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数学代写|复分析代写Complex analysis代考|Duality of Functions Defined in Lineally Convex Sets

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数学代写|复分析代写Complex analysis代考|Introduction to this section

Lineal convexity, a kind of complex convexity intermediate between usual convexity and pseudoconvexity, appears naturally in the study of Fantappiè transforms of analytic functionals. A set is called lineally convex if its complement is a union of complex hyperplanes. This property can be most conveniently defined in terms of the notion of dual complement: the dual complement of a set in $\mathbf{C}^n$ is the set of all hyperplanes that do not intersect the set. It is natural to add a hyperplane at infinity and consider $\mathbf{C}^n$ as an open subset of $\mathbf{P}^n$, complex projective space of dimension $n$. The definition of dual complement is then the same, and somewhat more natural: the set of all hyperplanes is again a projective space. In this setting, the dual complement is often called the projective complement. Indeed, Martineau (1966) called it le complémentaire projectif; the term dual complement used here was introduced by Andersson, Passare and Sigurdsson in a preprint from 1991 of their forthcoming book (2004).

We can now simply define a lineally convex set as a set which is the dual complement of its dual complement (here it becomes obvious that we should identify the hyperplanes in the space of all hyperplanes with the points in the original space). So this duality works well for sets. What about functions?

In convexity theory, a convenient dual object of a set is its support function as defined in Section 9.2. For functions, we have the Fenchel transformation, defined as well in Section 9.2.
Is there a duality for functions that generalizes the duality for sets defined by the dual complement? In this section we shall study such a duality. We call it the logarithmic transformation. It has many properties in common with the Fenchel transformation. However, there are some striking differences. The effective domain, defined by formula (Definition 9.2.5), of a Fenchel transform is always convex, but the effective domain of a logarithmic transform need not be lineally convex (Example 9.8.16). This is connected with the fact that the union of an increasing sequence of lineally convex sets is not necessarily lineally convex (Example 9.8.17). However, the interior of the effective domain of a logarithmic transform is always lineally convex (Theorem 9.8.14), and the transform is plurisubharmonic there (Theorem 9.8.18).

Working with functions defined on $\mathbf{P}^n$ is the same as working with functions defined on $\mathbf{C}^{1+n} \backslash{0}$ which are constant on complex lines, i.e., homogeneous of degree zero. For instance a plurisubharmonic function on an open subset of $\mathbf{P}^n$ can be pulled back to an open cone in $\mathbf{C}^{1+n} \backslash{0}$ and the pullback is plurisubharmonic for the $1+n$ coordinates there. However, I cannot define a duality for such functions. I have been led to consider instead functions defined on subsets of $\mathbf{C}^{1+n} \backslash{0}$ which are homogeneous in another sense: they satisfy $f(t z)=-\log |t|+f(z)$. Such functions are not pullbacks of functions on projective space, but the duality works for them. In a coordinate patch like $z_0=1$ we can identify them with functions on a subset of $\mathbf{P}^n$. Given any function $F$ on $\mathbf{C}^n$, we can define a function $f$ on $\mathbf{C}^{1+n} \backslash{0}$ by $f(z)=F\left(z_1 / z_0, \ldots, z_n / z_0\right)+c \log \left|z_0\right|$ when $z_0 \neq 0$ and $f(z)=+\infty$ when $z_0=0$, where $c$ is an arbitrary real constant; this function is homogeneous in the sense that $f(t z)=c \log |t|+f(z)$ , so we can choose any type of homogeneity. In other words, locally all kinds of homogeneity are equivalent, and there is no restriction in imposing the homogeneity we have here, viz. $c=-1$.

数学代写|复分析代写Complex analysis代考|Notation

Let $A$ be a subset of $\mathbf{C}^{1+n} \backslash{0}$, where $n \geqslant 1$. We shall say that $A$ is homogeneous if $t z \in A$ as soon as $z \in A$ and $t \in \mathbf{C} \backslash{0}$. To any homogeneous subset $A$ of $\mathbf{C}^{1+n} \backslash{0}$ we define its dual complement $A^$ as the set of all hyperplanes passing through the origin which do not intersect $A$. Since any such hyperplane has an equation $\zeta \cdot z=\zeta_0 z_0+\cdots+\zeta_n z_n=0$ for some $\zeta \in \mathbf{C}^{1+n} \backslash{0}$, we can define $$A^=\left{\zeta \in \mathbf{C}^{1+n} \backslash{0} ; \zeta \cdot z \neq 0 \text { forevery z } \in \mathrm{A}\right}$$

Strictly speaking, we should have two copies of $\mathbf{C}^{1+n} \backslash{0}$ (a Greek and a Latin one), and consider $A^$ as a subset of the dual (i.e., the Greek) space. A homogeneous set is called lineally convex if $\mathbf{C}^{1+n} \backslash A$ is a union of complex hyperplanes passing through the origin. A dual complement $A^$ is always lineally convex, and we always have $A^{* } \supset A$. The set $A^{ }$ is called the lineally convex hull of $A$. A set $A$ is lineally convex if and only if $A=A^{ *}$.

The operation of taking the dual complement is an example of a Galois correspondence, and the operation of taking the lineally convex hull defines a cleistomorphism in the ordered set of all subsets of $\mathbf{C}^{1+n} \backslash{0}$. For the general definitions of these concepts, see Section 9.3.

We shall write $z=\left(z_0, z^{\prime}\right)=\left(z_0, z_1, \ldots, z_n\right)$ for points in $\mathbf{C}^{1+n} \backslash{0}$, with $z_0 \in \mathbf{C}$ and $z^{\prime}=\left(z_1, \ldots, z_n\right) \in \mathbf{C}^n$. Homogeneous sets in $\mathbf{C}^{1+n} \backslash{0}$ correspond to subsets of projective $n$ space $\mathbf{P}^n$, and we can transfer the notions of dual complement and lineal convexity to $\mathbf{P}^n$. In the open set where $z_0 \neq 0$ we can use $z^{\prime}$ as coordinates in $\mathbf{P}^n$.
We shall denote by
$$Y_\zeta=\left{z \in \mathbf{C}^{1+n} \backslash{0} ; \zeta \cdot z=0\right}, \quad \zeta \in \mathbf{C}^{1+n} \backslash{0}$$
the hyperplane defined by $\zeta$. Then the dual complement can be conveniently defined as
$$A^=\left{\zeta ; Y_\zeta \cap A=\emptyset\right}$$ and its set-theoretical complement in $\mathbf{C}^{1+n} \backslash{0}$ is $$\complement A^=\left(\mathbf{C}^{1+n} \backslash{0}\right) \backslash A^*=\left{\zeta ; Y_\zeta \cap A \neq \emptyset\right}$$

数学代写|复分析代写Complex analysis代考|Notation

$t \in \mathbf{C} \backslash 0$. 对于任何齐次子集 $A$ 的 $\mathbf{C}^{1+n} \backslash 0$ 我们定义它的对偶补码一个从作为所有通过原点但不相交

$A \wedge=\mid$ left ${|z e t a|$ in $\backslash$ mathbf ${C} \wedge{1+n} \mid$ backslash ${0} ;|z e t a| c d o t ~ z \mid$ neq 0 |text ${$ 永远 $z} \mid$ in $|\operatorname{mathrm}{A}|$ right $}$

$z^{\prime}=\left(z_1, \ldots, z_n\right) \in \mathbf{C}^n$. 齐次套入 $\mathbf{C}^{1+n} \backslash 0$ 对应于投影的子集 $n$ 空间 $\mathbf{P}^n$ ，我们可以将对偶裉码

$Y_{-} \mid$zeta $=\backslash$ left ${z \mid$ in $\backslash$ mathbf ${c} \wedge{1+n} \mid$ backslash ${0} ; \mid$ zeta $\mid$ cdot $z=0 \mid$ right $}, \mid$ quad $\mid$ zeta $\backslash$ in $\mid$ mathbf ${c} \wedge{1+n} \mid$ backslash ${0}$

$$A \Lambda=\backslash \text { left }\left{\text { zeta; } Y_{-} _ \text {Izeta } \mid \text { cap A }=\text { lemptyset } \backslash \text { right }\right}$$

$$A^=\left{\zeta ; Y_\zeta \cap A=\emptyset\right}$$ and its set-theoretical complement in $\mathbf{C}^{1+n} \backslash{0}$ is $$\complement A^=\left(\mathbf{C}^{1+n} \backslash{0}\right) \backslash A^*=\left{\zeta ; Y_\zeta \cap A \neq \emptyset\right}$$

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