W. V. D. Hodge

Some Enumerative Results in the Theory of Forms

In a recent note I attempted to obtain the postulation formula for the Grassmannian of k -spaces in [ n ] by the consideration of forms of a certain type in k + 1 sets of r + 1 homogeneous variables, which I called k -connexes. My attempt was not entirely successful; I obtained a formula for k -connexes which suggested what the required postulation formula should be, but was unable to prove it. D. E. Littlewood has now written a paper to show that my problem is intimately connected with the theory of invariant matrices, and has thereby established the truth of the postulation formula which I had conjectured. Littlewoods proof requires a considerable knowledge of the theory of invariant matrices, and this paper results from an attempt to re-write his proof in a form which is intelligible to a student not having this specialized knowledge. Prof. H. W. Turnbull has pointed out to me the importance of the so-called k -connexes in the theory of forms, particularly in connexion with the Gordan-Capelli series, and for this reason I am taking the k -connexes as the principal topic of this paper, leaving the deduction of certain postulation formulae which are the more immediate concern of a geometer to the end.

Differential forms on a Kähler manifold

While a number of special properties of differential forms on a Kahler manifold have been mentioned in the literature on complex manifolds, no systematic account has yet been given of the theory of differential forms on a compact Kahler manifold. The purpose of this paper is to show how a general theory of these forms can be developed. It follows the general plan of de Rhams paper (2) on differential forms on real manifolds, and frequent use will be made of results contained in that paper. For convenience we begin by giving a brief account of the theory of complex tensors on a complex manifold, and of the differential geometry associated with a Hermitian, and in particular a Kahlerian, metric on such a manifold.

Differential forms in algebraic geometry

Before considering more general spaces we shall first discuss (1) the r-dimensional projective space Π r . In this space we shall consider a homogeneous coordinate system (Z0, Z1, ... , Z r ). Let U α be that part of Π r in which Z α ≠ 0. In U α we may then introduce non-homogeneous coordinates zαi = Zι/Zα (ι≠α). Any two distinct sets U α and U β will overlap and in U α ∩ U β we have the transformation law

A theorem on algebraic correspondences

z_\alpha ^i = \frac{z_{\beta ^i} } {z_{\beta} ^\alpha }\left( {i \ne \alpha ,\beta } \right)\,\,;\,z_\alpha ^\beta = \frac{1} {{z_{\beta} ^\alpha }}\,\,.

Principles of Geometry

(1.1)

Methods of algebraic geometry

The expression for the number of linearly independent line complexes of degree n in space of r dimensions was derived by Sisam as a particular case in the determination of the number of linearly independent hyperconnexes of given degrees. The purpose of this note is to generalize Sisams result by determining the number of linearly independent algebraic forms of a special type to which we give the name k -connex.

The Theory and Applications of Harmonic Integrals

In his chapter on correspondences between algebraic curves Prof. Baker has raised a problem concerning the possibility, when we are given the equations of Hurwitz for a correspondence between two algebraic curves, of obtaining therefrom a reduction of the everywhere finite integrals on either curve into complementary regular defective systems of integrals. The problem is stated as an unproved theorem, an exact formulation of which is given below. The object of the present note is to give a proof of this theorem on the lines of Prof. Bakers chapter.

The Intersection Formulae for a Grassmannian Variety

In my book The Theory and Applications of Harmonic Integrals (hereinafter referred to as H. I. ), Chapter iv is devoted to the application of the theory to algebraic varieties. In order to introduce harmonic integrals on an algebraic variety I have to assign a metric to the variety; this metric is not related to the variety in any invariant sense, and indeed its introduction is extremely artificial. Nevertheless it turns out that we can deduce from the properties of the harmonic integrals a number of properties of the variety which are birationally invariant, that is, do not depend on the choice of metric. This remarkable fact seems to indicate either that the invariant properties in question should be obtainable directly without the introduction of the harmonic integrals, or that the metric introduced is not so artificial as it first appeared to be. This note is intended to examine this question.