Zariski Open Set Research Articles

Plücker formula and the equisingularity set of a linear system

We study the process of resolution of linear systems on 2-dimensional regular local rings by quadratic transformations and we give an effective characterization of the Zariski open set of equisingular elements in the linear system. We give a description of the behaviour at the first infinitesimal neighborhood by means of the Plücker Formulas.

Algorithms forb-Functions, Restrictions, and Algebraic Local Cohomology Groups ofD-Modules

Our purpose is to present algorithms for computing some invariants and functors attached to algebraic D-modules by using Grobner bases for differential operators. Let K be an algebraically closed field of characteristic zero and let X be a Zariski open set of K with a positive integer n. We fix a coordinate system x = (x1, . . . , xn) of X and write ∂ = (∂1, . . . , ∂n) with ∂i := ∂/∂xi. We denote by DX the sheaf of algebraic linear differential operators on X (cf. [3], [6]). Let M be a coherent left DX-module and u a section of M. Suppose that f = f(x) ∈ K[x] is an arbitrary non-constant polynomial of n variables. If M is holonomic, then for each point p of Y := {x ∈ X | f(x) = 0}, there exist a germ P (x, ∂, s) of DX [s] at p and a polynomial b(s) ∈ K[s] of one variable so that P (x, ∂, s)(f u) = b(s)f u (1.1)

On the nilpotent * - Fourier transform

LetG be a nilpotent Lie group. The adapted nilpotent Fourier transformℰ was introduced by D. Arnal and J. C. Cortet,ℰ:L(G) →C ∞(V,L(∝2d )), whereL(G) is the Schwartz space ofG andV × ∝2k is aG-invariant Zariski open set ing * the dual of the Lie algebra ofG. We prove the surjectivity of this transformation, which allows us to extend it to distribution spaces.

Classification des alg�bres de Lie filiformes de dimension 8

In this work the authors classify the filiform Lie algebras (i.e., Lie algebras that are nilpotent with an adjoint derivation of maximal order) of dimension m=8 over the field of complex numbers. These algebras, introduced by M. Vergne in her thesis ["Varietes des algebres de Lie nilpotentes'', These de 3 eme cycle, Univ. Paris, Paris, 1966; BullSig(110) 1967:299], form a Zariski-open set in the variety N m of nilpotent Lie algebra laws of C n . For each m≤6 there exists a filiform algebra which gives the only rigid law (i.e., the orbit is open under the natural action of the linear group GL m (C) ) in N m . In N 7 there exists a one-parameter family of filiform algebras but there is no rigid law. For m=8 the authors enumerate six continuous families with one complex parameter and fourteen algebras, one of which is rigid in N 8 ; this is the most remarkable fact. The methods use perturbations, which are the analogue of deformations in the language of nonstandard analysis, and similarity invariants for a nilpotent matrix. Notice that we do not have for the moment an example of a Lie algebra which is nilpotent and rigid in the variety of all the Lie algebras in dimension m ; such an algebra cannot be filiform.

Vector fields on some class of complete symmetric varieties

In the previous papers [6], [7], we show that the set of an algebraic homogeneous space G/P fixed under the action of a maximal torus T can be canonically identified with the coset W1 = W/W1 of Weyl group W. We find a T invariant Zariski open set near each element w ∊ W1 and introduce a very nice local coordinate system such that we can express the maximal torus action explicitly. As a result, we become able to apply the study of J. B. Carrell and D. Lieberman [2], [3] to the space G/P and investigate the numerical properties of its characteristic classes and cycles.

Quotients by 𝐶* and 𝑆𝐿(2,𝐶) actions

Let ρ : C ∗ × X → X \rho :{{\mathbf {C}}^{\ast }} \times X \to X be a meromorphic action of C ∗ {{\mathbf {C}}^{\ast }} on a compact normal analytic space. We completely classify C ∗ {{\mathbf {C}}^{\ast }} -invariant open U ⊆ X U \subseteq X with a compact analytic space U / T U/T as a geometric quotient for a wide variety of actions, including all algebraic actions. As one application, we settle affirmatively a conjecture of D \text {D} . Mumford on compact geometric quotients by SL(2 , C ) {\text {SL(2}},{\mathbf {C}}) of Zariski open sets of ( P C 1 ) n {({\mathbf {P}}_{\mathbf {C}}^1)^n} .

Fourier transforms of smooth functions on certain nilpotent Lie groups

The authors consider irreducible representations π ϵ N ̂ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels K φ ( x, y) of the trace class operations π φ = ∝ N φ( n) π n dn, regarding the π as modeled in L 2( R k ) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels K φ ( x, y, λ) can be interpreted in a useful way. The variables ( x, y, λ) run through a Zariski open set in R n , n = dim N. The authors show there is a polynomial map u = A( x, y, λ) that is a birational isomorphism A: R n → R n with the following properties. The Fourier transforms F 1 φ = K φ ( x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ( u) such that Fφ ∘ A = F 1 φ. On the rationalized parameter space a function f( u) is of the form F φ = f ⇔ f is Schwartz class on R n . If polynomial operators T ϵ P( N) are transferred to operators T ̃ on R n such that F(Tφ) = T ̃ (Fφ), P(N) is transformed isomorphically to P( R n ).

On the zeros of polynomials over arbitrary finite dimensional algebras

In [3] it was shown that a polynomial of degree n with coefficients in an associative division algebra, which is d-dimensional over its center, has either infinitely many or at most nd zeros. In this paper we raise the same question for arbitrary m-ary F-algebras A which are d-dimensional over the algebraically closed field F. Our main result states that in the affine space of m-ary algebras of dimension d there is a non-empty Zariski-open set whose elements A have the following property: in the space of polynomial of precise degree n with coefficients in A there is a non-empty Zariski-open set whose elements have precisely nd zeros. It is shown that all simple algebras, all semi-simple associative algebras, all semisimple Jordan algebras (char F≠2), all semi-simple Lie algebras (char F=0), and the generic algebra possess this property.

Extension theorems for reductive group actions on compact Kaehler manifolds

It is obvious that the topological closure of an orbit of an algebraic group acting algebraically on a projective manifold over tE is an algebraic set containing the orbit as a Zariski open set. This article treats the above situation when the group is a connected reductive complex Lie group acting holomorphically on a compact Kaehler manifold. Recall (cf. § II below) that a connected complex reductive Lie group, G, has the structure of a linear algebraic group, and this algebraic structure is compatible with the underlying analytic structure. Let G be any projective manifold in which G is Zariski open and which induces the above algebraic structure on G. A complex connected Lie group G is said to act projeetively on a compact Kaehler manifold X if G acts holomorphically on X and the Lie algebra of holomorphic vector-fields that G generates on X is annihilated by every holomorphic one form on X. This definition is justified in § II. Note (cf. §III) that G acts projectively on X if either H~(X, Q)=0, or G is semi-simple, or if every generator of the solvable radical of G has a fixed point on X, or if G is linear algebraic acting algebraically on a projective X. The main result of the paper (cf. § II) where G is as above for reduct i~ G is: Proposition. Let G be a complex connected reduetive Lie group acting projeetively on a compact Kaehler X. Let q~: Y ~ X be a hotomorphic map where Y is a normal reduced complex space. Consider the equivariant map q~:G× Y ~ X , ~ extends meromorphically (in the sense of Remmert) to CJ × Y. Taking Y to be a point one gets the analog of the result mentioned in the opening sentence. Another simple corollary is the classical result that the linear algebraic structure chosen on G which makes it algebraic is unique; and in fact that any reductive connected subgroup of a linear algebraic group over ~ is an algebraic subgroup. As a further application of the techniques used a new proof of an improved form of a fixed point theorem (cf. [20]) of the author is given: Proposition. Let S be a complex solvable Lie group acting holomorphically on a compact Kaehler manifold X. The following are equivalent:

Extension theorems for reductive group actions on compact Kaehler manifolds

It is obvious that the topological closure of an orbit of an algebraic group acting algebraically on a projective manifold over tE is an algebraic set containing the orbit as a Zariski open set. This article treats the above situation when the group is a connected reductive complex Lie group acting holomorphically on a compact Kaehler manifold. Recall (cf. § II below) that a connected complex reductive Lie group, G, has the structure of a linear algebraic group, and this algebraic structure is compatible with the underlying analytic structure. Let G be any projective manifold in which G is Zariski open and which induces the above algebraic structure on G. A complex connected Lie group G is said to act projeetively on a compact Kaehler manifold X if G acts holomorphically on X and the Lie algebra of holomorphic vector-fields that G generates on X is annihilated by every holomorphic one form on X. This definition is justified in § II. Note (cf. §III) that G acts projectively on X if either H~(X, Q)=0, or G is semi-simple, or if every generator of the solvable radical of G has a fixed point on X, or if G is linear algebraic acting algebraically on a projective X. The main result of the paper (cf. § II) where G is as above for reduct i~ G is: Proposition. Let G be a complex connected reduetive Lie group acting projeetively on a compact Kaehler X. Let q~: Y ~ X be a hotomorphic map where Y is a normal reduced complex space. Consider the equivariant map q~:G× Y ~ X , ~ extends meromorphically (in the sense of Remmert) to CJ × Y. Taking Y to be a point one gets the analog of the result mentioned in the opening sentence. Another simple corollary is the classical result that the linear algebraic structure chosen on G which makes it algebraic is unique; and in fact that any reductive connected subgroup of a linear algebraic group over ~ is an algebraic subgroup. As a further application of the techniques used a new proof of an improved form of a fixed point theorem (cf. [20]) of the author is given: Proposition. Let S be a complex solvable Lie group acting holomorphically on a compact Kaehler manifold X. The following are equivalent:

Complex-Analytic Properties of Certain Zariski Open Sets on Algebraic Varieties

Complex-Analytic Properties of Certain Zariski Open Sets on Algebraic Varieties

On the Zeta Function of a Hypersurface: IV. A Deformation Theory for Singular Hypersurfaces

-may be described in terms of an alternating product [3, Th. 20.1], the most important term of which comes from S, the subspace of R consisting of elements satisfying certain growth conditions. On the other hand if f is defined over an algebraic number field then for almost all primes, S R. In [2, ? 5] we considered the variation of St as f ran through a rationally parametrized family of forms whose generic element represents (in characteristic zero) a non-singular hypersurface in general position. In the present article we extend this theory to the case of families of hypersurfaces whose generic elements may be singular. We show that many of the preceding results remain valid and, in particular, that St has a basis depending rationally on the parameters, and that this basis has essential singularities only at parameter values for which the dimension of S is less than the generic dimension and that St St for all f in some Zariski open set (which may be empty for some primes). We find that the variation of the endomorphism a* of M is again given by a solution matrix of a system of linear differential equations (Equation 6.5 below). It is known from the work of Katz [4], that in the non-singular case, these differential equations are (with very minor modifications) the same as the classical differential equations satisfied by the elements of the nth deRham space (holomorphic n forms modulo exact ones) of the complement H' in projective n-space (characteristic zero) of the algebraic set defined by