10.1 Global potentials and positive cohomology classesCFDF01A8B5A7

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Let $X$ be a connected compact Kähler manifold, and let $T$ be a closed positive $(1,1)$-current on $X$. By theorem 5.4.3, every point of $X$ has a coordinate neighborhood on which $T=\mathrm{dd}^{\mathrm{c}}u$ with $u$ plurisubharmonic. On the overlap of two such neighborhoods the two local potentials differ by a pluriharmonic function, and in general no choice of the local potentials makes them agree. For instance, let $X$ have positive dimension and let $T=\omega $ be a Kähler form. A global potential $u$ would be plurisubharmonic on $X$, so it would attain its maximum, being upper semicontinuous on the compact space $X$. The set where it does so is closed, and it is also open by the maximum principle (proposition 3.4.7), applied on coordinate balls, where $u$ is subharmonic (theorem 5.9.1). Thus $u$ would be constant and $\omega =\mathrm{dd}^{\mathrm{c}}u=0$. The obstruction is the cohomology class of $T$: by the $\mathrm{dd}^{\mathrm{c}}$-lemma (theorem 10.2.15), if $\theta $ is a smooth closed form in the class of $T$, then $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ for a quasi-plurisubharmonic function $\varphi $ on $X$, unique up to an additive constant. We prove this by solving a Laplace equation for the trace of $T$, after developing the elliptic theory that this requires from the beginning; the same argument shows that the real $(1,1)$-cohomology of $X$ is finite dimensional. Closed positive currents in the class of $\theta $ therefore correspond, up to additive constants, to the functions in $\mathrm{PSH}(X,\theta )$ (definition 6.14.3), and the subsequent constructions works with such potentials.

Once the additive constant is fixed by normalizing the supremum, the local compactness and integrability theorems for plurisubharmonic functions become uniform statements on $X$, and the upper envelope $V_\theta $ of the nonpositive potentials has minimal singularities. With these tools and the Demailly approximation of section 6.13 we study the positivity of a class (pseudo-effective, nef, big or Kähler) and the places where it fails. Siu's decomposition (theorem 10.4.8) splits off the divisorial part of a current. The minimal multiplicities of a class and its non-nef locus locate the points at which the positive currents in the class are forced to be singular, and the non-Kähler locus does the same for Kähler currents; we also compare these notions on $X$ and on a modification of $X$. A mixed version of the scalar equation gives the mixed Hodge index theorem (theorem 10.4.3) and inequalities between intersection numbers.

The second half of the chapter concerns a holomorphic line bundle $L$ on $X$. If $h_0$ is a smooth metric on $L$ with curvature form $c_1(L,h_0)$, a singular metric $h_0\mathrm e^{-\varphi }$ with $\varphi \in \mathrm{PSH}(X,c_1(L,h_0))$ is the global form of a plurisubharmonic weight; the metrics defined by finitely many holomorphic sections of powers of $L$ are examples with analytic singularities. We prove the $L^2$ estimate for $\bar\partial $ on line bundles over Kähler manifolds and combine it with the local extension theorems of section 6.1 to construct global sections with a prescribed value at a point (theorem 10.7.9) or a prescribed restriction to a compact submanifold (theorem 10.8.18), with constants that depend only on the geometry. We also describe the weighted spaces of global sections through their Bergman kernels. A related gluing construction extends strictly positive potentials from a submanifold (theorem 10.8.26). In the last section we leave the Kähler setting and prove Josefson's theorem on a compact complex manifold: every pluripolar set lies in the pole set of a single quasi-plurisubharmonic function (theorem 10.9.19).

Historical remarks and references for this chapter are collected in subsection 25.1.14.