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

This article is cited in 4 scientific papers (total in 4 papers)

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.

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

Bibliographic databases:

UDC: 519.9
MSC: 34M45, 32Q15, 32Q40
Received: 12.12.1969

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}


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    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  mathnet  crossref  mathscinet  zmath
    2. I. M. Dektyarev, “Parabolic mappings of differentiable manifolds”, Funct. Anal. Appl., 13:4 (1979), 291–292  mathnet  crossref  mathscinet  zmath
    3. I. M. Dektyarev, “Tests for equivalence of hyperbolic manifolds”, Funct. Anal. Appl., 15:4 (1981), 292–293  mathnet  crossref  mathscinet  zmath  isi
    4. I. M. Dektyarev, “Capacity and deficient divisors”, Funct. Anal. Appl., 22:1 (1988), 53–55  mathnet  crossref  mathscinet  zmath  isi
  • Успехи математических наук Russian Mathematical Surveys
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