Cycle Spaces and Intersection Theory Eric M. Friedlander
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Algebraic Cycles from a Computational Point of View
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 392 (2008) 128–140 www.elsevier.com/locate/tcs Algebraic cycles from a computational point of view Carlos Simpson CNRS, Laboratoire J. A. Dieudonne´ UMR 6621, Universite´ de Nice-Sophia Antipolis, 06108 Nice, France Abstract The Hodge conjecture implies decidability of the question whether a given topological cycle on a smooth projective variety over the field of algebraic complex numbers can be represented by an algebraic cycle. We discuss some details concerning this observation, and then propose that it suggests going on to actually implement an algorithmic search for algebraic representatives of classes which are known to be Hodge classes. c 2007 Elsevier B.V. All rights reserved. Keywords: Algebraic cycle; Hodge cycle; Decidability; Computation 1. Introduction Consider the question of representability of a topological cycle by an algebraic cycle on a smooth complex projective variety. In this paper we point out that the Hodge conjecture would imply that this question is decidable for varieties defined over Q. This is intuitively well-known to specialists in algebraic cycles. It is interesting in the context of the present discussion, because on the one hand it shows a relationship between an important question in algebraic geometry and the logic of computation, and on the other hand it leads to the idea of implementing a computer search for algebraic cycles as we discuss in the last section. This kind of consideration is classical in many areas of geometry. For example, the question of deciding whether a curve defined over Z has any Z-valued points was Hilbert’s tenth problem, shown to be undecidable in [10,51, 39]. -
On Classification Problem of Loday Algebras 227
Contemporary Mathematics Volume 672, 2016 http://dx.doi.org/10.1090/conm/672/13471 On classification problem of Loday algebras I. S. Rakhimov Abstract. This is a survey paper on classification problems of some classes of algebras introduced by Loday around 1990s. In the paper the author intends to review the latest results on classification problem of Loday algebras, achieve- ments have been made up to date, approaches and methods implemented. 1. Introduction It is well known that any associative algebra gives rise to a Lie algebra, with bracket [x, y]:=xy − yx. In 1990s J.-L. Loday introduced a non-antisymmetric version of Lie algebras, whose bracket satisfies the Leibniz identity [[x, y],z]=[[x, z],y]+[x, [y, z]] and therefore they have been called Leibniz algebras. The Leibniz identity combined with antisymmetry, is a variation of the Jacobi identity, hence Lie algebras are antisymmetric Leibniz algebras. The Leibniz algebras are characterized by the property that the multiplication (called a bracket) from the right is a derivation but the bracket no longer is skew-symmetric as for Lie algebras. Further Loday looked for a counterpart of the associative algebras for the Leibniz algebras. The idea is to start with two distinct operations for the products xy, yx, and to consider a vector space D (called an associative dialgebra) endowed by two binary multiplications and satisfying certain “associativity conditions”. The conditions provide the relation mentioned above replacing the Lie algebra and the associative algebra by the Leibniz algebra and the associative dialgebra, respectively. Thus, if (D, , ) is an associative dialgebra, then (D, [x, y]=x y − y x) is a Leibniz algebra. -
Hodge-Theoretic Invariants for Algebraic Cycles
HODGE-THEORETIC INVARIANTS FOR ALGEBRAIC CYCLES MARK GREEN∗ AND PHILLIP GRIFFITHS Abstract. In this paper we use Hodge theory to define a filtration on the Chow groups of a smooth, projective algebraic variety. Assuming the gen- eralized Hodge conjecture and a conjecture of Bloch-Beilinson, we show that this filtration terminates at the codimension of the algebraic cycle class, thus providing a complete set of period-type invariants for a rational equivalence class of algebraic cycles. Outline (1) Introduction (2) Spreads; explanation of the idea p (3) Construction of the filtration on CH (X)Q (4) Interpretations and proofs (5) Remarks and examples (6) Appendix: Reformation of the construction 1. Introduction Some years ago, inspired by earlier work of Bloch, Beilinson (cf. [R]) proposed a series of conjectures whose affirmative resolution would have far reaching conse- quences on our understanding of the Chow groups of a smooth projective algebraic variety X. For any abelian group G, denoting by GQ the image of G in G⊗Z Q, these P conjectures would have the following implications for the Chow group CH (X)Q: (I) There is a filtration p 0 p 1 p (1.1) CH (X)Q = F CH (X)Q ⊃ F CH (X)Q p p p+1 p ⊃ · · · ⊃ F CH (X)Q ⊃ F CH (X)Q = 0 whose successive quotients m p m p m+1 p (1.2) Gr CH (X)Q = F CH (X)Q=F CH (X)Q may be described Hodge-theoretically.1 The first two steps in the conjectural filtration (??) are defined classically: If p 2p 0 : CH (X)Q ! H (X; Q) is the cycle class map, then 1 p F CH (X)Q = ker 0 : Setting in general m p p m p F CH (X) = CH (X) \ F CH (X)Q ; ∗Research partially supported by NSF grant DMS 9970307. -
Rigidity of Some Classes of Lie Algebras in Connection to Leibniz
Rigidity of some classes of Lie algebras in connection to Leibniz algebras Abdulafeez O. Abdulkareem1, Isamiddin S. Rakhimov2, and Sharifah K. Said Husain3 1,3Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia. 1,2,3Laboratory of Cryptography,Analysis and Structure, Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia. Abstract In this paper we focus on algebraic aspects of contractions of Lie and Leibniz algebras. The rigidity of algebras plays an important role in the study of their varieties. The rigid algebras generate the irreducible components of this variety. We deal with Leibniz algebras which are generalizations of Lie algebras. In Lie algebras case, there are different kind of rigidities (rigidity, absolutely rigidity, geometric rigidity and e.c.t.). We explore the relations of these rigidities with Leibniz algebra rigidity. Necessary conditions for a Lie algebra to be Leibniz rigid are discussed. Keywords: Rigid algebras, Contraction and Degeneration, Degeneration invariants, Zariski closure. 1 Introduction In 1951, Segal I.E. [11] introduced the notion of contractions of Lie algebras on physical grounds: if two physical theories (like relativistic and classical mechanics) are related by a limiting process, then the associated invariance groups (like the Poincar´eand Galilean groups) should also be related by some limiting process. If the velocity of light is assumed to go to infinity, relativistic mechanics ”transforms” into classical mechanics. This also induces a singular transition from the Poincar´ealgebra to the Galilean one. Another ex- ample is a limiting process from quantum mechanics to classical mechanics under ~ → 0, arXiv:1708.00330v1 [math.RA] 30 Jul 2017 that corresponds to the contraction of the Heisenberg algebras to the abelian ones of the same dimensions [2]. -
Algebraic Cycles on Generic Abelian Varieties Compositio Mathematica, Tome 100, No 1 (1996), P
COMPOSITIO MATHEMATICA NAJMUDDIN FAKHRUDDIN Algebraic cycles on generic abelian varieties Compositio Mathematica, tome 100, no 1 (1996), p. 101-119 <http://www.numdam.org/item?id=CM_1996__100_1_101_0> © Foundation Compositio Mathematica, 1996, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 100: 101-119,1996. 101 © 1996 KluwerAcademic Publishers. Printed in the Netherlands. Algebraic cycles on generic Abelian varieties NAJMUDDIN FAKHRUDDIN Department of Mathematics, the University of Chicago, Chicago, Illinois, USA Received 9 September 1994; accepted in final form 2 May 1995 Abstract. We formulate a conjecture about the Chow groups of generic Abelian varieties and prove it in a few cases. 1. Introduction In this paper we study the rational Chow groups of generic abelian varieties. More precisely we try to answer the following question: For which integers d do there exist "interesting" cycles of codimension d on the generic abelian variety of dimension g? By "interesting" cycles we mean cycles which are not in the subring of the Chow ring generated by divisors or cycles which are homologically equivalent to zero but not algebraically equivalent to zero. As background we recall that G. Ceresa [5] has shown that for the generic abelian variety of dimension three there exist codimen- sion two cycles which are homologically equivalent to zero but not algebraically equivalent to zero. -
Deformations of Bimodule Problems
FUNDAMENTA MATHEMATICAE 150 (1996) Deformations of bimodule problems by Christof G e i ß (M´exico,D.F.) Abstract. We prove that deformations of tame Krull–Schmidt bimodule problems with trivial differential are again tame. Here we understand deformations via the structure constants of the projective realizations which may be considered as elements of a suitable variety. We also present some applications to the representation theory of vector space categories which are a special case of the above bimodule problems. 1. Introduction. Let k be an algebraically closed field. Consider the variety algV (k) of associative unitary k-algebra structures on a vector space V together with the operation of GlV (k) by transport of structure. In this context we say that an algebra Λ1 is a deformation of the algebra Λ0 if the corresponding structures λ1, λ0 are elements of algV (k) and λ0 lies in the closure of the GlV (k)-orbit of λ1. In [11] it was shown, us- ing Drozd’s tame-wild theorem, that deformations of tame algebras are tame. Similar results may be expected for other classes of problems where Drozd’s theorem is valid. In this paper we address the case of bimodule problems with trivial differential (in the sense of [4]); here we interpret defor- mations via the structure constants of the respective projective realizations. Note that the bimodule problems include as special cases the representa- tion theory of finite-dimensional algebras ([4, 2.2]), subspace problems (4.1) and prinjective modules ([16, 1]). We also discuss some examples concerning subspace problems. -
For a = Zg(O). (Iii) the Morphism (100) Is Autk-Equivariant. We
HITCHIN’S INTEGRABLE SYSTEM 101 (ii) The composition zg(K) → Z → zg(O) is the morphism (97) for A = zg(O). (iii) The morphism (100) is Aut K-equivariant. We will not prove this theorem. In fact, the only nontrivial statement is that (99) (or equivalently (34)) is a ring homomorphism; see ???for a proof. The natural approach to the above theorem is based on the notion of VOA (i.e., vertex operator algebra) or its geometric version introduced in [BD] under the name of chiral algebra.21 In the next subsection (which can be skipped by the reader) we outline the chiral algebra approach. 3.7.6. A chiral algebra on a smooth curve X is a (left) DX -module A equipped with a morphism ! (101) j∗j (A A) → ∆∗A where ∆ : X→ X × X is the diagonal, j :(X × X) \ ∆(X) → X. The morphism (101) should satisfy certain axioms, which will not be stated here. A chiral algebra is said to be commutative if (101) maps A A to 0. Then (101) induces a morphism ∆∗(A⊗A)=j∗j!(A A)/A A→∆∗A or, which is the same, a morphism (102) A⊗A→A. In this case the chiral algebra axioms just mean that A equipped with the operation (102) is a commutative associative unital algebra. So a commutative chiral algebra is the same as a commutative associative unital D D X -algebra in the sense of 2.6. On the other hand, the X -module VacX corresponding to the Aut O-module Vac by 2.6.5 has a natural structure of → chiral algebra (see the Remark below). -
JOINT CNU-USTC-SUST SEMINAR on P-ADIC DEFORMATION of ALGEBRAIC CYCLE CLASSES AFTER BLOCH-ESNAULT-KERZ
JOINT CNU-USTC-SUST SEMINAR ON p-ADIC DEFORMATION OF ALGEBRAIC CYCLE CLASSES AFTER BLOCH-ESNAULT-KERZ 1. Introduction Our joint seminar program is to understand the paper [4] that study the p-adic variation of Hodge conjecture of Fontaine-Messing ([6], see also [4, Conjecture 1.2]) which states roughly as follows: For simplicity, let X=W be a smooth pro- jective scheme, with W = W (k) the Witt ring of a perfect field k of characteristic 1 p > 0. If the crystalline cycle class of an algebraic cycle Z1 in the special fiber X1 = X × k has the correct Hodge level under the canonical isomorphism of Berthelot-Ogus ∗ ∼ ∗ (1.0.1) Hcris(X1=W ) = HdR(X=W ); then it can be lifted to an algebraic cycle Z of X. This problem is known when the dimension of an cycle is zero dimensional (trivial by smoothness) and codi- mension one case (Betherlot-Ogus [2, Theorem 3.8]). The article [4] proved that, under some technical condition arising from integral p-adic Hodge theory, once the Hodge level is correct, the cycle can be formally lifted to the p-adic formal scheme X = \ lim "X ⊗ W=pn of X, and it remains to be an open problem that · −!n whether there exists some formal lifting which is algebraic. The p-adic variation of Hodge conjecture is a p-adic analogue of the infinites- imal variation of Hodge conjecture ([4, Conjecture 1.1]): in this conjecture the base ring is the formal disk, i.e., the spectrum of k[[t]] where k is a character- istic zero field, and the Berthelot-Ogus isomorphism (1.0.1) is replaced by the trivialization of Gauß-Manin connection over the formal disk ([13, Proposition 8.9]) ∗ ∼ ∗ (1.0.2) (HdR(X1=k) ⊗ k[[t]]; id ⊗ d) = (HdR(X=k[[t]]); rGM): It can be shown that the infinitesimal variation of Hodge conjecture is equivalent to the original version of Grothendieck on variation of Hodge conjecture ([11], page 103 footnote), which is indeed a direct consequence of Hodge conjecture after Deligne's global invariant cycle theorem. -
A Computable Functor from Graphs to Fields
A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation MILLER, RUSSELL et al. “A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS.” The Journal of Symbolic Logic 83, 1 (March 2018): 326–348 © 2018 The Association for Symbolic Logic As Published http://dx.doi.org/10.1017/jsl.2017.50 Publisher Cambridge University Press (CUP) Version Original manuscript Citable link http://hdl.handle.net/1721.1/116051 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS RUSSELL MILLER, BJORN POONEN, HANS SCHOUTENS, AND ALEXANDRA SHLAPENTOKH Abstract. We construct a fully faithful functor from the category of graphs to the category of fields. Using this functor, we resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure , there exists a countable field with the same essential computable-model-theoretic propeSrties as . Along the way, we developF a new “computable category theory”, and prove that our functorS and its partially- defined inverse (restricted to the categories of countable graphs and countable fields) are computable functors. 1. Introduction 1.A. A functor from graphs to fields. Let Graphs be the category of symmetric irreflex- ive graphs in which morphisms are isomorphisms onto induced subgraphs (see Section 2.A). Let Fields be the category of fields, with field homomorphisms as the morphisms. Using arithmetic geometry, we will prove the following: Theorem 1.1. -
Voevodsky's Univalent Foundation of Mathematics
Voevodsky's Univalent Foundation of Mathematics Thierry Coquand Bonn, May 15, 2018 Voevodsky's Univalent Foundation of Mathematics Univalent Foundations Voevodsky's program to express mathematics in type theory instead of set theory 1 Voevodsky's Univalent Foundation of Mathematics Univalent Foundations Voevodsky \had a special talent for bringing techniques from homotopy theory to bear on concrete problems in other fields” 1996: proof of the Milnor Conjecture 2011: proof of the Bloch-Kato conjecture He founded the field of motivic homotopy theory In memoriam: Vladimir Voevodsky (1966-2017) Dan Grayson (BSL, to appear) 2 Voevodsky's Univalent Foundation of Mathematics Univalent Foundations What does \univalent" mean? Russian word used as a translation of \faithful" \These foundations seem to be faithful to the way in which I think about mathematical objects in my head" (from a Talk at IHP, Paris 2014) 3 Voevodsky's Univalent Foundation of Mathematics Univalent foundations Started in 2002 to look into formalization of mathematics Notes on homotopy λ-calculus, March 2006 Notes for a talk at Standford (available at V. Voevodsky github repository) 4 Voevodsky's Univalent Foundation of Mathematics Univalent Foundations hh nowadays it is known to be possible, logically speaking, to derive practically the whole of known mathematics from a single source, the Theory of Sets ::: By so doing we do not claim to legislate for all time. It may happen at some future date that mathematicians will agree to use modes of reasoning which cannot be formalized in the language described here; according to some, the recent evolutation of axiomatic homology theory would be a sign that this date is not so far. -
Arxiv:1611.02437V2 [Math.CT] 2 Jan 2017 Betseiso Structures
UNIFIED FUNCTORIAL SIGNAL REPRESENTATION II: CATEGORY ACTION, BASE HIERARCHY, GEOMETRIES AS BASE STRUCTURED CATEGORIES SALIL SAMANT AND SHIV DUTT JOSHI Abstract. In this paper we propose and study few applications of the base ⋊ ¯ ⋊ F¯ structured categories X F C, RC F, X F C and RC . First we show classic ¯ transformation groupoid X//G simply being a base-structured category RG F . Then using permutation action on a finite set, we introduce the notion of a hierarchy of ⋊ ⋊ ⋊ base structured categories [(X2a F2a B2a) ∐ (X2b F2b B2b) ∐ ...] F1 B1 that models local and global structures as a special case of composite Grothendieck fibration. Further utilizing the existing notion of transformation double category (X1 ⋊F1 B1)//2G, we demonstrate that a hierarchy of bases naturally leads one from 2-groups to n-category theory. Finally we prove that every classic Klein geometry F ∞ U is the Grothendieck completion (G = X ⋊F H) of F : H −→ Man −→ Set. This is generalized to propose a set-theoretic definition of a groupoid geometry (G, B) (originally conceived by Ehresmann through transport and later by Leyton using transfer) with a principal groupoid G = X ⋊ B and geometry space X = G/B; F which is essentially same as G = X ⋊F B or precisely the completion of F : B −→ ∞ U Man −→ Set. 1. Introduction The notion of base structured categories characterizing a functor were introduced in the preceding part [26] of the sequel. Here the perspective of a functor as an action of a category is considered by using the elementary structure of symmetry on objects especially the permutation action on a finite set. -
Ringel-Hall Algebras Andrew W. Hubery
Ringel-Hall Algebras Andrew W. Hubery 1. INTRODUCTION 3 1. Introduction Two of the first results learned in a linear algebra course concern finding normal forms for linear maps between vector spaces. We may consider a linear map f : V → W between vector spaces over some field k. There exist isomorphisms V =∼ Ker(f) ⊕ Im(f) and W =∼ Im(f) ⊕ Coker(f), and with respect to these decompositions f = id ⊕ 0. If we consider a linear map f : V → V with V finite dimensional, then we can express f as a direct sum of Jordan blocks, and each Jordan block is determined by a monic irreducible polynomial in k[T ] together with a positive integer. We can generalise such problems to arbitrary configurations of vector spaces and linear maps. For example f g f g U1 −→ V ←− U2 or U −→ V −→ W. We represent such problems diagrammatically by drawing a dot for each vector space and an arrow for each linear map. So, the four problems listed above corre- spond to the four diagrams A: B : C : D : Such a diagram is called a quiver, and a configuration of vector spaces and linear maps is called a representation of the quiver: that is, we take a vector space for each vertex and a linear map for each arrow. Given two representations of the same quiver, we define the direct sum to be the representation produced by taking the direct sums of the vector spaces and the linear maps for each vertex and each arrow. Recall that if f : U → V and f 0 : U 0 → V 0 are linear maps, then the direct sum is the linear map f ⊕ f 0 : U ⊕ U 0 → V ⊕ V 0, (u,u0) 7→ (f(u), f 0(u0)).