PomeTheorem 5.9.1.2A4F77263F48 Let $\Omega \subset \mathbb {C}^ n$ be a domain, $n\ge 1$. Every $u\in \mathrm{PSH}(\Omega )$ is subharmonic and locally integrable on $\Omega $, and $\mathrm{dd}^{\mathrm{c}}u\geq 0$. Conversely, a real distribution $T$ on $\Omega $ with $\mathrm{dd}^{\mathrm{c}}T\geq 0$ has a unique plurisubharmonic representative, that is, there is exactly one $u\in \mathrm{PSH}(\Omega )$ with $\langle T,\chi \rangle =\int _\Omega u\chi \, \mathrm dV_{2n}$ for every test function $\chi $. This representative is the canonical subharmonic representative of $T$ given by theorem 3.4.12, and it satisfies (3.4.13).
5.9 Positive Hessians and holomorphic maps484A05EF6DF1
484A05EF6DF1
We now compare plurisubharmonicity with the distributional condition $\mathrm{dd}^{\mathrm{c}}u\geq 0$, where $(u_{j\bar k})$ is the complex Hessian of (5.2.9). The next theorem shows that every almost-everywhere class satisfying this condition contains exactly one plurisubharmonic function, namely its canonical representative. We then treat smooth functions, regularization and holomorphic maps, and define plurisubharmonic functions on complex manifolds.
Proof. Let $u\in \mathrm{PSH}(\Omega )$. Being upper semicontinuous, $u$ is bounded above on compact sets. Fix $x\in \Omega $ and $r>0$ with $\overline{B(x,r)}\subset \Omega $, and let $M$ be an upper bound for $u$ on $\overline{B(x,r)}$. For every unit vector $a\in \mathbb {C}^ n$ the closed disc $\{ x+ta:|t|\leq r\} $ lies in $\Omega $, and the restriction $t\mapsto u(x+ta)$ is subharmonic, possibly identically $-\infty $, on a neighborhood of this disc. The monotonicity of circle means in theorem 3.4.5 therefore gives
\[ u(x)\leq \frac1{2\pi }\int _0^{2\pi }u(x+r\mathrm{e}^{\mathrm{i}\theta }a)\, \mathrm d\theta . \]
With $\tau $ as in (5.8.3), we integrate this inequality over $a\in S^{2n-1}$ with respect to $\tau $; by Tonelli's theorem, applied to the nonnegative function $M-u$, and by the invariance of $\tau $ under $a\mapsto \mathrm{e}^{\mathrm{i}\theta }a$, the right-hand side becomes the sphere mean $\int _{S^{2n-1}}u(x+ra)\, \mathrm{d}\tau (a)$. Thus $u$ satisfies the submean inequality of definition 3.4.2 at every point, so $u$ is subharmonic. Since $\Omega $ is connected and $u\not\equiv -\infty $, proposition 3.4.8 shows that $u\in L^1_{\mathrm{loc}}(\Omega )$. Since, moreover, the restriction of $u$ to each connected component of each affine complex-line section of $\Omega $ is subharmonic, possibly identically $-\infty $, lemma 5.8.2 gives $\mathrm{dd}^{\mathrm{c}}u\ge 0$.
Conversely, let $T$ be a real distribution with $\mathrm{dd}^{\mathrm{c}}T\geq 0$. Then $\Delta T=4\sum _ jT_{j\bar j}\geq 0$, so by theorem 3.4.12 $T$ has a canonical subharmonic representative $v$, which is locally integrable and satisfies (3.4.13). Since $\mathrm{dd}^{\mathrm{c}}v=\mathrm{dd}^{\mathrm{c}}T\geq 0$, proposition 5.4.4 shows that the restriction of $v$ to each connected component of each affine complex-line section of $\Omega $ is subharmonic or identically $-\infty $. As $v$ is upper semicontinuous and locally integrable, hence not identically $-\infty $, it is plurisubharmonic. If $w$ is another plurisubharmonic representative of $T$, then $w$ is a locally integrable subharmonic function by the first part, and $w=v$ almost everywhere; by the uniqueness in theorem 3.4.12, $w=v$ everywhere. $\square$
The condition $\mathrm{dd}^{\mathrm{c}}u\geq 0$ depends only on the almost-everywhere class of $u$, whereas plurisubharmonicity is a pointwise property, and theorem 5.9.1 singles out one function in each class. For example, the function equal to $1$ at one point $x_0$ and to $0$ elsewhere is upper semicontinuous and has zero distributional complex Hessian, but it violates the submean inequality on complex lines at $x_0$; the plurisubharmonic representative of its class is the zero function.
For smooth functions, plurisubharmonicity is a pointwise condition on the complex Hessian, and every plurisubharmonic function is, locally, the decreasing limit of smooth plurisubharmonic functions. In the next proposition, $\rho _\varepsilon $ is the radial mollifier of section 2.5 and, as there, $\Omega _\varepsilon =\{ x\in \Omega :\operatorname {dist}(x,\mathbb C^ n\setminus \Omega )>\varepsilon \} $ for an open set $\Omega \subset \mathbb C^ n$ and $\varepsilon >0$.
Proposition 5.9.2.06C722B7D837 Let $\Omega \subset \mathbb {C}^ n$ be open, $n\geq 1$.
A real function $u$ of class $C^2$ on $\Omega $ is plurisubharmonic on each connected component of $\Omega $ if and only if the Hermitian matrix $(u_{j\bar k}(z))$ is positive semidefinite at every point $z\in \Omega $.
Let $u$ be a function on $\Omega $ which is plurisubharmonic on each connected component of $\Omega $, and let $\varepsilon >0$. Then $u*\rho _\varepsilon $ is smooth on $\Omega _\varepsilon $ and plurisubharmonic on each connected component of $\Omega _\varepsilon $, and $u*\rho _\varepsilon (x)$ decreases to $u(x)$ as $\varepsilon \downarrow 0$, for every $x\in \Omega $.
Proof. (1) Let $\Omega _0$ be a connected component of $\Omega $. We show that both conditions are equivalent to $\mathrm{dd}^{\mathrm{c}}u\geq 0$ on $\Omega _0$. Since $u$ is of class $C^2$, integration by parts shows that its distributional derivatives $u_{j\bar k}$ are the continuous functions $\partial _{z_ j}\partial _{\bar z_ k}u$, so for $\xi \in \mathbb {C}^ n$ the distribution $\sum _{j,k}u_{j\bar k}\xi _ j\bar\xi _ k$ is the continuous real function $z\mapsto \sum _{j,k}u_{j\bar k}(z)\xi _ j\bar\xi _ k$. A continuous real function is a positive distribution exactly when it is nonnegative: if it were negative at a point, it would be negative on a ball around that point, and its integral against a nonnegative test function supported in this ball and positive at its center would be negative. Hence, by the matrix criterion of section 5.4, $\mathrm{dd}^{\mathrm{c}}u\geq 0$ on $\Omega _0$ if and only if $(u_{j\bar k}(z))$ is positive semidefinite at every $z\in \Omega _0$.
If $u$ is plurisubharmonic on $\Omega _0$, then $\mathrm{dd}^{\mathrm{c}}u\geq 0$ there by theorem 5.9.1. Conversely, if $\mathrm{dd}^{\mathrm{c}}u\geq 0$ on $\Omega _0$, then by theorem 5.9.1 the class of $u$ has a plurisubharmonic representative $v$ on $\Omega _0$ satisfying the ball-average formula in (3.4.13). Since $v=u$ almost everywhere and $u$ is continuous, $v(x)$ is the limit of the averages of $u$ over the balls $B(x,r)$ as $r\to 0$, that is, $v(x)=u(x)$. Hence $u=v$ is plurisubharmonic on $\Omega _0$.
(2) By theorem 5.9.1, applied on each connected component of $\Omega $, the function $u$ is subharmonic and locally integrable on $\Omega $, and $\mathrm{dd}^{\mathrm{c}}u\geq 0$. By proposition 2.5.3(1), $u*\rho _\varepsilon $ is smooth on $\Omega _\varepsilon $, and its derivatives are the mollifications of the distributional derivatives of $u$. Thus, for $\xi \in \mathbb {C}^ n$, the function $\sum _{j,k}(u*\rho _\varepsilon )_{j\bar k}\xi _ j\bar\xi _ k$ is the mollification of the distribution $\sum _{j,k}u_{j\bar k}\xi _ j\bar\xi _ k$, which is positive by the matrix criterion; so it is nonnegative by proposition 2.5.3(3). By (1), $u*\rho _\varepsilon $ is plurisubharmonic on each connected component of $\Omega _\varepsilon $. Finally, since $u$ is subharmonic on $\Omega $, theorem 3.4.12(2) shows that $u*\rho _\varepsilon (x)\downarrow u(x)$ as $\varepsilon \downarrow 0$, for every $x\in \Omega $. $\square$
Let $u$ be a smooth real function on an open set $\Omega \subset \mathbb C^ n$. We call $u$ strictly plurisubharmonic if its complex Hessian matrix $(u_{j\bar k})$ is positive definite everywhere. For a general $u$, strict plurisubharmonicity means that every point has a neighborhood on which $u-\varepsilon |z|^2$ is plurisubharmonic for some $\varepsilon >0$; on a complex manifold we require this in local holomorphic coordinates around each point, and $\varepsilon $ may depend on the point and the coordinates.
For smooth $u$ the two definitions agree. Since $(|z|^2)_{j\bar k}=\delta _{jk}$, the complex Hessian of $u-\varepsilon |z|^2$ is $(u_{j\bar k})-\varepsilon I$. The smallest eigenvalue $\lambda (z)=\min _{|\xi |=1}\sum _{j,k}u_{j\bar k}(z)\xi _ j\bar\xi _ k$ of the matrix $(u_{j\bar k}(z))$ is a continuous function of $z$, as the minimum over the compact unit sphere of a jointly continuous function of $(z,\xi )$. If the matrix is positive definite everywhere, then $\lambda $ is positive, hence bounded below by some $\varepsilon >0$ on a closed ball around each point, and on the open ball the smooth function $u-\varepsilon |z|^2$ has a positive semidefinite complex Hessian, so it is plurisubharmonic by proposition 5.9.2(1). Conversely, the plurisubharmonicity of $u-\varepsilon |z|^2$ on a ball around a point gives $(u_{j\bar k})\geq \varepsilon I$ there, by the same criterion.
Proposition 5.9.3.F6423BCE3BB1 Let $D\subset \mathbb {C}^ m$ and $\Omega \subset \mathbb {C}^ n$ be domains, let $F\colon D\to \Omega $ be a holomorphic map, and let $u\in \mathrm{PSH}(\Omega )$. Then $u\circ F$ is either identically $-\infty $ or plurisubharmonic on $D$. More generally, if $D\subset \mathbb {C}^ m$ is only assumed to be open, this alternative holds on each connected component of $D$.
Proof. The statement for open $D$ follows by applying the statement for domains to each connected component of $D$. If $n=0$, then $\Omega $ is a point, $u$ is a real constant, and so is $u\circ F$. If $m=0$, then $D$ is a point and $u\circ F$ is a single value in $[-\infty ,\infty )$; it is either $-\infty $ or a real number, that is, a plurisubharmonic function on the point. Now let $m,n\geq 1$. By theorem 5.9.1, $u$ is the canonical subharmonic representative of a locally integrable function with $\mathrm{dd}^{\mathrm{c}}u\geq 0$, so proposition 5.4.4 applies: $u\circ F$ is either identically $-\infty $, or locally integrable and subharmonic, in particular upper semicontinuous. In the second case, the restriction of $u\circ F$ to a connected component of an affine complex-line section of $D$ is the composition of $u$ with a holomorphic map from an open subset of $\mathbb {C}$, so it is subharmonic or identically $-\infty $ by the same proposition. Hence $u\circ F$ is plurisubharmonic. $\square$
Let $X$ be a complex manifold. A function $u\colon X\to [-\infty ,\infty )$ is plurisubharmonic if its expression in every holomorphic chart with connected domain is plurisubharmonic in the sense of definition 5.8.1; we write $\mathrm{PSH}(X)$ for the set of such functions.
On a domain $\Omega \subset \mathbb C^ n$, plurisubharmonicity is a local property: if every point of $\Omega $ has an open neighborhood on which $u$ is plurisubharmonic, then $u\in \mathrm{PSH}(\Omega )$. Indeed, upper semicontinuity is a local condition. Let $\Lambda $ be a connected component of an affine complex-line section of $\Omega $. Near each point of $\Lambda $, the restriction of $u$ to $\Lambda $ is subharmonic or identically $-\infty $, hence subharmonic; since subharmonicity is a local property, the restriction of $u$ to $\Lambda $ is subharmonic. Finally, $u$ is locally integrable by theorem 5.9.1, hence not identically $-\infty $.
It follows that it suffices to check the condition on $X$ for the charts of one atlas with connected chart domains. Indeed, the transition maps are biholomorphic, so by proposition 5.9.3 the expression of $u$ in another chart with connected domain is, on a connected neighborhood of each point, plurisubharmonic or identically $-\infty $. The second alternative would make the expression of $u$ in a chart of the atlas identically $-\infty $ on a nonempty open set, which is impossible because plurisubharmonic functions are locally integrable. Hence the expression of $u$ in the other chart is plurisubharmonic near each point, and therefore on its connected domain.
Consequently plurisubharmonicity is also a local property on $X$: if every point of $X$ has a neighborhood on which $u$ is plurisubharmonic, then $u\in \mathrm{PSH}(X)$, because the expression of $u$ in a chart with connected domain is then plurisubharmonic near each point. Since a plurisubharmonic function on $X$ is locally integrable in charts, it is not identically $-\infty $ on any connected component of $X$. On an open set $\Omega \subset \mathbb C^ n$, regarded as a complex manifold, a function is plurisubharmonic exactly when it is plurisubharmonic on each connected component of $\Omega $: each component is a connected chart domain, and conversely the locality just proved applies.
Limits of plurisubharmonic functions may degenerate to the constant $-\infty $. On a connected manifold, whether this happens can be decided locally.
Lemma 5.9.4.598B886D700E Let $X$ be a connected complex manifold, and let $v\colon X\to [-\infty ,\infty )$ be a function such that every point of $X$ has an open neighborhood on which $v$ is plurisubharmonic or identically $-\infty $. Then $v$ is plurisubharmonic on $X$ or identically $-\infty $ on $X$.
Proof. Let $A$ be the set of points of $X$ having an open neighborhood on which $v\equiv -\infty $, and $B$ the set of points having an open neighborhood on which $v$ is plurisubharmonic. Both sets are open, and they cover $X$ by hypothesis. They are disjoint: if $x\in A\cap B$, with open neighborhoods $W_1$ on which $v\equiv -\infty $ and $W_2$ on which $v$ is plurisubharmonic, then the restriction of $v$ to the nonempty open set $W_1\cap W_2$ is plurisubharmonic and identically $-\infty $, which is impossible because plurisubharmonic functions are locally integrable. Since $X$ is connected, either $X=A$, and then $v\equiv -\infty $, or $X=B$, and then $v$ is plurisubharmonic near every point, hence on $X$, because plurisubharmonicity is a local property. $\square$
Every closed positive $(1,1)$-current $T$ on a complex manifold of positive dimension has local plurisubharmonic potentials: every point has a neighborhood on which $T=\mathrm{dd}^{\mathrm{c}}u$ for a plurisubharmonic function $u$. Indeed, on a coordinate ball inside the neighborhood given by theorem 5.4.3, the canonical local potential is the canonical subharmonic representative of a class with positive complex Hessian, hence plurisubharmonic by theorem 5.9.1.
Proposition 5.9.5.7EBEE004A7FA Let $X$ be a complex manifold. Nonnegative multiples, finite sums and maxima of nonempty finite families of plurisubharmonic functions on $X$ are plurisubharmonic; here the zero multiple of a function is the zero function. Moreover, let $u\in \mathrm{PSH}(X)$, let $D\subset X$ be open, and let $v\in \mathrm{PSH}(D)$. If
\[ \varlimsup _{D\ni y\to x}v(y)\le u(x)\qquad \text{for every }x\in \partial D, \]
where $\partial D$ is the boundary of $D$ in $X$, then the function equal to $\max \{ u,v\} $ on $D$ and to $u$ on $X\setminus D$ is plurisubharmonic on $X$. This boundary inequality holds in particular if
\[ \varliminf _{\substack {D\ni y\to x\\ v(y)>-\infty }}(u(y)-v(y))\ge 0 \qquad \text{for every }x\in \partial D. \]
Proof. In local coordinates, upper semicontinuity and the submean inequality on complex lines are preserved by nonnegative multiples, sums and maxima; for sums we use that circle means are additive on functions with values in $[-\infty ,\infty )$. It remains to check that the resulting functions are not identically $-\infty $ on any component. For $cu$ with $c>0$ this holds because $cu=-\infty $ exactly where $u=-\infty $; a sum of plurisubharmonic functions is locally integrable by theorem 5.9.1, and a maximum dominates each of its members. This proves the first assertion.
Let $w$ denote the glued function. On $D$ it equals $\max \{ u,v\} $, which is plurisubharmonic by the first part, and on the open set $X\setminus \overline D$ it equals $u$. At $x\in \partial D$ we have $w(x)=u(x)$, and the upper semicontinuity of $u$ together with the boundary inequality gives
\[ \varlimsup _{y\to x}w(y)\leq \max \Bigl \{ \varlimsup _{y\to x}u(y),\ \varlimsup _{D\ni y\to x}v(y)\Bigr \} \leq u(x)=w(x). \]
Hence $w$ is upper semicontinuous. To verify the line condition, work in a chart and restrict $w$ to a connected component of an affine complex-line section. At points of $D$ and of $X\setminus \overline D$, the restriction satisfies the submean inequality on small circles, because $w$ is plurisubharmonic near such points. At a point $x\in \partial D$ we have $w\geq u$ everywhere and $w(x)=u(x)$, so each small circle mean of $w$ is at least the corresponding circle mean of $u$, which is at least $u(x)=w(x)$; if the restriction of $u$ is identically $-\infty $, then $w(x)=u(x)=-\infty $ and there is nothing to prove. Thus the restriction of $w$ is upper semicontinuous and satisfies the submean inequality on small circles at every point; that is, it is subharmonic. Finally $w\geq u$, so $w$ is not identically $-\infty $ in any chart, and $w$ is plurisubharmonic.
For the sufficient condition, fix $x\in \partial D$ and $\varepsilon >0$. By hypothesis, $x$ has a neighborhood $W$ such that $v(y)<u(y)+\varepsilon $ for all $y\in W\cap D$ with $v(y)>-\infty $. Hence $v\leq u+\varepsilon $ on $W\cap D$, the inequality being trivial where $v=-\infty $, and the upper semicontinuity of $u$ gives
\[ \varlimsup _{D\ni y\to x}v(y)\leq \varlimsup _{y\to x}u(y)+\varepsilon \leq u(x)+\varepsilon . \]
Letting $\varepsilon \downarrow 0$ gives the boundary inequality. $\square$
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