, Chow-Künneth decompositions of the diagonal, motivic cohomology via higher Chow groups, the Hodge conjecture for certain fourfolds, an effective version of Nori’s connectivity theorem, Beilinson's Hodge and Tate conjecture for open complete intersections.
		Three lectures on the Hodge conjecture J. Lewis; 7.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521545471 (297 words)

	Half Twists and the Cohomology of Hypersurfaces (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		Abstract: A Hodge structure V of weight k on which a CM eld acts de nes, under certain conditions, a Hodge structure of weight k 1, its half twist.
		In this paper we consider hypersurfaces in projective space with a cyclic automorphism which de nes an action of a cyclotomic eld on a Hodge substructure in the cohomology.
		We determine when the half twist exists and relate it to the geometry and moduli of the hypersurfaces.
	citeseer.ist.psu.edu /473376.html (363 words)

	Résumé
		The class map on rational algebraic cycles is split into the composition of the Abel–Jacobi map and another map which we prove is surjective.
		The conjecture is thus reduced to the surjectivity of the Abel–Jacobi map.
		Colwell, J., The Conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication by a nonmaximal order.
	math.ucsd.edu /~jcolwell/cv.html (778 words)

	Absolute Hodge cycles (Site not responding. Last check: 2007-10-22)
		If one is to believe the Hodge conjecture then any integral class of type
		Even this (weak) consequence of the Hodge conjecture is not known and led to the definition (see [
		The latter is an infinitesimal analogue of the notion of absolute Hodge cycles and should prove easier to study in relation with the Hodge conjecture.
	www.imsc.res.in /~kapil/work/node10.html (195 words)

	Read This: What's Happening in the Mathematical Sciences, Volume 5
		Having just introduced the concepts of manifold and higher dimensions in the discussion of the Poincaré conjecture, Cipra writes "The Hodge conjecture concerns the analysis of high-dimensional manifolds defined by systems of algebraic equations.
		It says, very roughly, that everything you always wanted to know about algebraically defined manifolds (but were afraid to ask) is to be found in the theory of calculus." These two sentences are a good start towards capturing the general nature of the Hodge conjecture.
		The first article traces the grand epic which starts with Taniyama's conjecture in 1955 that every elliptic curve is modular, goes through the proof of Fermat's last theorem, and continues to this day in the framework of the Langlands program.
	www.maa.org /reviews/whatshap5.html (923 words)

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