Higher Dimensional Varieties Research Articles

On the differential properties of algebraic morphisms into Grassmannians

In the thesis [Pe1] it was introduced, studied and applied a general theory of Weierstrass loci for vector bundles on a smooth curve. The results, proofs, background, examples and motivations of this thesis are contained in [Pe2]. We believe that this theory, at least in characteristic 0, is the ‘right’ one. The aim of this paper is to introduce and study an extension of [Pe1] to the case of higher dimensional varieties. At least two possible theories seem to be useful and natural; see the discussion just after Remark 1.1 and Section 4. We strongly prefer the ‘symmetric’ one (see Definition 1.5). In the first section we introduce the general theory and give the main general results. In the second section we study in details the case of ). Here, except at the first step we are able to work only with the ‘symmetric’ definition.

Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry

We propose a new higher dimensional version of the McKay correspondence which enables us to understand the “Hodge numbers” assigned to singular Gorenstein varieties by physicists. Our results lead to the conjecture that string theory indicates the existence of some new cohomology theory H st ∗(X) for algebraic varieties with Gorenstein singularities. We give a formal mathematical definition of the Hodge numbers h st p, q ( X) inspired from the consideration of strings on orbifolds and from this new (conjectural) version of the McKay correspondence. The numbers h st p, q ( X) are expected to give the spectrum of orbifoldized Landau-Ginzburg models and mirror duality relations for higher dimensional Calabi-Yau varieties with Gorenstein toroidal or quotient singularities.

Discrete symmetry groups of vertex models in statistical mechanics

We analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations. They can be seen as generated by involutions corresponding respectively to two kinds of transformations onq×q matrices: the inversion of theq×q matrix and an (involutive) permutation of the entries of the matrix. We show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics. This approach enables one to analyze two-dimensionalq4-state vertex models as simply as three-dimensional vertex models, or higher-dimensional vertex models. Various examples of birational symmetries of vertex models are analyzed. A particular emphasis is devoted to a three-dimensional vertex model, the 64-state cubic vertex model, which exhibits a polynomial growth of the complexity of the calculations. A subcase of this general model is seen to yield integrable recursion relations. We also concentrate on a specific two-dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes Yang-Baxter integrable. It is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces, or higher-dimensional algebraic varieties.

From integrability to weak chaos

We analyze birational transformations obtained from very simple algebraic calculations, namely taking the inverse of q × q matrices and permuting some of the entries of these matrices. We concentrate on 4 × 4 matrices and elementary transpositions of two entries. This analysis brings out six classes of birational transformations. Three classes correspond to integrable mappings, their iteration yielding elliptic curves. The iterations corresponding to the three other classes are included in higher dimensional non-trivial algebraic varieties. For many initial conditions in the parameter space these orbits lie on (transcendental) curves, and finally explode in these higher dimensional varieties. These transformations act on fifteen (or q 2−1) variables, however one can associate to them remarkably simple non-linear recurrences bearing on a single variable. The study of these last recurrences gives a complementary understanding of these amazingly regular non-integrable mappings.

ALMOST INTEGRABLE MAPPINGS

We analyze birational transformations obtained from very simple algebraic calculations, namely taking the inverse of q × q matrices and permuting some of the entries of these matrices. We concentrate on 4 × 4 matrices and elementary transpositions of two entries. This analysis brings out six classes of birational transformations. Three classes correspond to integrable mappings, their iteration yielding elliptic curves. Generically, the iterations corresponding to the three other classes are included in higher dimensional non-trivial algebraic varieties. Nevertheless some orbits of the parameter space lie on (transcendental) curves. These transformations act on fifteen (or q2 − 1) variables, however one can associate to them remarkably simple non-linear recurrences bearing on a single variable. The study of these last recurrences gives a complementary understanding of these amazingly regular non-integrable mappings, which could provide interesting tools to analyze weak chaos.

Global generation of pluricanonical and adjoint linear series on smooth projective threefolds

The purpose of this paper is to show how the cohomological techniques developed by Kawamata, Reid, Shokurov, and others lead to some effective and practical results of Reider-type on freeness of linear series on smooth complex projective threefolds. In recent years, there has been a great deal of interest in the geometric properties of pluricanonical and adjoint linear systems on surfaces and higherdimensional varieties. Among other things, one wants to understand as explicitly as possible when the systems in question are base-point free or very ample. Modem work in this area goes back to Kodaira [Kod] and Bombieri [Bmb], who studied pluricanonical maps of surfaces of general type. More recently, many of their results have emerged as special cases of a celebrated theorem of Reider [Rdr]. Reider uses vector bundle techniques to show that, if B is a nef line bundle on a surface X such that B2 > 5, then KX + B is globally generated unless there exist certain special curves C c X such that B * C < 1; he also obtains analogous criteria for KX + B to be very ample. A cohomological approach to these theorems, based on Miyaoka's vanishing theorem for Zariski decompositions, was given by Sakai [Sak2].

Parabolic sheaves on higher dimensional varieties

Parabolic sheaves on higher dimensional varieties

Rational points on K3 surfaces: A new canonical height

A fundamental tenet of Diophantine Geometry is that the geometric properties of an algebraic variety should determine its basic arithmetic properties. This is certainly true for curves, where the sign of the Euler characteristic of C determines whether the set of rational points on C is finite (X (C) 0). For higher dimensional varieties there are some precise conjectures due to Bombieri, Lang, and Vojta [15] which predict when the rational points on a variety should be finite or degenerate (i.e. not Zariski dense), and some conjectures of Manin et al. [-2, 5] on the distribution of rational points in those cases when they are Zariski dense. But except for abelian varieties, their subvarieties, and some Fano varieties (varieties for which the anticanonical bundle is ample), there are very few general theorems. In this paper we will study the rational points on a certain class of K 3 surfaces defined over a number field K. The moduli space of marked algebraic K3 surfaces is a countable union of 19 dimensional quasi-projective varieties. We are going to look at the 18 dimensional family studied by Wehler 1-17]. Wehler's family consists of K3 surfaces S whose automorphism group Aut(S) contains a subgroup d isomorphic to the free product 7/,2"~ 2 of two cyclic groups of order 2. We will use the geometric information provided by this infinite automorphism group to study the K-rat ional points on S. For any point PeS, we can look at the orbit of P under the action of d,

Rational Points on Symmetric Products of a Curve

Of course, Faltings' theorem [3] says that the left-hand side of (1) is actually bounded, independently of X. However, Mumford's result has certain uniformity properties which, when combined with Faltings' theorem, help one to bound the number of points in C(K). (See [15]. This idea is due to Parshin.) In this paper we will give estimates similar to (1) for certain higher dimensional varieties. For each integer d 1, let Cd) be the d-fold symmetric product of the curve C. Thus C(d) is a smooth projective variety of dimension d. As above, let

Extremal rays on higher dimensional projective varieties

Extremal rays on higher dimensional projective varieties

Correspondences for Hecke Rings and (Co-)Homology Groups on Smooth Compactifications of Siegel Modular Varieties

We show that the Hecke rings act on the $l$-adic cohomology groups of suitable non-singular projective toroidal compactifications of the higher dimensional modular varieties. We extend the fixed point theory of Lefschetz to the correspondences for the Hecke rings on those compactifications. We treat here the Siegel modular case.

Differential Operators on an Affine Curve

Let X denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero. Denote by Dx the sheaf of differential operators on X, and D(X)=Γ(X,Dx), the ring of global differential operators on X. The following is established: THEOREM. D(X) is a finitely generated k-algebra, and a noetherian ring. Furthermore, D(X) has a unique minimal non-zero ideal J, and D(X)/J is a finite-dimensional k-algebra. Let X ˜ denoted the normalisation of X, and π: X ˜ →X the projection map. The main technique is to compare D(X) with D( X ˜ ). THEOREM. The following are equivalent: (i) π is injective, (ii) D(X) is a simple ring, (iii) D(X) is Morita equivalent to D( X ˜ ), (iv) the categories DX-Mod and D X ˜ -Mod are equivalent, (v) gr D(X) is noetherian, (vi) the global homological dimension of D(X) is 1. For higher-dimensional varieties the techniques produce examples of varieties X for which D(X) is right but not left noetherian.

Relative Zariski decomposition on higher dimensional algebraic varieties

Relative Zariski decomposition on higher dimensional algebraic varieties

On extremal rays of the higher dimensional varieties

On extremal rays of the higher dimensional varieties

On the extremal ray of higher dimensional varieties

On the extremal ray of higher dimensional varieties

The topology of complex projective varieties after S. Lefschetz

AFTER THE topology of complex algebraic curves, i.e. the genus of Riemannian surfaces, had been understood mathematicians like Picard [12] and PoincarC [ 12a] went on to the next dimension and began to investigate the topology of complex algebraic surfaces. From 1915 on Lefschetz continued their work and extended it to higher dimensional varieties. In 1924 he published his famous exposition [L] of this work. When it was written knowledge of topology was still primitive and Lefschetz “made use most uncritically of early topology g la PoincarC and even of his own later developments”?. This makes it nowadays rather difficult to understand the topological parts of [L] properly. But that is not the only difficulty: Implicitly Lefschetz quite often appeals to geometric intuition where we would like to see a more precise argument. Thus there is some temptation to discard Lefschetz’s original “proofs” and adopt instead the more recent methods which have been employed to obtain many of his results, using Hodge’s theory of harmonic differential forms or Morse theory or sheaf theory and spectral sequences. But none of these very elegant methods yields Lefschetz’s full geometric insight, e.g. they do not show us the famous “vanishing cycles”. The first attempt to rewrite the topological part of [L] using modern singular homology theory was made more than twenty years ago by Wallace [16]. But the details of his presentation are too complicated to popularize Lefschetz’s original methods. Wallace leaves the realm of algebraic geometry far too early when he makes Lefschetz’s intuitive arguments precise. Furthermore he does not give a complete picture of Lefschetz’s achievements. In the following I make a new attempt to present Lefschetz’s almost sixty year old investigations rigorously but as geometrically as he did in [L]. For topologists Lefschetz is usually interesting for the work he did in pure topology after he had completed [L]. But [Ll has at least “a unique historical interest in being almost the first account of the topology of a construct of importance in general mathematics which is not trivial” (Hedge). We may furthermore speculate how much of the contributions of Poincare, Lefschetz and others to algebraic topology we owe to the difficulties they encountered with the topology of algebraic varieties. The necessary prerequisites in algebraic geometry can be found in the first two chapters of Shafarevich’s book[l3]. The main tool from differential topology is Ehresmann’s fibration theorem, which for the convenience of the reader is stated in 3.0. (Strangely enough this theorem is not included in the standard textbooks.) As far as homology theory is concerned a textbook like Dold’s[6] will amply suffice. Furthermore some basic facts about the fundamental group and the homotopy lifting theorem for fibre bundles will be used.

Valuative Criteria for Families of Vector Bundles on Algebraic Varieties

The object of this paper is to derive separation and completeness properties for the families of vector bundles on a nonsingular projective variety X over a fixed algebraically closed field k. By a vector bundle on X we mean a torsion-free coherent sheaf on X. If Xis a curve, such a sheaf must be locally free; thus this definition corresponds with the usual notion of vector bundle. On higher dimensional varieties, it appears that the category of locally free sheaves is too restrictive; for example, in this category, bundles do not in general have complete flags, whereas in the category of torsion-free sheaves, complete flags always exist (Proposition 1). Let S be an algebraic scheme over k. We define a family of vector bundles on X over S to be a coherent sheaf E on X x S, flat over S, such that for each s e S the induced sheaf E. on P`'(s) is torsion-free. We consider two such families E and E' to be equivalent if there is an invertible sheaf L on S such that E _ E' ( p*(L). Starting from the concept of a family of vector bundles, we are led to define a contravariant functor 'F3@ from the category of algebraic schemes over k to the category of sets, in the following way: If S is an algebraic scheme, let '0B,(S) be the set of equivalence classes of families of vector bundles on X over S. Then if T > S is a morphism of algebraic schemes and E a family of vector bundles over S, (g x X)*(E) will be a family of vector bundles over T. It is natural to ask whether the functor F3SX is representable; that is, is there an algebraic scheme V and a family of vector bundles E over V such that Hom (S, V) -= B,(S) for all S? Such a V would be a natural parameter space for the bundles on X, and E would be a universal family. It turns out that this is too much to expect. First of all, there are too many bundles to be parameterized by an algebraic scheme: we must break the functor up into separate parts corresponding to certain natural invariants, for example, the Hilbert polynomial. Next, we must throw away some

Semicontinuity of Kodaira dimension

with the convention K(X, L) = °° if L has no nontrivial sections for all i > 0. In the particular case when X is nonsingular and L = ^ is the canonical bundle, the invariant K(X) = K(X, £2) is called the canonical (or Kodaira) dimension of X and is the fundamental invariant in the classification of surfaces. Recent works by Ueno [4] and Iitaka [1], [2] have studied K(X, L) for higher dimensional varieties. A fundamental open question is the behavior of K(X, L) under deformations of (X, L). When X is a smooth surface the plurigenera (and hence the Kodaira dimension) are deformation invariant [1], and Iitaka has constructed a family of threefolds Xt with K(X0) = 0 and K(Xt) = ~ t* 0. Our main result is

Areas of projections of analytic sets

If V is a pure 1-dimensional analytic subvariety of C, then the area of V is the sum of the areas, counting multiplicity, of the projections of V onto the n coordinate lines [5, 6]. If V is a pure 1-dimensional variety in the unit ball in C which passes through the origin, then [5] the area of V is at least ~. Together these theorems imply that a 1variety through 0 in the unit ball has the property that the sum of the areas of its projections on the coordinate lines, taken with multiplicity, is bounded below by ft. We shall generalize this result by showing that the conclusion is valid without counting multiplicities. A similar property of higher dimensional varieties will also be obtained. The proof depends upon a one variable result (Theorem l) which relates the area of the image of a function analytic in the unit disc to its LZ-norm on the circle. As an application we extend a result of Nishino [4] to the effect that a family of varieties is locally finite if each family of coordinate slices is such. This in turn will be applied to complete a theorem of Rothstein [7] on characterizing analytic sets in C as those whose coordinate slices are analytic sets.

A Note on the Nakai-Moisezon Test for Ampleness of a Divisor

Let V be a nonsingular variety defined over an algebraically closed field. Call a divisor D on V arithmetically positive if the intersection number (DiY) is strictly positive for all i-dimensional subvarieties Y of V, i , r, where r is the dimension of V. Call D ample if some multiple nD of i is linearly equivalent to a hyperplane section for some projective embedding of V, i. e., if the rational map of V defined by i nD I is biregular. Clearly if D is ample, it is arithmetically positive. Conversely, the NakaiMoisezon test states, if D is arithmetically positive, it is ample. This test was first discovered by Nakai [4] for nonsingular surfaces. Then Moisezon outlined a proof for higher dimensional nonsingular varieties in [2], and in [3] he suggested a definition of the intersection number (Di Y) on singular varieties and remarked that with that definition the test continues to be valid. Independently, Nakai [5] extended the test to projective algebraic schemes. In Section 1 below, we give a proof of the test, which, in our opinion, is simpler and more self-contained although based on that of Nakai [5]. Since we have stated the test for nonsingular varieties, we cannot immediately apply induction on the dimension r of V. Therefore, we prove the following slightly stronger theorem, in substance stated by Moisezon [2], from which the test follows immediately by taking U V: