A8A4EA7F273E
The nonclosed error in proposition 10.15.1 destroys the cohomological bound on masses. On a compact Kähler manifold, by contrast, the non-pluripolar mass of a positive current in a closed class is bounded by a top intersection number. Pulling a class back to a Kähler resolution of a subvariety therefore gives a finite invariant: the supremum of the non-pluripolar masses of positive currents in the pulled-back class. It measures positivity along every subvariety, including subvarieties along which a given modification of $X$ fails to be an isomorphism.
In this section and the next, $X$ belongs to Fujiki's class $\mathcal C$. Then every irreducible reduced subvariety $Y\subset X$ has a resolution $\pi :Z\to Y$ with $Z$ a compact Kähler manifold: closed subspaces of a space in class $\mathcal C$ are again in class $\mathcal C$, and the Kähler modification theorem applies to $Y$. These classical facts are discussed in section 25.18. In particular, a Kähler model of $Y$ is obtained directly, not as a proper transform under one fixed modification of $X$.
Two such resolutions $Z_1,Z_2$ of $Y$ admit a common Kähler refinement. Indeed, apply embedded resolution to the closure in $Z_1\times Z_2$ of the graph of the induced bimeromorphic map. The resulting strict transform is smooth, and it is Kähler because the ambient resolution is a sequence of blowups of the compact Kähler product along smooth centers. A blowup of a compact Kähler manifold along a smooth center is again Kähler: add to the pulled-back Kähler form a small positive multiple of a curvature form of $\mathcal O(-E)$, which is positive in the vertical directions along the exceptional divisor $E$.
For a closed positive $(1,1)$-current $S$ on a compact Kähler manifold $Z$ of dimension $k>0$, write $\langle S^ k\rangle $ for its non-pluripolar Monge–Ampère measure, that is, $\mathrm{MA}^{\mathrm{ext}}_\eta (u)$ for any decomposition $S=\eta +\mathrm{dd}^{\mathrm{c}}u$ with $\eta $ smooth and closed (theorem 13.14.3). For a class $b\in H^{1,1}_{\mathrm{BC}}(Z,\mathbb R)$, set
63C31A84C03F
10.16.1
\begin{equation} \mathcal R_ Z(b)=\sup _{\substack {S\in b\\ S\geq 0}} \int _ Z\langle S^ k\rangle , \qquad \sup \varnothing =0. \end{equation}
For a class $a\in H^{1,1}_{\mathrm{BC}}(X,\mathbb R)$ and an irreducible reduced subvariety $Y\subset X$ of dimension $k>0$, let $j:Y\hookrightarrow X$ be the inclusion, write $f=j\circ \pi $, and define
E4E79DEE021C
10.16.2
\begin{equation} \mathcal R(a;Y)=\mathcal R_ Z(f^*a). \end{equation}
Proposition 10.16.3.5D137EECD5CC The quantities in (10.16.1) and (10.16.2) are finite, and the latter is independent of the Kähler resolution. If $\eta \in b$ is smooth, $\kappa $ is Kähler and $C>0$ satisfies $C\kappa -\eta >0$, then
A4CFA8A325BE
10.16.4
\begin{equation} \mathcal R_ Z(b+t\{ \kappa \} ) \leq (C+t)^ k\int _ Z\kappa ^ k \qquad (t\geq 0). \end{equation}
Let $\mu :X'\to X$ be a modification and $Y'$ the proper transform of a subvariety $Y$ that meets the locus where $\mu $ is an isomorphism. Then
\[ \mathcal R(\mu ^*a;Y')=\mathcal R(a;Y). \]
Proof.
Step 1: Kähler forms. Let $\lambda $ be a Kähler form on $Z$ and $q\in \mathrm{PSH}(Z,\lambda )$ with $q\leq 0$. The truncations $q_ j=\max (q,-j)$ have mass $\int _ Z\lambda ^ k$ by theorem 11.1.7, or by Stokes' theorem when $k=1$. Restricting their Monge–Ampère measures to $\{ q>-j\} $ and passing to the increasing limit gives
\[ \int _ Z\langle (\lambda +\mathrm{dd}^{\mathrm{c}}q)^ k\rangle \leq \int _ Z\lambda ^ k. \]
The smooth current $\lambda $ attains equality.
Step 2: the mass bound. Put $A=C\kappa -\eta $. For a positive current $S$ in $b+t\{ \kappa \} $, the current $S+A$ is positive and belongs to $(C+t)\{ \kappa \} $. Since local non-pluripolar products are positive and multilinear, $\langle S^ k\rangle \leq \langle (S+A)^ k\rangle $, and Step 1 applied to the Kähler form $(C+t)\kappa $ yields (10.16.4). In particular $\mathcal R_ Z(b)$ is finite. The definition involves only the class, so the choice of its smooth representative or of the comparison form does not change the supremum.
Step 3: modifications. Let $\rho :W\to Z$ be a modification between compact Kähler manifolds. By proposition 5.43.18, every positive current in $\rho ^*b$ is the pullback of a unique positive current in $b$. Their non-pluripolar measures correspond on the locus where $\rho $ is an isomorphism, and neither charges the exceptional analytic sets. Hence their total masses agree and
\[ \mathcal R_ W(\rho ^*b)=\mathcal R_ Z(b). \]
Applying this identity on a common Kähler refinement shows that $\mathcal R(a;Y)$ is independent of the resolution. The same argument compares resolutions of $Y$ and $Y'$, since their maps to $X$ agree on a dense open set and therefore everywhere on a common refinement.
$\square$
On a compact Kähler manifold, positivity of this mass has a classical interpretation. We use two facts from Kähler pluripotential theory, discussed in section 25.18: a class containing a positive current with positive non-pluripolar top mass is big, and in a big class a current with minimal singularities attains the volume, the largest non-pluripolar top mass; for a nef class this volume is the top intersection number. For a nef class, the Kähler current used here can also be obtained from the following normalized diagonal construction.
Proposition 10.16.5.BA31DCC7CCB7 Let $Z$ be a compact Kähler manifold of dimension $k>0$ and $b\in H^{1,1}_{\mathrm{BC}}(Z,\mathbb R)$. Then
\[ \mathcal R_ Z(b)>0\quad \Longleftrightarrow \quad b\text{ is big}. \]
If $b$ is nef, then
1AB8FDFCA122
10.16.6
\begin{equation} \mathcal R_ Z(b)=\int _ Zb^ k. \end{equation}
Suppose that $b$ is nef with $I=\int _ Zb^ k>0$. Fix a Kähler form $\kappa $, let $p_1,p_2:Z\times Z\to Z$ be the projections and $\Delta \subset Z\times Z$ the diagonal, and put
\[ M=k\int _ Zb^{k-1}\wedge \{ \kappa \} . \]
Then there are $d>0$ and a closed positive $(k,k)$-current $\Theta $ in $(p_1^*b+p_2^*b)^ k$ with $\Theta \geq d[\Delta ]$ such that
5827E195F217
10.16.7
\begin{equation} K_ b=M^{-1}(p_1)_*(\Theta \wedge p_2^*\kappa )\in b, \qquad K_ b\geq (d/M)\kappa . \end{equation}
Proof.
If $\mathcal R_ Z(b)>0$, some positive representative has positive non-pluripolar top mass, so $b$ is big by the first fact above. Conversely, if $S\in b$ satisfies $S\geq \delta \kappa $ with $\delta >0$, then $S-\delta \kappa $ is positive and closed, and expanding the non-pluripolar product gives $\langle S^ k\rangle \geq \delta ^ k\kappa ^ k$. Hence $\mathcal R_ Z(b)>0$.
Suppose now that $b$ is nef. For $t>0$, the class $b+t\{ \kappa \} $ is Kähler. Adding $t\kappa $ to an arbitrary positive representative of $b$ and using the Kähler mass bound of Step 1 in the proof of proposition 10.16.3 yields
\[ \mathcal R_ Z(b)\leq \int _ Z(b+t\{ \kappa \} )^ k. \]
Letting $t$ decrease to zero gives the upper bound in (10.16.6). In particular, equality holds if $I=0$.
Let $I>0$, and choose $L>0$ such that $L\{ \kappa \} -b$ is Kähler. Since intersection numbers of nef classes are nonnegative,
\[ 0\leq \int _ Zb^{k-1}\wedge (L\{ \kappa \} -b) =LM/k-I, \]
so $M>0$. The class $B=p_1^*b+p_2^*b$ is nef and satisfies
\[ \int _{Z\times Z}B^{2k}={2k\choose k}I^2>0. \]
The classical mass concentration theorem along the diagonal gives $\Theta \in B^ k$ with $\Theta \geq d[\Delta ]$ for some $d>0$. When $B^ k\wedge p_2^*\{ \kappa \} $ is pushed forward by $p_1$, only the term containing exactly one factor $p_1^*b$ survives. Thus
\[ (p_1)_*(B^ k\wedge p_2^*\{ \kappa \} )=Mb, \qquad (p_1)_*([\Delta ]\wedge p_2^*\kappa )=\kappa . \]
Since $\Theta -d[\Delta ]$ is positive, these identities prove (10.16.7); the $\partial \bar\partial $-lemma for currents on the compact Kähler manifold $Z$ identifies the cohomology class computed in this way with the Bott–Chern class $b$. Therefore $b$ is big, and by the second fact above a current with minimal singularities in $b$ has non-pluripolar mass $\int _ Zb^ k$. This gives the remaining lower bound.
$\square$
When $Y$ is smooth, the criterion descends to $Y$. Write $a_ Y=j^*a$ for the restricted class, let $\pi :Z\to Y$ be a Kähler modification with Kähler form $\kappa $, and choose a Hermitian metric $h$ on $Y$ and $C>0$ with $\pi ^*h\leq C\kappa $. If $K\in \pi ^*a_ Y$ satisfies $K\geq \delta \kappa $ with $\delta >0$, then descent of potentials (proposition 5.43.18) makes $\pi _*K$ a positive representative of $a_ Y$, and
77857992F798
10.16.8
\begin{equation} \pi _*K\geq (\delta /C)\pi _*\pi ^*h=(\delta /C)h. \end{equation}
Conversely, let $S$ be a strict positive representative of $a_ Y$. On a coordinate ball in the locus where $\pi $ is an isomorphism, $\pi ^*S$ dominates a positive multiple of the closed form $\kappa $, so the local expansion of the non-pluripolar product shows that $\pi ^*S$ has positive non-pluripolar mass on that ball. Consequently, $\mathcal R(a;Y)>0$ if and only if the restricted class $a_ Y$ is big. The Hermitian metric $h$ enters only through inequalities between currents; the products are compared with the closed form $\kappa $.
Proposition 10.16.9.FE6CD72EA03A Let $W$ be a connected compact Kähler manifold of dimension $k>0$, let $g:W\to X$ be holomorphic, and suppose $\dim g(W)<k$. Then for every class $a\in H^{1,1}_{\mathrm{BC}}(X,\mathbb R)$,
\[ \mathcal R_ W(g^*a)=0. \]
Proof.
If the image is a point, then $g^*a=0$ and (10.16.6) gives the assertion. Otherwise, choose a Kähler resolution $r:Q\to g(W)$ and resolve the graph of the meromorphic lift of $g$ to $Q$. This gives a Kähler modification $\rho :\widehat W\to W$ and a holomorphic map $F:\widehat W\to Q$ with $g\circ \rho =r\circ F$, where $r$ is also viewed as a map into $X$.
Choose a smooth representative $\eta \in a$, a Kähler form $\kappa _ Q$ on $Q$, and $C>0$ such that $C\kappa _ Q-r^*\eta \geq 0$. Since $\rho ^*g^*\eta =F^*r^*\eta $, the forms
\[ P=CF^*\kappa _ Q, \qquad A=P-\rho ^*g^*\eta \]
are closed and semipositive. Since $\dim Q<k$, we have $P^ k=0$. For any positive current $S\in g^*a$, the current $\rho ^*S+A$ is positive and lies in the nef class $\{ P\} $. Hence
\[ \int _{\widehat W}\langle (\rho ^*S)^ k\rangle \leq \int _{\widehat W}\langle (\rho ^*S+A)^ k\rangle \leq \mathcal R_{\widehat W}(\{ P\} ) =\int _{\widehat W}P^ k=0. \]
Modification invariance (Step 3 in the proof of proposition 10.16.3) now gives the result. If the class has no positive representative, its invariant is zero by definition.
$\square$
Corollary 10.16.10.3BC40D26B752 Let $a$ be a nef class on $X$. For every positive-dimensional irreducible reduced subvariety $Y\subset X$,
\[ \mathcal R(a;Y)=\int _ Ya^{\dim Y}. \]
If $\int _ Xa^ n>0$, then
43D7AA1ADEB6
10.16.11
\begin{equation} \bigcup _{\substack {Y\subset X\ \mathrm{irreducible},\ \dim Y>0\\ \mathcal R(a;Y)=0}}Y =\operatorname {Null}(a)=E_{\mathrm{nK}}(a). \end{equation}
Proof.
The pullback of $a$ to a Kähler resolution of $Y$ is nef: pull back smooth representatives with arbitrarily small negative part, and bound the pullback of $\omega $ by a multiple of a Kähler form on the resolution. Since the intersection number $\int _ Ya^{\dim Y}$ is defined through the same pullback, (10.16.6) gives the first assertion. The first equality in (10.16.11) then follows from the definition of the null locus, and the second is the classical theorem on nef and big classes recalled earlier in this chapter.
$\square$
When $X$ is Kähler and $a$ is nef, the numerical restricted volume (10.14.1) along $Y$ is also $\int _ Ya^{\dim Y}$, so the two restricted invariants agree. Outside the nef case this comparison does not identify them: (10.16.2) is a supremum over all positive currents on a resolution of $Y$, and these currents need not extend to $X$.
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.