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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
$\displaystyle\int_0^r\biggl (\int_{X_t}f^*\omega\biggr)\,dt$, 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.
UDC:
519.9
MSC: 34M45,
32Q15,
32Q40 Received: 12.12.1969
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