scholarly journals On the Differential Forms on Algebraic Varieties

1952 ◽  
Vol 4 ◽  
pp. 73-78
Author(s):  
Yûsaku Kawahara
Keyword(s):  
Algebraic Geometry ◽  
Differential Form ◽  
Differential Forms ◽  
Multiple Point ◽  
Algebraic Varieties ◽  
Complete Variety ◽  
Case Characteristic

In the book “Foundations of algebraic geometry” A. Weil proposed the following problem ; does every differential form of the first kind on a complete variety U determine on every subvariety V of U a differential form of the first kind? This problem was solved affirmatively by S. Koizumi when U is a complete variety without multiple point. In this note we answer this problem in affirmative in the case where V is a simple subvariety of a complete variety U (in §1). When the characteristic is 0 we may extend our result to a more general case but this does not hold for the case characteristic p≠0 (in §2).

scholarly journals A Note on the Differential Forms on Everywhere Normal Varieties

1951 ◽  
Vol 2 ◽  
pp. 93-94
Author(s):  
Yûsaku Kawahara
Keyword(s):  
Algebraic Geometry ◽  
Differential Form ◽  
Differential Forms ◽  
Multiple Point ◽  
Algebraic Varieties ◽  
Simple Point ◽  
Multiple Points ◽  
Complete Variety ◽  
Normal Variety ◽  
Normal Varieties

A. Weil proposed in his book “Foundations of algebraic geometry” several problems concerning differential forms on algebraic varieties, S. Koizumi has proved that if ω is a differential form on a complete variety U without multiple point, which is finite at every point of IT, then ω is the differential form of the first kind. The following example shows that on everywhere normal varieties with multiple points this statement holds no more; that is: A differential form on a everywhere normal variety which is finite on every simple point of its variety is not always the differential form of the first kind.

scholarly journals Remarks on the Differential Forms of the First Kind on Algebraic Varieties

1953 ◽  
Vol 6 ◽  
pp. 37-40
Author(s):  
Yûsaku Kawahara
Keyword(s):  
Differential Form ◽  
Differential Forms ◽  
Algebraic Varieties ◽  
Simple Point ◽  
Generic Point ◽  
Complete Variety ◽  
Field Of Definition ◽  
Common Field

A differential form co on a complete variety Un is said to be of the first kind if it is finite at every simple point of any variety which is birationally equivalent to U. Let k be a common field of definition for Uand ω, and let Pbe a generic point of Uover k. If ω is of the first kind, then ω(P) is of course a differential form of the first kind belonging to the extension k(P) of k.

scholarly journals POINCARÉ AND SOBOLEV INEQUALITIES FOR DIFFERENTIAL FORMS IN HEISENBERG GROUPS AND CONTACT MANIFOLDS

2020 ◽  
pp. 1-52
Author(s):  
ANNALISA BALDI ◽  
BRUNO FRANCHI ◽  
PIERRE PANSU
Keyword(s):  
Differential Form ◽  
Scale Invariance ◽  
Differential Forms ◽  
Sobolev Inequalities ◽  
Exact Form ◽  
Heisenberg Groups ◽  
Contact Manifolds ◽  
Bounded Geometry ◽  
Exterior Differential ◽  
Rumin Complex

Abstract In this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups $\mathbb{H}^{n}$ , where the word ‘contact’ is meant to stress that de Rham’s exterior differential is replaced by the exterior differential of the so-called Rumin complex $(E_{0}^{\bullet },d_{c})$ , which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of $\mathbb{H}^{n}$ . In addition, we construct smoothing operators for differential forms on sub-Riemannian contact manifolds with bounded geometry, which act trivially on cohomology. For instance, this allows us to replace a closed form, up to adding a controlled exact form, with a much more regular differential form.

scholarly journals On singular sets of flat holomorphic mappings with isolated singularities

1978 ◽  
Vol 70 ◽  
pp. 47-80
Author(s):  
Hideo Omoto
Keyword(s):  
Algebraic Geometry ◽  
Singular Point ◽  
Critical Points ◽  
Euler Number ◽  
Holomorphic Mappings ◽  
Algebraic Varieties ◽  
Isolated Singularities ◽  
General Fiber ◽  
Singular Sets ◽  
Algebraic Maps

In [4] B. Iversen studied critical points of algebraic mappings, using algebraic-geometry methods. In particular when algebraic maps have only isolated singularities, he shows the following relation; Let V and S be compact connected non-singular algebraic varieties of dimcV = n, and dimc S = 1, respectively. Suppose f is an algebraic map of V onto S with isolated singularities. Then it follows thatwhere χ denotes the Euler number, μf(p) is the Milnor number of f at the singular point p, and F is the general fiber of f : V → S.

scholarly journals On the Theory of Differential Forms on Algebraic Varieties

1957 ◽  
Vol 11 ◽  
pp. 13-39
Author(s):  
Yûsaku Kawahara
Keyword(s):  
Function Field ◽  
Projective Variety ◽  
Hyperplane Section ◽  
Differential Forms ◽  
Classical Case ◽  
Algebraic Varieties ◽  
Simple Point ◽  
Perfect Field ◽  
Linearly Independent ◽  
Normal Varieties

Let K be a function field of one variable over a perfect field k and let v be a valuation of K over k. Then is the different-divisor (Verzweigungsdivisor) of K/k(x), and (x)∞ is the denominator-divisor (Nennerdivisor) of x. In §1 we consider a generalization of this theorem in the function fields of many variables under some conditions. In §2 and §3 we consider the differential forms of the first kind on algebraic varieties, or the differential forms which are finite at every simple point of normal varieties and subadjoint hypersurfaces which are developed by Clebsch and Picard in the classical case. In §4 we give a proof of the following theorem. Let Vr be a normal projective variety defined over a field k of characteristic 0, and let ω1, …, ωs be linearly independent simple closed differential forms which are finite at every simple point of Vr. Then the induced forms on a generic hyperplane section are also linearly independent.

scholarly journals Multi-symplectic magnetohydrodynamics: II, addendum and erratum

2015 ◽  
Vol 81 (6) ◽  
Author(s):  
G. M. Webb ◽  
J. F. McKenzie ◽  
G. P. Zank
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Differential Form ◽  
Compatibility Condition ◽  
Conservation Law ◽  
Differential Forms ◽  
Geometric Theory ◽  
Momentum Equation ◽  
Plasma Phys

A recent paper by Webb et al. (J. Plasma Phys., vol. 80, 2014, pp. 707–743) on multi-symplectic magnetohydrodynamics (MHD) using Clebsch variables in an Eulerian action principle with constraints is further extended. We relate a class of symplecticity conservation laws to a vorticity conservation law, and provide a corrected form of the Cartan–Poincaré differential form formulation of the system. We also correct some typographical errors (omissions) in Webb et al. (J. Plasma Phys., vol. 80, 2014, pp. 707–743). We show that the vorticity–symplecticity conservation law, that arises as a compatibility condition on the system, expressed in terms of the Clebsch variables is equivalent to taking the curl of the conservation form of the MHD momentum equation. We use the Cartan–Poincaré form to obtain a class of differential forms that represent the system using Cartan’s geometric theory of partial differential equations

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