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Guseva, and more recently by my student R. I will present a reformulation of the conjecture in terms of the behavior of a real singular foliation. Next, I will present a recent work in collaboration with A. Figalli, L. Rifford and A. Parusinski, where we show that the strong version of the conjecture holds in the analytic category and in dimension 3.

Gal Binyamini - Point counting for foliations over number fields. Using this theory we prove the Wilkie conjecture for sets defined using leafs of foliations under a certain assumption about the algebraicity locus. This statement and its generalizations have many applications in diophantine geometry following the Pila-Zannier strategy.

I will focus mostly on the proof of the main statement, which uses a combination of differential-algebraic methods related to foliations with some ideas from complex geometry and value distribution theory. If time permits I will briefly discuss the applications to counting algebraic points and diophantine geometry at the end. Jean-Pierre Demailly - Existence of logarithmic and orbifold jet differentials - Notes. Abstract - Given a projective algebraic orbifold, one can define associated logarithmic and orbifold jet bundles. These bundles describe the algebraic differential operators that act on germs of curves satisfying ad hoc ramification conditions.

Holomorphic Morse inequalities can be used to derive precise cohomology estimates and, in particular, lower bounds for the dimensions of spaces of global jet differentials. A striking consequence is that, under suitable geometric hypotheses, the corresponding entire curves must satisfy nontrivial algebraic differential equations.

The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics.

It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory , and other 20th-century techniques not available to Fermat. An important property of the ring of integers is that it satisfies the fundamental theorem of arithmetic , that every positive integer has a factorization into a product of prime numbers , and this factorization is unique up to the ordering of the factors. This may no longer be true in the ring of integers O of an algebraic number field K.

A prime element is an element p of O such that if p divides a product ab , then it divides one of the factors a or b. This property is closely related to primality in the integers, because any positive integer satisfying this property is either 1 or a prime number. However, it is strictly weaker. If factorizations into prime elements are permitted, then, even in the integers, there are alternative factorizations such as. In general, if u is a unit , meaning a number with a multiplicative inverse in O , and if p is a prime element, then up is also a prime element.

Numbers such as p and up are said to be associate. Requiring that prime numbers be positive selects a unique element from among a set of associated prime elements. When K is not the rational numbers, however, there is no analog of positivity. This leads to equations such as. For this reason, one adopts the definition of unique factorization used in unique factorization domains UFDs.

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In a UFD, the prime elements occurring in a factorization are only expected to be unique up to units and their ordering. However, even with this weaker definition, many rings of integers in algebraic number fields do not admit unique factorization. There is an algebraic obstruction called the ideal class group.

When the ideal class group is trivial, the ring is a UFD. When it is not, there is a distinction between a prime element and an irreducible element.

These are the elements that cannot be factored any further. Every element in O admits a factorization into irreducible elements, but it may admit more than one. This is because, while all prime elements are irreducible, some irreducible elements may not be prime. This means that the number 9 has two factorizations into irreducible elements,. Unlike the situation with units, where uniqueness could be repaired by weakening the definition, overcoming this failure requires a new perspective.

If I is an ideal in O , then there is always a factorization. In particular, this is true if I is the principal ideal generated by a single element. This is the strongest sense in which the ring of integers of a general number field admits unique factorization. In the language of ring theory, it says that rings of integers are Dedekind domains.

When O is a UFD, every prime ideal is generated by a prime element. Otherwise, there are prime ideals which are not generated by prime elements. Historically, the idea of factoring ideals into prime ideals was preceded by Ernst Kummer's introduction of ideal numbers. These are numbers lying in an extension field E of K. This extension field is now known as the Hilbert class field. By the principal ideal theorem , every prime ideal of O generates a principal ideal of the ring of integers of E. A generator of this principal ideal is called an ideal number. Kummer used these as a substitute for the failure of unique factorization in cyclotomic fields.

These eventually led Richard Dedekind to introduce a forerunner of ideals and to prove unique factorization of ideals. An ideal which is prime in the ring of integers in one number field may fail to be prime when extended to a larger number field.

Algebraic methods in the global theory of complex spaces | UNIVERSITY OF NAIROBI LIBRARY

Consider, for example, the prime numbers. The corresponding ideals p Z are prime ideals of the ring Z. However, when this ideal is extended to the Gaussian integers to get p Z [ i ] , it may or may not be prime. A complete answer to the question of which ideals remain prime in the Gaussian integers is provided by Fermat's theorem on sums of two squares.

Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory. Class field theory accomplishes this goal when K is an abelian extension of Q i. Unique factorization fails if and only if there are prime ideals that fail to be principal. The object which measures the failure of prime ideals to be principal is called the ideal class group.

Defining the ideal class group requires enlarging the set of ideals in a ring of algebraic integers so that they admit a group structure. This is done by generalizing ideals to fractional ideals.

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All ideals of O are also fractional ideals. If I and J are fractional ideals, then the set IJ of all products of an element in I and an element in J is also a fractional ideal. This operation makes the set of non-zero fractional ideals into a group. The quotient of the group of non-zero fractional ideals by this subgroup is the ideal class group. Therefore, the ideal class group makes two fractional ideals equivalent if one is as close to being principal as the other is. The ideal class group is generally denoted Cl K , Cl O , or Pic O with the last notation identifying it with the Picard group in algebraic geometry.

The number of elements in the class group is called the class number of K. The ideal class group has another description in terms of divisors. These are formal objects which represent possible factorizations of numbers. The divisor group Div K is defined to be the free abelian group generated by the prime ideals of O.

The kernel of div is the group of units in O , while the cokernel is the ideal class group. In the language of homological algebra , this says that there is an exact sequence of abelian groups written multiplicatively ,. These are called real embeddings and complex embeddings , respectively.

Principal stress is laid on the study of the properties of the algebra of convergent power series in variables over and its quotient algebras — the so-called analytic algebras, the foundations of which were laid by K. The local theory comprises the theory of normalization, the study of singular points, local properties of analytic functions and mappings, etc. The most important results obtained in this field refer to the case of algebraically closed fields [1] , [4] , [7]. There appears the important concept of a coherent analytic sheaf , which continues to play a leading part in the global theory.

In particular, the structure sheaf of the analytic space and the sheaf of ideals of any analytic set are coherent for any algebraically closed. The case has also been thoroughly studied. Global analytic geometry studies the properties of analytic functions, mappings and other analytic objects, defined "globally" on the entire analytic space, as well as the geometrical properties of these spaces. In the process of studying complex-analytic spaces natural classes of them were isolated. These include, first, the class of Stein spaces cf. Stein space , which can be roughly described as the class of spaces with a sufficiently large amount of global holomorphic functions.

Stein spaces are the most natural multi-dimensional generalizations of the domains of the complex plane considered in the classical theory of functions of one complex variable. This class of spaces in fact coincides with the class of analytic subspaces of the spaces. Its algebraic analogue is the class of affine algebraic varieties cf. Affine variety. For a domain holomorphic completeness is equivalent with the fact that is a domain of holomorphy , i.

The boundary of a domain of holomorphy is pseudo-convex, i. The problem of the validity of the converse theorem cf. Levi problem gave rise to a number of investigations and yielded a new characterization of Stein spaces. The class of compact spaces is, in a certain sense, the opposite case. The following generalization of the classical theorem of Liouville is valid: Functions which are holomorphic on a reduced compact space are constant on each connected component of this space and therefore form a finite-dimensional vector space. A generalization of this theorem are the finiteness theorems , which confirm the finite dimensionality of the homology groups with values in a coherent analytic sheaf.

Holomorphically-convex complex spaces, -complete, -pseudo-convex, -pseudo-concave spaces, which are generalizations of Stein spaces, and compact spaces are also considered cf.

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Holomorphically-convex complex space. These classes of complex spaces have their analogues in the theory of holomorphic mappings. Thus, to compact spaces correspond proper holomorphic mappings; to holomorphically complete spaces correspond Stein mappings, etc. The corresponding generalization of finiteness theorems are theorems of coherence of direct images of coherent analytic sheaves under holomorphic mappings, the first and most important one of which for proper mappings was demonstrated by H.

Grauert [6a]. An important role in the theory of complex spaces is played by holomorphic mappings of a special kind — the so-called modifications cf. Modification , i. One says that is obtained from by "contracting" the subset on , while is obtained from by "blowing up" the subset into. Of special interest are analytic subsets that can be contracted into a point exceptional analytic sets ; these were characterized by H. Grauert [6b]. A natural problem in analytic geometry is the following problem of resolution of singularities: Is it possible to "blow up" an analytic space so that the entire space becomes smooth?

It should be noted that modifications in algebraic geometry were studied as early as the 19th century, while modifications in analytic geometry were introduced by Behnke and Stein in in the context of the concept of a Riemannian domain. Another natural object of study, which is also closely connected with the ideas of algebraic geometry, are meromorphic functions on complex spaces and their generalizations — meromorphic mappings a mapping which yields an operation inverse to a modification may serve as an example; cf.

Meromorphic function ; Meromorphic mapping. Meromorphic functions on a reduced compact complex space form a field of transcendence degree this was first demonstrated by C. Siegel in for the smooth case. Spaces for which Moishezon spaces form a class which is very close to the class of projective algebraic varieties; they are characterized by the fact that they are modifications of smooth projective algebraic varieties.

A number of criteria for the projectivity of a compact complex space are known [3] , [6b] , [13]. Studies of automorphic functions in several complex variables have made a major contribution to the development of this subject. The theory of deformations of analytic structures cf. Deformation is concerned with the problem of classification of analytic objects of a given type e. In the former case one speaks of the global moduli problem, while in the latter one speaks of the local moduli problem. An example of the global moduli problem is the problem of the classification of all complex structures on a compact Riemann surface cf.

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