Complex Algebraic Geometry Research Articles

The Number of Embeddings of Minimally Rigid Graphs

Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with $n$ vertices. We show that, modulo planar rigid motions, this number is at most ${{2n-4}\choose {n-2}} \approx 4^n$. We also exhibit several families which realize lower bounds of the order of $2^n$, $2.21^n$ and $2.28^n$. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley--Menger variety ${\it CM}^{2,n}(C)\subset P_{{{n}\choose {2}}-1}(C)$ over the complex numbers $C$. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with $2n-4$ hyperplanes yields at most $deg({\it CM}^{2,n})$ zero-dimensional components, and one finds this degree to be $D^{2,n}=\frac{1}{2}{{2n-4}\choose {n-2}}$. The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences. The same approach works in higher dimensions. In particular, we show that it leads to an upper bound of $2 D^{3,n}= {({2^{n-3}}/({n-2}})){{2n-6}\choose{n-3}}$ for the number of spatial embeddings with generic edge lengths of the $1$-skeleton of a simplicial polyhedron, up to rigid motions. Our technique can also be adapted to the non-Euclidean case.

Sets of Sets: A Cognitive Obstacle

Lawrence Brenton (brenton turing.math.wayne.edu) was educated at the University of Pennsylvania (BA, 1968), and the University of Washington (PhD, 1974). He held posts as :>+.Xr, na post-doctoral research fellow at the University of Bonn, Germany and as a visiting professor at Tulane University before joining the faculty of Wayne State University, where he has served for twenty-six years. His research interests are several complex variables and algebraic geometry, especially the theory of singularities of analytic varieties. His current passion is directing the WSU Undergraduate Research Program.

The Brauer group of analytic K3 surfaces

We show that for complex analytic K3 surfaces any torsion class in H2(X,O∗ X) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to Grothendieck. In our situation, we use twistor spaces to deform a given analytic K3 surface to suitable projective K3 surfaces, and then stable bundles and hyperholomorphy conditions to pass back and forth between the members of the twistor family. In analogy to the isomorphism Pic(X) ∼= H1(X,O∗ X), Grothendieck investigated in [8] the possibility of interpreting classes in H2(X,O∗ X) as geometric objects. He observed that the Brauer group Br(X), parameterizing equivalence classes of sheaves of Azumaya algebras on X, naturally injects into H2(X,O∗ X). It is not difficult to see that Br(X) ⊂ H2(X,O∗ X) is contained in the torsion part of H2(X,O∗ X) and Grothendieck asked: Is the natural injection Br(X ) ⊂ H (X ,O∗ X )tor an isomorphism? This question is of interest in various geometric categories, e.g. X might be a scheme, a complex space, a complex manifold, etc. It is also related to more recent developments in the application of complex algebraic geometry to conformal field theory. Certain elements in H2(X,O∗ X) have been interpreted as so-called B-fields, and those are used to construct super conformal field theories associated to Ricci-flat manifolds. Thus, understanding the geometric meaning of the cohomological Brauer group Br′(X) := H2(X,O∗ X)tor is also of interest for the mathematical interpretation of string theory and mirror symmetry. An affirmative answer to Grothendieck’s question has been given only in very few special cases: • If X is a complex curve, then H2(X,O∗ X) = 0. Hence, Br(X) = Br ′(X) = H2(X,O∗ X) = 0 (see [8, Cor.2.2] for the general case of a curve). • For smooth algebraic surfaces the surjectivity has been proved by Grothendieck [8, Cor.2.2] and for normal algebraic surfaces a proof was given more recently by Schroer [14]. • Hoobler [9] and Berkovich [3] gave an affirmative answer for abelian varieties of any dimension and Elencwajg and Narasimhan gave another proof for complex tori [6].

An algebraic proof of Zak's inequality for the dimension of the Gauss image

A classical theorem in complex algebraic geometry states that, for any smooth projective variety, the Gauss map is finite; in particular, a smooth variety and its Gauss image have the same dimension (with the obvious exception of a linear space). Furthermore, even when the variety is not smooth, Zak proved a lower bound on the dimension of its Gauss image in terms of the dimension of its singular locus. Our purpose in this note is to reinterpret Gauss maps within a more general algebraic framework, and thus recover Zak’s bound on the dimension of the Gauss image as a special case of an interesting new bound on the analytic spread of a module of Kahler differentials. This connects that classical subject in complex geometry to recent research in commutative algebra concerning integral closures of modules. In particular, we give a new, purely algebraic proof of Zak’s theorem. We recall a precise version of Zak’s theorem. Let X be an irreducible projective variety of dimension d defined over an algebraically closed field k, considered with a fixed embedding X ⊂ P(V ) for some finite dimensional k-vector space V . The Gauss map is the rational map from X to the Grassmannian of projective d-planes in P(V ) assigning to each smooth k-point of X the projective tangent plane there,

Real Factorization of Multivariate Polynomials with Integer Coefficients

In the recent papers, a new efficient probabilistic semi-numerical absolute (i.e., complex) factorization algorithm for multivariate polynomials with integer coefficients is given. It is based on a simple property of the monomials arising after a generic linear change of coordinates for bivariate polynomials and on a deep result of complex algebraic geometry. Here, we consider the a priori simpler problem of factorization over the field of reals. We briefly review our algorithm for complex factorization and adapt it to solving the problem on the field of reals. This allows us to spare a significant part of the computations and to improve the range of tractability. The method provides factors with approximative coefficients and eventually exact factors in a suitable real algebraic extension of \(\mathbb{Q}\). Bibliography: 15 titles.

Unitary tridiagonalization inM(4, ⫳)

A question of interest in linear algebra is whether alln xn complex matrices can be unitarily tridiagonalized. The answer for alln ≠ 4 (affirmative or negative) has been known for a while, whereas the casen = 4 seems to have remained open. In this paper we settle then = 4 case in the affirmative. Some machinery from complex algebraic geometry needs to be used.

CONWAY POLYNOMIAL OF A FIBERED SOLVABLE LINK

We compute Kauffman's potential function (equivalent to the Conway polynomial) for any solvable fibered link. Solvable links are links that can be built from the unknot by iterated cabling and summing. Links that arise in complex algebraic geometry are generally of this type.

The geometry of coherent states

We examine the geometry of the state space of a relativistic quantum field. The mathematical tools used involve complex algebraic geometry and Hilbert space theory. We consider the Kähler geometry of the state space of any quantum field theory based on a linear classical field equation. The state space is viewed as an infinite-dimensional complex projective space. In the case of boson fields, a special role is played by the coherent states, the totality of which constitutes a nonlinear submanifold 𝒞 of the projective Fock space PF. We derive the metric on 𝒞 induced from the ambient Fubini–Study metric on PF. Arguments from differential geometric, algebraic, and Kählerian points of view are presented, leading to the result that the induced metric is flat, and that the intrinsic geometry of 𝒞 is Euclidean. The coordinates for the single-particle Hilbert space of solutions are shown to be complex Euclidean coordinates for 𝒞. A transversal intersection property of complex projective lines in PF with 𝒞 is derived, and it is shown that the intrinsic geodesic distance between any two coherent states is strictly greater than the corresponding geodesic distance in the ambient Fubini–Study geometry. The functional metric norm of a difference field is shown to give the intrinsic geodesic distance between two coherent states, and the metric overlap expression is shown to measure the angle subtended by two coherent states at the vacuum, which acts as a preferred origin in the Euclidean geometry of 𝒞. Using the flatness of 𝒞 we demonstrate the relationship between the manifold complex structure on PF and the quantum complex structure viewed as an active transformation on the single-particle Hilbert space. These properties of 𝒞 hold independently of the specific details of the single-particle Hilbert space. We show how 𝒞 arises as the affine part of its compactification obtained by setting the vacuum part of the state vector to zero. We discuss the relationship between unitary orbits and geodesics on 𝒞 and on PF. We show that for a Fock space in which the expectation of the total number operator is bounded above, the coherent state submanifold is Kähler and has finite conformal curvature.

Riemann Surfaces, Plane Algebraic Curves and Their Period Matrices

The aim of this paper is to present theoretical basis for computing a representation of a compact Riemann surface as an algebraic plane curve and to compute a numerical approximation for its period matrix. We will describe a program C ars (Semmler et al., 1996) that can be used to define Riemann surfaces for computations. C ars allows one also to perform the Fenchel–Nielsen twist and other deformations on Riemann surfaces.Almost all theoretical results presented here are well known in classical complex analysis and algebraic geometry. The contribution of the present paper is the design of an algorithm which is based on the classical results and computes first an approximation of a polynomial representing a given compact Riemann surface as a plane algebraic curve and further computes an approximation for a period matrix of this curve. This algorithm thus solves an important problem in the general case. This problem was first solved, in the case of symmetric Riemann surfaces, in Seppälä (1994).

Hypercomplex Algebraic Geometry

It is well-known that sums and products of holomorphic functions are holomorphic, and the holomorphic functions on a complex manifold form a commutative algebra over C . The study of complex manifolds using algebras of holomorphic functions upon them is called complex algebraic geometry. The purpose of this paper is to develop an analogue of complex algebraic geometry, in which the complex numbers C are replaced by the quaternions H . The natural quaternionic analogue of a complex manifold is called a hypercomplex manifold. A class of H -valued functions on hypercomplex manifolds will be defined, called q-holomorphic functions, that are analogues of holomorphic functions on complex manifolds. Now, the set of holomorphic functions on a complex manifold is a commutative algebra over C . Therefore one asks: does the set of q-holomorphic functions on a hypercomplex manifold have an analogous algebraic structure, and if so, what is it? We shall show that the q-holomorphic functions on a (noncompact) hypercomplex manifold do indeed possess a rich algebraic structure. To describe it, we shall introduce a theory of quaternionic algebra, which is a quaternionic analogue of real linear algebra. This theory is built on three building blocks: AH-modules, the analogues of vector spaces, AH-morphisms, the analogues of linear maps, and the quaternionic tensor product, the analogue of tensor product of real vector spaces. As far as the author can tell, these ideas seem to be new. They enable us to construct algebraic structures over H as though H were a commutative field. Quaternionic algebra describes the algebraic structure of hypercomplex manifolds in a remarkable way, and it seems to be the natural language of hypercomplex algebraic geometry, the algebraic geometry of hypercomplex manifolds. We believe that quaternionic algebra is worth studying for its own sake. It has many similarities with linear algebra over R or C , which is why the analogies between complex and quaternionic theories work so well, but there are also deep differences, which give quaternionic algebra a flavour all of its own. Quillen [12] has given a sheaf-theoretic interpretation of the ideas of quaternionic algebra, based on a previous version of this paper. He finds a contravariant equivalence between a class of AH-modules and regular sheaves on a real form of CP. Regular sheaves on CP are equivalent to representations of the Kronecker quiver, and out of such a representation Quillen constructs an AH-module. Under Quillen’s equivalence stable AH-modules correspond to regular vector bundles over the real form of CP.

Weierstrass points and ramification loci on singular plane curves

Let X be a smooth compact RIemann surface (or a smooth projective curve) of genus g. A classicaltopic of study in Complex Analysis and Algebraic Geometry was the study of Weierstrass points of X. For a survey and the history of the subject up to 1986, see [EH]. For another survey containing the main definitions and results on Weierstrass points on singular Gorenstein curves, see [G]. For the case of a base field with positive characteristic,see [L]. Since Weierstrass points are special points on a curve, they have been very useful to study moduli problems. In particular, some subvarieties of the moduli space of smooth genus g curves are defined by the existence of suitable Weierstrass points. Several papers were devoted to the study of Weierstrass points on some interesting classes of projective curves (e.g. smooth plane curves and k-gonal curves). Our paper belong to this set of papers. We consider singular plane curves with ordinary cusps or nodes as only singularities.We believe that our paper gives a non-trivial contribution to the understanding of the existence of certain types of Weierstrass points and osculating points on these curves. In the firstsection we make easy extensions of [K2], Th. 1.1, to the case of singular curves. In the second section we use deformation theory to show the existence of several pairs (C,P) with C in integral nodal plane curve, P e Creg, such that the tangent line D of C at P has high order of contact with C at P (see Theorems 2.1 and 2.2 for precise statements). Calling X the normalization of C and seen Pas a point of X, these pairs (C,P) satisfiesthe conditions of Proposition 1.1 below and hence P is a Weierstrass point of X. In the third section we consider the Weierstrass semigroup of the Weierstrass points obtained in this way. Here the main aim is to give a recipe to extract from the numerical calculations in [K2] as much informations as possible for the Weierstrass semigroup of the pair (X,P). The case of a total inflection point for nodal plane curves was considered in detailsin [CK]. Our recipe (see 3.1) gives

Inverse Eigenvalue Problems for Symmetric Toeplitz Matrices

The inverse eigenvalue problems for symmetric Toeplitz matrices with complex-valued (IEPSCTM) and real-valued (IEPSRTM) entries are studied. The main tools are complex and real algebraic geometry. In the complex case it is shown that the IEPSCTM is solvable for most spectra and always solvable for $n\leqq 4$. In the real case the natural decomposition of the space of all $n \times n$ real symmetric Toeplitz matrices to a finite number of connected components of matrices with a simple spectrum is given. It is then shown that the solvability of the IEPSRTM for all spectra can be deduced if the corresponding map to the IEPSRTM has a nonzero degree for at least one component. This is the case for $n\leqq 4$, which gives an alternative proof to Delsarte and Genin’s results. The IEPSRTM for odd Toeplitz matrices with real-valued entries is also considered.

Computing singular solutions to polynomial systems

A method to generate accurate approximations to the singular solutions of a system of (complex) polynomial equations is presented. This method is established in a context of polynomial continuation; thus, all solutions are generated, with the singular solutions being approximated more accurately than by standard implementations. The theorem on which the method is based is proven using results from several complex variables and algebraic geometry. No special conditions on the derivatives of the system, such as restrictions on the rank of the Jacobian matrix at solutions, are required. A specific implementation is given and the results of numerical experiments in solving four test problems are presented.

Introduction to string theory: its structure and its uses

Over five years ago experiments at CERN confirmed that the weak (radioactive) interactions of elementary particles are mediated by gauge particles that are heavy relatives of the photon, namely the quantum of light and radio wave propagation. Gauge particles have to belong to a pattern given by the structure of a compact Lie group. Mathematicians listed such patterns at the beginning of the century and it seems that nature favours one of the ‘exceptional’ possibilities when nuclear forces are included. Twenty years ago a picture of elementary particles as quanturns of the excitations of a one-dimensional string was developed. Consistency with the principles of relativity and quantum mechanics seemed to require the aforementioned exceptional gauge structure as well as gravitational forces in Einstein’s formulation. Thus a simple ‘string’ principle promised to explain and unify all the diverse fundamental forces of nature: electromagnetic, weak, nuclear and gravitational. Unfortunately, there remain detailed questions still to be resolved. Nevertheless, the theory possesses rich mathematical structure encompassing Lie algebras and infinite-dimensional generalizations and complex algebraic geometry in a way which sheds valuable new perspectives on modern pure mathematics. At the same time it has unexpected applications in describing and classifying the modes of phase transition in two-dimensional materials, a classical problem in statistical physics.

Complex algebraic geometry and calculation of multiplicities for induced representations of nilpotent Lie groups

Let $G$ be a connected, simply connected nilpotent Lie group, $H$ a Lie subgroup, and $\sigma$ an irreducible unitary representation of $H$. In a previous paper, the authors and G. Grelaud gave an explicit direct integral decomposition (with multiplicities) of $\operatorname {Ind} (H \uparrow G, \sigma )$. One consequence of that work was that the multiplicity function was either a.e. infinite or a.e. bounded. In this paper, it is proved that if the multiplicity function is bounded, its parity is a.e. constant. The proof is algebraic-geometric in nature and amounts to an extension of the familiar fact that for almost all polynomials over $R$ of fixed degree, the parity of the number of roots is a.e. constant. One consequence of the methods is that if $G$ is a complex nilpotent Lie group and $H$ a complex Lie subgroup, then the multiplicity is a.e. constant.

Complex Algebraic Geometry and Calculation of Multiplicities for Induced Representations of Nilpotent Lie Groups

Let $G$ be a connected, simply connected nilpotent Lie group, $H$ a Lie subgroup, and $\sigma$ an irreducible unitary representation of $H$. In a previous paper, the authors and G. Grelaud gave an explicit direct integral decomposition (with multiplicities) of $\operatorname {Ind} (H \uparrow G, \sigma )$. One consequence of that work was that the multiplicity function was either a.e. infinite or a.e. bounded. In this paper, it is proved that if the multiplicity function is bounded, its parity is a.e. constant. The proof is algebraic-geometric in nature and amounts to an extension of the familiar fact that for almost all polynomials over $R$ of fixed degree, the parity of the number of roots is a.e. constant. One consequence of the methods is that if $G$ is a complex nilpotent Lie group and $H$ a complex Lie subgroup, then the multiplicity is a.e. constant.

Bosonization on higher genus Riemann surfaces

We prove the equivalence between certain fermionic and bosonic theories in two spacetime dimensions. The theories have fields of arbitrary spin on compact surfaces with any number of handles. Global considerations require that we add new topological terms to the bosonic action. The proof that our prescription is correct relies on methods of complex algebraic geometry.

Angular momentum, convex Polyhedra and Algebraic Geometry

The three families of classical groups of linear transformations (complex, orthogonal, symplectic) give rise to the three great branches of differential geometry (complex analytic, Riemannian and symplectic). Complex analytic geometry derives most of its interest from complex algebraic geometry, while symplectic geometry provides the general framework for Hamiltonian mechanics.These three classical groups “intersect” in the unitary group and the three branches of differential geometry correspondingly “intersect” in Kähler geometry, which includes the study of algebraic varieties in projective space. This is the basic reason why Hodge was successful in applying Riemannian methods to algebraic geometry in his theory of harmonic forms.

Complex analysis and algebraic geometry

Complex analysis and algebraic geometry

A connectedness principle in the geometry of positive curvature

The main purpose of this paper is to develop a connectedness principle in the geometry of positive curvature. In the form, this is a surprising analog of the classical connectedness principle in complex algebraic geometry. The connectedness principle, when applied to totally geodesic immersions, provides not only a uniform formulation for the classical Synge theorem, the Frankel theorem and a recent theorem of Wilking for totally geodesic submanifolds, but also new connectedness theorems for totally geodesic immersions in the geometry of positive curvature. However, the connectedness principle may apply in certain cases which do not require the existence of totally geodesic immersions.