RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Uspekhi Mat. Nauk: Year: Volume: Issue: Page: Find

 Uspekhi Mat. Nauk, 1970, Volume 25, Issue 6(156), Pages 53–84 (Mi umn5427)

Problems of value distribution in dimensions higher than unity

I. M. Dektyarev

Abstract: Consider two $n$-dimensional complex manifolds $X$ and $M$, where $M$ is assumed to be compact. Suppose that on $M$ a form $\omega$ is give, which defines an element of volume, and on $X$ a function $\tau$ with isolated critical points and such that the domain $X_r=\{x:\tau(x)<r\}$ is relatively compact for all $r$. For each point we construct on $M\setminus a$ a form $\lambda_a$ of bidegree $(n-1, n-1)$ with certain special properties which allow us to use a more or less standard techniques to prove the following “first main theorem”: if a holomorphic map $f:X\to M$ is non-degenerate for at least one point, then
$$T(r)=N(r, a)+\int_{\partial X_r}d^c\tau \wedge f^*\lambda_a -\int_{X_r}f^*\lambda_a \wedge dd^c\tau,$$
where $T(r)$ denotes the integral $\displaystyle\int_0^r(\int_{X_t}f^*\omega) dt$, and $N(r, a)$ the integral $\displaystyle\int_0^r n(X_t,a) dt$; here $n(X_t, a)$ is the number of points (including multiplicities) $x\in X_t$ such that $f(x)=a$. Under various conditions on the exhaustion $\tau$ and the mapping $f$ we obtain various theorems which assert that when these conditions hold, then the quantity grows for almost all $a\in M$ (over some subsequence of numbers $r$) at the same rate as $T(t)$.
We also consider the case of real manifolds and smooth maps. Here we obtain analogous results, though by different methods.

Full text: PDF file (3394 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 1970, 25:6, 51–82

Bibliographic databases:

UDC: 519.9
MSC: 34M45, 32Q15, 32Q40

Citation: I. M. Dektyarev, “Problems of value distribution in dimensions higher than unity”, Uspekhi Mat. Nauk, 25:6(156) (1970), 53–84; Russian Math. Surveys, 25:6 (1970), 51–82

Citation in format AMSBIB
\Bibitem{Dek70} \by I.~M.~Dektyarev \paper Problems of value distribution in dimensions higher than unity \jour Uspekhi Mat. Nauk \yr 1970 \vol 25 \issue 6(156) \pages 53--84 \mathnet{http://mi.mathnet.ru/umn5427} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=316753} \zmath{https://zbmath.org/?q=an:0207.37803|0223.32007} \transl \jour Russian Math. Surveys \yr 1970 \vol 25 \issue 6 \pages 51--82 \crossref{https://doi.org/10.1070/RM1970v025n06ABEH001268} 

• http://mi.mathnet.ru/eng/umn5427
• http://mi.mathnet.ru/eng/umn/v25/i6/p53

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. I. M. Dektyarev, “Structure of defective sets in the multidimensional theory of value distribution”, Funct. Anal. Appl., 6:2 (1972), 112–118
2. I. M. Dektyarev, “Parabolic mappings of differentiable manifolds”, Funct. Anal. Appl., 13:4 (1979), 291–292
3. I. M. Dektyarev, “Tests for equivalence of hyperbolic manifolds”, Funct. Anal. Appl., 15:4 (1981), 292–293
4. I. M. Dektyarev, “Capacity and deficient divisors”, Funct. Anal. Appl., 22:1 (1988), 53–55
•  Number of views: This page: 201 Full text: 60 References: 23 First page: 1