Multivariable value distribution theory. Application to holomorphic curves

Abstract

I. M. Dektyarev UDC 517.547.28 IntroductionThis paper presents the author's approach to the multivariable value distribution theory. In shortened form, the results of this paper appeared in [4][5][6][7][8][9].The manifolds considered in this paper are assumed to be orientable, their orientation being chosen once and for all. The orientation of the boundary of any domain is assumed to be induced by that of the latter (the coherent orientation) and the orientation of complex manifolds is compatible with the complex structure. All manifolds, mappings, functions, and forms are assumed to be smooth (to the extent that is necessary or makes sense):An elementary fact in the multivariable value distribution theory is the following corollary to the Stokes theorem. Let D be a smoothly bounded domain in a real n-dimensional manifold X, let w be the normalized (by unity) volume form on a compact n-dimensional manifold Y, and let for some point a 6 Y, an (n -1 )-form A= be given with the following properties. Outside the point a, the form A= is smooth and satisfies d As = w, and near that point, the form A~ can be represented as the sum A'+ A", where ,k' is a form which has a smooth continuation to a, and A" is a closed form. If F : X + Y is a smooth map with the property that the inverse image of the point a is a discrete subset in D and does not meet the boundary OD, then the number of points in D which belong to F-l(a) is equal (counting with multiplicities and the orientation) to vol(D) -fad F*A~. Here vol(D) denotes the volume of the image of the domain D with the number of sheets and the orientation taken into account. In what follows, after some general lemmas, we shall study only complex manifolds, and therefore, no problems concerning orientation will arise. This is the so-called nonintegrated first main theorem. To get its integrated version, one must specify a function r on X having compact domains of lower values and having no critical points outside some compact set. Taking as Dt the domains {x 6 X : r(x) < t} and writing for them the identity of the nonintegrated first main theorem, we "integrate" it from -0o to r. To avoid the difficulties which can arise here, one must impose some restrictions on the mapping F and on the singularity of the form Aa near the point a. If one considers dimension-decreasing mappings, then extra problems concerning counting of the number of inverse images arise.We shall be interested in application of the general theory to the study of holomorphic curves. In this case, inverse images of points are replaced by inverse images of intersections with the divisor. However, we can consider a fiber bundle rr : W --+ Y over Y, the total space W consisting of the pairs {a point, a divisor passing through this point}. This space is a submanifold in the Cartesian product Y x Z with Z denoting the manifold of all divisors under consideration. The map F induces the fiber bundle F*Tr over X and the mapping from the induced fiber space to the manifold W. We now apply the general theory to the mappi...

Аналоги Существенно Особых Точек Для Последовательностей Полиномиальных Отображений

“…Рассмотрим последовательность отображений F k : C 2 → C 2 ⊂ CP 2 , задаваемых формулами F k (z 1 , z 2 ) = (1 : z k 1 : z k 2 ), где (w 0 : w 1 : w 2 ) -однородные координаты в пространстве CP 2 . Очевидно, что для любого положительного ε и любой точки w (0) пространства CP 2 (включая расположенные на «бесконечно удаленной» прямой {w 0 = 0}) в шаре радиуса √ 2 + ε пространства C 2 найдется такая последовательность точек z (1) , . .…”

“…n , задающих отображение F k . Если для некоторого шара B(r, a) пространства C n радиуса r с центром в точке a последовательность произведений {d k relvol(B(r, a), F k )} стремится к бесконечности, то для любой точки w (0) ∈ CP n и любого положительного ε найдется такая последовательность точек z (1) , . .…”

“…. , принадлежащих шару B(r + ε, a), что последовательность F 1 (z (1) , a), F k )} ограничена снизу положительной константой, то для любой точки w (0) ∈ CP n найдется такая последовательность точек z (1) , . .…”

“…ДОКАЗАТЕЛЬСТВО ЛЕММЫ 2 проводится точно так же, но вместо функций γ w (y) из [1] воспользуемся функцией ψ(y) = (4π) −1 ln(1 + y 2 ). Она также удовлетворяет нужному уравнению dd c (ψ(y)µ) = ω, но вместо ее оценки вне геодезического шара радиуса ρ с центром в w (0) следует воспользоваться неравенством для ее значений внутри шара радиуса ρ 1 пространства C n .…”