Claire voisin, hodge theory and complex algebraic geometry. Polynomial automorphisms and the jacobian conjecture. Steven zucker, the hodge conjecture for cubic fourfolds, compositio math. Ii, cambridge studies in advanced mathematics, vol. Due to lefschetz principle it is su cient in this case to deal only with the complex jacobian conjecture if jacf const. Resolution of singularities in characteristic p hodge conjecture standard conjectures on algebraic cycles though these are not so urgent since deligne proved the weil conjectures. Taking the real and imaginary parts of the equations above, we see that the following polynomials in ra 1,a 2,b 1,b 2,c 1,c 2,d 1,d 2 cut out su 2. The absolute anabelian geometry of canonical curves shinichi mochizuki received. A rudins theorem is the assertion that any polynomial injection between affine spaces of the same dimension has a polynomial inverse, and the inverse is also given by polynomials. The authors thank the referee for an alternative proof of the injectivity of f in theorem 1, and we have included this proof as remark 2. It is suitable for graduate students and research mathematicians interested in algebraic geometry. The authors have confined themselves to fundamental concepts and geometrical methods, and do not give detailed developments of geometrical properties, but geometrical meaning has been emphasised throughout. It is a principally polarized abelian variety, of dimension g, and hence, over the complex numbers, it is a complex torus.
Holding a phd 2003 from tohoku university, japan, his main research focuses are on affine algebraic geometry and polynomial ring theory. The absolute anabelian geometry of canonical curves. In mathematics, the jacobian conjecture is a famous unsolved problem on polynomials in several variables. Gang han submitted on 14 nov 2017, last revised 15 nov 2017 this version, v2 abstract. This article is to present the following results that reduces the jacobian conjecture to an existence problem of integral points on af. The jacobian variety is named after carl gustav jacobi, who proved the complete version abeljacobi theorem, making the injectivity statement of niels abel into an isomorphism. If the conjecture is true, it provides a simple way to check whether a given polynomial mapping has an inverse. This is stronger than jacobian conjecture, but i cannot prove the other way round. Based on the results people have obtained, we try to prove the jacobian conjecture, but there is a gap in the proof.
Generalized asymptotic expansions of cornishfisher type hill, g. The jacobian conjecture states that any locally invertible polyno mial system in cn is globally. This has involved line tangents to objects such as spheres, triangles, or line segments, or classifying degenerate con. Abhyankar, expansion techniques in algebraic geometry, tata inst. Let a 1 and a 2 be the real and imaginary parts of a, respectively, and similarly for b,c,d. This book, the first in the field, collects many of the results scattered throughout the literature. Lecture 1 algebraic geometry notes x3 abelian varieties given an algebraic curve x, we saw that we can get a jacobian variety jx. If k r, then equivalent formulations of the real jacobian conjecture are also given, cf. This conjecture was firstly posed by keller in 1939 for kn c, and put in smales 1998 list of mathematical problems for the next century. His technique, of using hecke operators to construct new cycles has been used by several others to give other examples. Algebraic geometry i, complex projective varieties. Motivated by some notorious open problems, such as the jacobian conjecture and the tame generators problem, the subject of polynomial automorphisms has become a rapidly growing field of interest.
His main research interests are in commutative algebra related to affine algebraic geometry. This is an exercise from an invitation to algebraic geometry. Algebraic geometry is a branch of mathematics that combines techniques of abstract algebra with the language and the problems of geometry. In dimension three to prove the real jacobian conjecture it is enough to assume that the set s f is a curve. It has a long history, going back more than a thousand years. Nobuharu onoda is a professor at university of fukui, japan. The property that the jacobian of the xk be a serretate canonical. Algebraic geometry in india 637 m v nori showed that the gri. It is a complex torus so that it has a natural group structure, and it also has the structure of a projective variety. Polynomial rings and affine algebraic geometry praag 2018. Real solutions, applications, and combinatorics frank sottile. Though the problem is inherently algebraic, it crops up in fields throughout mathematics including perturbation theory, quantum field theory and combinatorics.
Jacobian conjecture is an algebraic geometry problem. The jacobian conjecture in two variables is studied. This proceedings volume gathers works presented at the polynomial rings and affine algebraic geometry conference, which was held at tokyo metropolitan university on february 1216, 2018. The jacobian conjecture and injectivity conditions springerlink. Introduction let x, y be a coordinate system in c2.
So, this proof is said an optimization based method. Then f is a polynomial automorphism if and only if the jacobian of f is constant and nonzero. Eisenstein ideals and the rational torsion subgroups of modular jacobian varieties ohta, masami, journal of the mathematical society of japan, 20. It has now been four decades since david mumford wrote that algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and. This book, the first in the field, collects many of. The problem, which resides in the eld of algebraic geometry, is known for the large number of at.
Pdf an optimization approach to jacobian conjecture. Algebraic geometry is fairly easy to describe from the classical viewpoint. Moh 1 introduction the jacobian conjecture in its simplest form is the following. Pdf recent developments of affine algebraic geometry, especially the theory of open. Polynomial rings and affine algebraic geometry praag. This work provides a lucid and rigorous account of the foundations of modern algebraic geometry. Given the simplistic statement, how little is truly understood of that problem is simply shocking, and the first pages of the book really helped me dispel many. Injective polynomial maps and the jacobian conjecture. Proving finite generation of the canonical ring for general type used to be open though i think it was recently solved. A geometric approach to kellers jacobian conjecture. We first give a purely algebraic proof of this reduction result and we then expose a. The jacobian of a positive quartic has 15 such real 2torsion. Unfortunately, many contemporary treatments can be so abstract prime spectra of rings, structure sheaves, schemes, etale.
Proof of two dimensional jacobian conjecture by yucai su 2005. In this section we study algebraic 1tilings of affine space over algebraically closed fields. The book contains research and survey papers discussing recent progress on the jacobian conjecture and affine algebraic geometry and includes a large collection of open problems. It asserts that if is a mapping with and such that for all the real parts of all eigenvalues of are, then each solution of tends to zero if tends to infinity. This paper contains conditions that are equivalent to the jacobian conjecture jc in two variables and partial results toward establishing these conditions. It includes a discussion of the theorems of honda and tate concerning abelian varieties over finite fields and the paper of faltings in which he proves mordells conjecture. It states that if a polynomial function from an ndimensional space to itself has jacobian determinant which is a nonzero constant, then the function has a polynomial inverse. Topics covered include group actions and linearization, automorphism groups. From linear algebra we know that the jacobian conjecture is true if degf1. The problem, which resides in the eld of algebraic geometry, is known for the large number of attempted proofs that were found to contain errors see.
A geometric approach to the jacobian conjecture in c. Kontsevich, the jacobian conjecture is stably equivalent to the dixmier conjecture, mosc. It is no surprising to use algebraic and geometric methods in the proof, but the optimization method is really an extra auxiliary. Other restrictions on counterexamples are also obtained. Pdf lectures on geometry and topology of polynomials. G, tried to give a purely algebraic proof of the jacobian conjecture, but. Expansion techniques in algebraic geometry, tata inst. The jacobian conjecture is also known to hold at least for those.
Also, we state and prove several global univalence theorems and present a couple of applications of them. The jacobian conjecture is closely related to many questions of affine algebraic geometry concerning affine space, such as the cancellation. Jacobian, 402 jacobian matrix, 474 jacobian criterion, 248 jacobson radical, 163 k3 surfaces, 427 kernel, 34. The jacobian conjecture as a problem in combinatorics, in the monograph affine algebraic geometry, in honor of masayoshi miyanishi,edited by takayuki hibi, published by osaka university press 2007 7. Polynomial automorphisms and the jacobian conjecture arno. I work in nonlinear computational geometry, applying ideas from real algebraic geometry and computational algebraic geometry to solve geometric problems, typically in r3. The main theorem also relates to a conjecture of the aforementioned authors, involving the well known jacobian conjecture in algebraic geometry. Among the topics covered by this volume are algebraic curve theory, algebraic surface theory, moduli space, automorphic forms, mordellweil lattices, and automorphisms of hyperkahler manifolds. The jacobian conjecture says that a polynomial mapping p, q.
Apr 18, 2018 one of the aims of this article is to provide a class of polynomial mappings for which the jacobian conjecture is true. The articles in this volume explore a wide variety of problems that illustrate interactions between algebraic geometry and other branches of mathematics. A ne algebraic geometry study of a ne varieties almost always. Moreover, the real jacobian conjecture is true in dimension two see 5, 7. Yamabe in 1960 is the global asymptotic stability jacobian conjecture, also called the markusyamabe conjecture. If p is a point of c, then the curve c can be mapped to a subvariety of j. Jacobian conjecture and semialgebraic maps request pdf. Jacobian conjecture states that for a polynomial map p from kn to itself, where n. C is a polynomial map whose jacobian matrix has determinant 1, then f is an isomorphism 1. An introduction to the jacobian conjecture contents 1. Topological aspects of real algebraic geometry, of which i was the lead organizer.
Though the problem is inherently algebraic, it crops up in. Cubic equations for the hyperelliptic locus grushevsky, samuel, asian journal of mathematics. The jacobian conjecture, a reduction of the degree to the. Polynomial automorphisms and the jacobian conjecture emis. Indeed, the first use of algebraic geometry arises in the proof of lemma. Magnus, on polynomial solutions of a differential equation, math.
It is no surprising to use algebraic and geometric methods in the proof, but the optimization method is really an extra. The jacobian conjecture is a problem that was posed in 1939 by ott heinrich keller see 1. Jacobian conjecture from view points of the diophantine geometry and would be useful in attempting to understand the nature of this conjecture. Algebraic surfaces with vanishing betti numbers serve as a class of test examples for working hypothesis as well for conjectures like cancellation problem or jacobian conjecture, they appear naturally also when studying exotic structures on c n s or c.
On the formal degrees of squareintegrable representations of odd special orthogonal and metaplectic groups ichino, atsushi, lapid, erez, and mao, zhengyu, duke mathematical journal, 2017. The jacobian conjecture is also equivalent to the dixmier conjecture. Algebraic observations on the jacobian conjecture sciencedirect. In the complex dimension 2 case, we apply this method to prove some new results related the jacobian conjecture topics. Find materials for this course in the pages linked along the left. These two structures are in fact compatible with each other. If u and v are a jacobian pair of polynomials in k x, y which provide a counterexample then by a change of variables there is a jacobian pair which generate an ideal of the form. F is connected in zariski or in euclidean topology of. The jacobian conjecture is a longstanding open problem in algebraic geometry. The conjecture is equivalent to if c is the generic curve. Combinatorial approaches to the jacobian conjecture. Starkov is supported by russian science foundation under grant 171101229 and performed in. We show that the jacobian conjecture can be reduced to a weaker conjecture in which all fibers of coordinate functions are irreducible.
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