PomeProblem 24.4.3.1E16C9C72539 Let $\alpha $ be a smooth closed real $(1,1)$-form on $X$ whose class is nef with $\int _ X\alpha ^ n>0$, and let $h$ be a Hermitian form on $X$. Prove a positive lower bound for the root volume $\lambda _\mu (\alpha )$ of definition 12.2.1, and rule out the collapse of proposition 24.3.2 along a sequence of Gauduchon metrics minimizing the bigness threshold $\kappa _ h(\alpha )$.
24.4 Closed positivity and geometric equations8871793F78BB
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A closed Kähler current places the manifold in Fujiki's class $\mathcal C$, whereas a strictly positive current $\theta +\mathrm{dd}^{\mathrm{c}}u$ in a nonclosed background keeps the torsion $\mathrm{d}\theta $ of that background. This distinction links the passage from positive volume to bigness with the geometry of the manifold, and it limits what a scalar potential equation can prove. Throughout this section $\mu $ is a smooth strictly positive probability form on $X$, and we use balanced and pluriclosed metrics in the sense of definition 11.4.1. The geometric results from the literature used below are stated, with their precise hypotheses and sources, in section 25.22.
24.4.1 Nef classes, bigness and the Fujiki class806079150C7A
A compact complex manifold belongs to Fujiki's class $\mathcal C$ exactly when it carries a closed Kähler current, that is, a closed positive $(1,1)$-current bounded below by a positive multiple of a Hermitian form. The transcendental Grauert–Riemenschneider conjecture asks whether a closed nef real $(1,1)$-class $\alpha $ with
\[ \int _ X\alpha ^ n>0 \]
is big. For such a class, bigness is equivalent to $X\in \mathcal C$; the precise statements and sources are given in section 25.22.
The obstruction can be expressed through Monge–Ampère mass. Under the same hypotheses on $\alpha $, one has
\[ \alpha \text{ is big}\quad \Longleftrightarrow \quad v_+(h)<\infty \]
for a Hermitian metric $h$ on $X$, where $v_+(h)$ is the upper mass of definition 13.14.14. There is also a characterization of $\mathcal C$ by bounded mass together with positivity of an upper volume of some closed Bott–Chern class; that volume is different from the Gauduchon variational volume $\mathcal V_\mu $. Both results are stated in section 25.22.
The Hopf threefold fails bounded mass, but it is not a counterexample to the conjecture, since a counterexample would also need a closed nef class with positive top self-intersection. A proof of the conjecture must explain why this closed positivity forces bounded mass.
The elliptic threefold of theorem 12.19.1 is the complementary test case: it has bounded mass but no big closed class. By proposition 10.19.2, its nef cone is the nonnegative span of the classes of two smooth semipositive forms of rank one, and every top self-intersection vanishes. This example of zero lower mass therefore does not bear on the conjecture either. The explicit boundary calculation of proposition 10.19.8 describes its degeneration without identifying a nonclosed metric with a closed class.
The Gauduchon variational volume of definition 12.2.1 splits the conjecture into two possible steps:
018A7DA7A817
24.4.2
\begin{equation} \int _ X\alpha ^ n>0 \quad \stackrel{?}{\Longrightarrow }\quad \mathcal V_\mu (\alpha )>0 \quad \stackrel{?}{\Longrightarrow }\quad \alpha \text{ is big}. \end{equation}
Neither implication is known on arbitrary compact complex manifolds. The second concerns closed nef backgrounds; the projective counterexample of theorem 12.14.6 has a nonclosed background and does not decide it. On a compact Kähler manifold both implications follow from the usual volume formula for nef classes and the classical nef-and-big criterion. Without closedness, positivity of $\mathcal V_\mu $ cannot characterize $\mathcal C$, since every Hermitian metric $h$ on every compact complex manifold satisfies $\mathcal V_\mu (h)>0$.
The test metrics can be normalized by $\int _ Xh\wedge \gamma ^{n-1}=1$. By proposition 24.3.2, a positive lower bound for $D_\mu (\gamma )$ along one sequence minimizing the threshold would give the second implication. Solving a Gauduchon equation changes the volume density of a test metric while keeping the Aeppli class of $\gamma ^{n-1}$; since $\alpha $ is closed, it also keeps the pairing with $\alpha $. The missing estimate is control of the trace normalization and of the determinant denominator after this change. Prescribing a volume density alone does not give this estimate.
24.4.4 Positive lower mass and concentrationE1772510D012
Positive lower mass is an integrability property of the Gauduchon tests at an endpoint exponent. It is independent of finite upper mass: the threefold of theorem 12.19.1 has $v_-(h)=0$ for every Hermitian metric and carries a balanced metric of finite upper mass, while the normalized Hopf metric has $v_-=6$ and $v_+=\infty $.
Let $n\geq 2$, let $h$ be a Hermitian form and $\mathrm{d}V$ a smooth positive volume form. Recall the set $\mathscr G_ h$ of normalized Gauduchon currents (section 11.7) and the determinant measure $\Gamma _0$ of (12.7.1), and write
\[ \Gamma _0(T)=k_ T\, \mathrm{d}V,\qquad T\in \mathscr G_ h. \]
By theorem 12.9.2,
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24.4.5
\begin{equation} v_-(h)>0 \quad \Longleftrightarrow \quad \sup _{T\in \mathscr G_ h} \| k_ T\| _{L^{n/(n-1)}(\mathrm{d}V)}<\infty , \end{equation}
and the exact value of the supremum is $v_-(h)^{-1/n}$, including the value $+\infty $. Uniform estimates for every smaller exponent $1<q<n/(n-1)$ hold without either mass property. In the example of zero lower mass, the endpoint estimate fails already for smooth strictly positive tests. Positivity, the condition $\mathrm{dd}^{\mathrm{c}}T=0$ and the trace normalization alone therefore do not control concentration at the endpoint exponent.
Problem 24.4.6.81116FD94515 Let $n\geq 2$ and let $h$ be a Hermitian form on $X$. Find geometric conditions, beyond the metric identities in corollary 12.18.1, that imply the endpoint estimate in (24.4.5), and explain how they exclude the degeneration of zero lower mass constructed in section 12.19.
Corollary 12.18.1 proves the endpoint bound when some Hermitian metric $g$ satisfies $\mathrm{dd}^{\mathrm{c}}g=\mathrm{dd}^{\mathrm{c}}(g^2)=0$, which includes all compact Kähler manifolds and all compact complex surfaces. For a fixed smooth strictly positive density, theorem 12.7.3 gives a smooth maximizing current with bounded determinant density. However, proposition 12.18.2 produces maximizing sequences in fixed Kähler classes whose moments diverge at every exponent above the endpoint. Exact maximizers and maximizing sequences therefore require different estimates.
For principal elliptic threefolds over Kähler surfaces, theorem 12.17.7 gives a geometric sufficient condition: the sum of the squares of the two real curvature forms is nonnegative and not identically zero. A bounded cubic identity then gives positive lower mass, and the determinant bound transfers to every Hermitian background. These manifolds admit no pluriclosed metric; proposition 12.17.11, proposition 12.17.14 give explicit families. A primitive pair of curvature forms has the opposite sign, by proposition 12.17.9, so this criterion leaves the general problem open.
The balanced condition and bounded upper mass do not suffice, even together, as the threefold of theorem 12.19.1 shows. On this elliptic fibration, the identity (12.19.19) relates the failure of the lower bound to a vertical Dirichlet integral, and the horizontal curvature forms $\alpha ,\beta $ of section 12.19 have a positive exact product $\alpha \wedge \beta $, which excludes pluriclosed metrics and closed Kähler currents. These calculations give geometric information about this example, but not a classification of all degenerations of the lower mass.
For equations the consequence is exact: in a Hermitian background, corollary 15.27.5 characterizes positive lower mass by solvability of the normalized equation for all finite non-pluripolar measures. When the lower mass vanishes, even a positive density in $L^1$ bounded away from zero can admit no positive scaled subsolution. The remaining questions about equations concern specified measures or additional geometric hypotheses, and, separately, the effect of the exponential weight.
24.4.7 Primitive elliptic bundles148E1DFBEDF1
On a primitive elliptic bundle, the variation of the top-degree mass is a vertical Dirichlet integral. This gives the exact upper mass and a positive lower bound for separated potentials. The lower mass over all potentials remains a geometric question, since the collapse construction over a product of curves does not transfer directly to a base without curves.
Let $(B,\eta )$ be a connected compact Kähler surface and let $L\to B$ be a holomorphic line bundle such that
\[ \int _ Bc_1(L)\wedge \eta =0. \]
Choose a smooth Hermitian metric on $L$ whose curvature is primitive with respect to $\eta $. Such a metric exists: for an arbitrary metric, the trace of the curvature with respect to $\eta $ has mean zero by the condition above, and the scalar Poisson equation gives a smooth change of metric that cancels this trace. Let $c$ be the resulting smooth closed curvature representative of $-c_1(L)$, so that $\eta \wedge c=0$. Write $\pi :L\to B$ for the bundle projection and $w$ for a nonzero vector in a fiber, with norm taken in the chosen metric. On the complement $L^\times $ of the zero section, put
\[ t=\log |w|^2,\qquad \mathrm{dd}^{\mathrm{c}}t=\pi ^*c. \]
For $q>1$, form the quotient
\[ X=L^\times /\langle w\mapsto qw\rangle , \qquad \pi :X\longrightarrow B, \qquad T=2\log q, \qquad A=\int _ B\eta ^2, \]
where we use the same symbol $\pi $ for the induced projection. The action increases $t$ by $T$. It is free and properly discontinuous, and the compact annulus $1\leq |w|\leq q$ meets every orbit. Hence $X$ is a connected compact complex threefold; in this subsection it replaces the general manifold of the chapter. Over a trivializing open set $U\subset B$, a local holomorphic frame of $L$ identifies the fibration with $U\times (\mathbb C^*/q^{\mathbb Z})$. Since $\mathrm{d}t$ and $\mathrm d^{\mathrm c}t$ are invariant under the action, they descend to $X$, and we set
\[ D=\mathrm{d}t\wedge \mathrm d^{\mathrm c}t,\qquad \omega =\pi ^*\eta +D. \]
The form $D$ has rank one; it is semipositive and positive on the vertical line, and its kernel projects isomorphically onto the tangent space of $B$. Hence $\omega $ is Hermitian.
Proposition 24.4.8.8E5CAA6E5122 The metric $\omega $ is balanced and satisfies
\[ \mathrm{dd}^{\mathrm{c}}\omega =-\pi ^*(c^2)\geq 0,\qquad \int _ X\omega ^3=3TA. \]
For every smooth real $u$,
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24.4.9
\begin{equation} \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^3 =3TA-3\int _ X\mathrm{d}u\wedge \mathrm d^{\mathrm c}u\wedge \bigl (-\pi ^*(c^2)\bigr ). \end{equation}
In particular $v_+(\omega )=3TA$, and the canonical bundles satisfy $K_ X\simeq \pi ^*K_ B$. If $c_1(L)\ne 0$ in real cohomology, $X$ admits neither a pluriclosed metric nor a closed Kähler current. If $c_1(L)=0$ in real cohomology, the chosen $\omega $ is Kähler.
Proof. Write $a=\mathrm{d}t$ and $b=\mathrm d^{\mathrm c}t$. Then $\mathrm{d}a=0$, $\mathrm{d}b=\pi ^*c$, $\mathrm d^{\mathrm c}a=-\pi ^*c$ and $\mathrm d^{\mathrm c}b=0$. Consequently
\[ D^2=0,\qquad \mathrm{d}D=-a\wedge \pi ^*c, \qquad \mathrm{dd}^{\mathrm{c}}D=-\pi ^*(c^2), \]
\[ \mathrm{d}(\omega ^2)=-2\pi ^*(\eta \wedge c)\wedge a=0, \]
so $\omega $ is balanced. Since $c$ is primitive, its eigenvalues relative to $\eta $ are $\lambda $ and $-\lambda $ for a function $\lambda \geq 0$, so $c^2=-\lambda ^2\eta ^2$. This proves the asserted sign of $\mathrm{dd}^{\mathrm{c}}\omega $.
To compute the volume, fix an elliptic fiber $E=\pi ^{-1}(b_0)$, where $b_0\in B$. In a logarithmic fiber coordinate $x+\mathrm{i}y$, the real periods are $\log q$ and $2\pi $, while $t$ equals $2x$ up to an additive constant and $D|_ E=\pi ^{-1}\mathrm{d}x\wedge \mathrm{d}y$. Thus $\int _ ED=T$. Since $\omega ^3=3\pi ^*(\eta ^2)\wedge D$, the volume formula follows.
For the mass identity, expand the cube in (24.4.9). The terms with one and with three factors $\mathrm{dd}^{\mathrm{c}}u$ integrate to zero, because $\omega ^2$ is closed and $\mathrm{dd}^{\mathrm{c}}u$ is exact. For the remaining term, put $S=\mathrm{dd}^{\mathrm{c}}\omega $. Integration by parts gives
\[ \int _ X(\mathrm{dd}^{\mathrm{c}}u)^2\wedge \omega =\int _ Xu\, \mathrm{dd}^{\mathrm{c}}u\wedge S =-\int _ X\mathrm{d}u\wedge \mathrm d^{\mathrm c}u\wedge S, \]
where the last equality uses $\mathrm{d}S=0$. This proves the identity. Its last integral is nonnegative, and $u=0$ attains the upper bound $3TA$. Since smooth and bounded potentials have the same mass extrema, by theorem 12.15.4, this proves $v_+(\omega )=3TA$.
If $c_1(L)=0$ in real cohomology, then $c$ is exact and Stokes' theorem gives $\int _ Bc^2=0$. The pointwise identity $c^2=-\lambda ^2\eta ^2$ then forces $c=0$, and $\omega $ is closed. Otherwise $S=-\pi ^*(c^2)$ is nonzero, smooth and semipositive. For a pluriclosed metric $\gamma $, Stokes' theorem would give
\[ 0<\int _ X\gamma \wedge S =\int _ X\omega \wedge \mathrm{dd}^{\mathrm{c}}\gamma =0, \]
a contradiction. Likewise, a closed positive current $R\geq \varepsilon \omega $ with $\varepsilon >0$ would give $0<\int _ XR\wedge S=\int _ X\omega \wedge \mathrm{dd}^{\mathrm{c}}R=0$. All these pairings are pairings of currents with smooth forms.
Finally, let $w$ be a local holomorphic fiber coordinate given by a frame of $L$, and let $\sigma $ be a local holomorphic two-form on $B$. The form $\pi ^*\sigma \wedge \mathrm{d}w/w$ is invariant under the quotient action. Changing the frame of $L$ adds to $\mathrm{d}w/w$ a horizontal one-form, whose wedge product with $\pi ^*\sigma $ vanishes. These local sections of $K_ X$ therefore have the transition functions of $\pi ^*K_ B$, which proves $K_ X\simeq \pi ^*K_ B$. $\square$
For separated potentials, positivity bounds both the amplitude and the downward slope of the vertical derivative. The sharp variance bound under these two constraints determines how much mass such a potential can lose. For $M,T>0$, set
\[ F(M,T)= \begin{cases} T^2/12,& T\leq 2M,\\ M^2-4M^3/(3T),& T\geq 2M. \end{cases} \]
Lemma 24.4.10.987A41833787 If a smooth real $T$-periodic function $p$ has mean zero, $|p|<M$ and $p'>-1$, then
\[ \frac1T\int _0^ T p^2\, \mathrm{d}s<F(M,T). \]
The supremum of the left side over this class is $F(M,T)$.
Proof. For $p=0$ the inequality holds because $F(M,T)>0$. Otherwise set $a=\max p>0$, $-b=\min p<0$ and $r=a+b$. Integrating $-p'<1$ over a forward interval $I$ from a maximum to the next minimum gives $r<|I|\leq T$; also $r<2M$, since $|p|<M$. The function $G(s)=(a-s)(s+b)$ is nonnegative on the range of $p$. Since $p$ has mean zero and $-p'<1$, integration over $I$ gives
\[ abT-\int _0^ T p^2\, \mathrm{d}s =\int _0^ T G(p)\, \mathrm{d}s \geq -\int _ I G(p)p'\, \mathrm{d}s =\frac{r^3}{6}. \]
As $ab\leq r^2/4$, the average of $p^2$ is at most $r^2/4-r^3/(6T)$. This expression increases strictly for $0<r<T$, and its value at $r=\min \{ 2M,T\} $ is $F(M,T)$. Since $r<\min \{ 2M,T\} $, the strict inequality follows.
For sharpness, consider the following $T$-periodic profiles $p_0$. When $T\geq 2M$, let $p_0$ be constant equal to $M$ and to $-M$ on two intervals of length $(T-2M)/2$ each, joined by a descent of slope $-1$ and length $2M$, and close the period by an upward jump. When $T\leq 2M$, let $p_0(s)=T/2-s$ for $0\leq s<T$, with its upward jump at the end of the period. In both cases $p_0$ has mean zero, $|p_0|\leq M$, $p_0'\geq -1$ in the sense of distributions, and the average of $p_0^2$ is $F(M,T)$; both constructions apply when $T=2M$. Convolution with a nonnegative periodic approximate identity preserves the mean and the bound on the distributional derivative, and multiplication by $1-\varepsilon $ makes both inequalities strict. As the convolution scale and $\varepsilon $ tend to zero, the resulting smooth functions converge to $p_0$ in $L^2$, which proves that the supremum is $F(M,T)$. $\square$
Proposition 24.4.11.E63ACDB15696 Suppose $c_1(L)\ne 0$ in real cohomology, and put
\[ \kappa =-\frac{\int _ Bc^2}{A}>0. \]
Every smooth potential $u=f(t)+\pi ^*v$ with $f$ of period $T$, $v\in C^\infty (B,\mathbb R)$ and $\omega +\mathrm{dd}^{\mathrm{c}}u>0$ satisfies
\[ \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^3 >3TA\bigl (1-\kappa F(\kappa ^{-1/2},T)\bigr )>0. \]
Let $\lambda \geq 0$ be the nonnegative eigenvalue function of $c$ relative to $\eta $, so $c^2=-\lambda ^2\eta ^2$, and put $M_0=1/\max _ B\lambda $. For the subclass $v=0$, the exact mass infimum is
\[ 3TA\bigl (1-\kappa F(M_0,T)\bigr )>0. \]
It is approached by smooth potentials $f(t)$ with $\omega +\mathrm{dd}^{\mathrm{c}}f(t)>0$, and it is not attained by such potentials.
Proof. Put $p=f'$. The chain rule gives
\[ \omega +\mathrm{dd}^{\mathrm{c}}u =\pi ^*(\eta +pc+\mathrm{dd}^{\mathrm{c}}v)+(1+p')D. \]
Positivity on the vertical line implies $p'>-1$. Over every point of $B$, the function $t$ takes every value modulo $T$, so restriction to $\ker \partial t$ shows that $\eta +pc+\mathrm{dd}^{\mathrm{c}}v$ is a Kähler form on $B$ for every value of $p$. Its square has integral
\[ \int _ B(\eta +pc+\mathrm{dd}^{\mathrm{c}}v)^2=A(1-\kappa p^2)>0, \]
since primitivity removes the mixed term and Stokes' theorem removes the exact terms. Hence $|p|<\kappa ^{-1/2}$.
Only the vertical derivative enters (24.4.9), because $-\pi ^*(c^2)$ has top degree in the base directions. For a $T$-periodic function $F_0$, integration over a fiber gives $\int _ E F_0(t)D=\int _0^ T F_0(s)\, \mathrm{d}s$, and therefore
\[ \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^3 =3TA\left(1-\frac\kappa T\int _0^ T p^2\, \mathrm{d}s\right). \]
Now apply lemma 24.4.10 with $M=\kappa ^{-1/2}$. The formula for $F$ gives $F(M,T)<M^2$, which proves positivity of the bound.
For $v=0$, diagonalizing $c$ with respect to $\eta $ shows that the horizontal positivity condition is exactly $|p|<M_0$; together with $p'>-1$ it also suffices for positivity of the full form. Moreover $\kappa $ is the $\eta ^2$-average of $\lambda ^2$, so $\kappa \leq M_0^{-2}$. Lemma 24.4.10 with $M=M_0$ therefore proves the claimed positive lower bound and its nonattainment. The smooth approximating functions in its proof have mean zero and hence periodic primitives $f_\varepsilon $, and the potentials $f_\varepsilon (t)$ prove that the bound is sharp. $\square$
The proof of the first bound does not show that it is sharp for separated potentials over an arbitrary base: positivity of the classes $[\eta ]+p[c]$ alone does not provide one function $v$ that makes all the representatives $\eta +pc+\mathrm{dd}^{\mathrm{c}}v$ positive. For the threefold over $\mathbb P^1\times \mathbb P^1$ of section 12.19, where $A=2$, $\kappa =1$ and $T=4\pi $, the bound is $8$ and it is sharp.
A base without curves rules out the averages of horizontal curves used in the known collapse example. The following construction uses the period theorem for K3 surfaces and the description of their Kähler cone [Theorems 7.4.1 and 8.5.2 of the online version, HuyK3].
Example 24.4.12.8D6FF393E593 There is a K3 surface $B$ with $\operatorname {Pic}(B)=\mathbb Z[L]$, $c_1(L)^2=-4$, and a Kähler form $\eta $ orthogonal to $c_1(L)$. It has no compact curves. The associated $X$ has trivial canonical bundle, a balanced metric of finite upper mass, and no pluriclosed metric or closed Kähler current. It has no divisors, and every compact irreducible curve is an elliptic fiber.
Proof. In a hyperbolic plane of the K3 lattice, choose a basis $e,f$ with $e^2=f^2=0$ and $e\cdot f=1$, and put $\ell =e-2f$. This vector is primitive and has square $-4$. Choose a period in $\ell ^\perp $ outside the hyperplanes perpendicular to integral vectors not in $\mathbb Z\ell $. Such a period exists by Baire's theorem: the period domain in $\ell ^\perp $, of signature $(3,18)$, is nonempty and each excluded hyperplane is proper. The period theorem and the description of integral $(1,1)$-classes give the asserted Picard group. An irreducible curve would have nonzero class, because its Kähler area is positive, and hence square $-4m^2<-2$ for some integer $m\neq 0$, contradicting adjunction. Thus there are no curves.
The orthogonal complement of $\ell $ in $H^{1,1}(B,\mathbb R)$ has signature $(1,18)$. Choose a class of positive square there, with sign chosen so that it lies in the component of the positive cone containing the Kähler classes. Since there are no rational curves, the Kähler-cone theorem shows that this class is Kähler; choose $\eta $ in it. Proposition 24.4.8 then gives the assertions about the metric and the canonical bundle.
If a divisor $Z\subset X$ dominated $B$, the push-forward of its integration current would be $\pi _*[Z]=m$ for a positive generic degree $m$. Stokes' theorem and the projection formula would then give
\[ 0=\int _ X[Z]\wedge \mathrm{dd}^{\mathrm{c}}\omega =m\int _ B(-c^2)>0. \]
Thus no divisor dominates $B$. The image of any other irreducible divisor would be a curve or a point; the base has no curves, and a fiber has dimension one, so both alternatives are impossible. Likewise, the image of an irreducible compact curve is a point, and the curve is therefore an entire elliptic fiber. $\square$
Put $V=3TA$. The root volumes of complementary tests can be computed explicitly for one radial family. For smooth real $T$-periodic functions $a,g$, consider
\[ \Gamma =a(t)\eta ^2+g(t)\eta \wedge D. \]
Proposition 24.4.13.3D0A0BF3A375 The form $\Gamma $ is strictly positive and $\mathrm{dd}^{\mathrm{c}}$-closed exactly when $a$ is a positive constant and $g>0$. For such a form put $\bar g=T^{-1}\int _0^ Tg(s)\, \mathrm ds$ and $d_\Gamma =\int _ X\omega \wedge \Gamma $. Its positive square root and volume are
\[ \gamma =\sqrt a\, \eta +\frac{g(t)}{2\sqrt a}D, \qquad \int _ X\gamma ^3=\frac32TA\sqrt a\, \bar g \leq \frac{d_\Gamma ^{3/2}}{\sqrt V}. \]
Equality holds exactly when $\bar g=2a$, without requiring $g$ to be constant. In particular every degree-one test in this family has root volume at most $V^{-1/2}$.
Proof. Since $\eta \wedge c=0$, the form $g\eta \wedge D$ is closed. The chain rule gives
\[ \mathrm{dd}^{\mathrm{c}}(a\eta ^2)=a''D\wedge \eta ^2, \]
the $a'c\wedge \eta ^2$ term having excess base degree. Thus $\mathrm{dd}^{\mathrm{c}}\Gamma =0$ forces $a''=0$, and periodicity makes $a$ constant. In a positive coframe with $\eta =\sigma _1+\sigma _2$, $D=\sigma _3$, the complementary coefficient matrix of $\Gamma $ is $\operatorname {diag}(g,g,2a)$. Hence positivity is exactly $a,g>0$. The displayed $\gamma $ is positive and squares to $\Gamma $. Multiplication and fiber integration give
\[ d_\Gamma =TA(a+\bar g),\qquad \gamma ^3=\tfrac 32\sqrt a\, g\eta ^2\wedge D. \]
For fixed $d_\Gamma >0$ set $s=d_\Gamma /(TA)$. Then $0<a<s$, $\bar g=s-a$, and $\sqrt a(s-a)$ has its unique maximum at $a=s/3$. Substitution proves the bound and its equality condition. $\square$
For the divisor-free threefold of example 24.4.12, theorem 11.8.1 gives divergent transverse pairings against normalized Gauduchon tests. The radial family above has bounded root volume at fixed degree, even when $g$ concentrates arbitrarily strongly. Large transverse pairings do not by themselves give unbounded root volumes or a sequence of potentials with masses tending to zero.
Problem 24.4.14.B341DA2B6EFE Determine the lower Monge–Ampère mass (definition 13.14.14) of the primitive elliptic bundles above, for instance of their balanced metrics $\omega $ from proposition 24.4.8, in particular for the bundles of example 24.4.12. Does it vanish whenever $c_1(L)\neq 0$ in real cohomology?
The estimates for separated potentials do not answer this question; the known collapse over $\mathbb P^1\times \mathbb P^1$ uses potentials outside that class. On the K3 example every compact curve is vertical, so averaging over such curves cannot produce the horizontal directions used in lemma 12.19.10. A proof of collapse would therefore need a different construction, while positive lower mass would require a bound for all admissible potentials or for all normalized Gauduchon tests.
24.4.15 Null loci and the singularities of the Chern–Ricci flowF6F63D20E784
Return to the general manifold $X$ of dimension $n$. Let $\omega _0$ be a Hermitian metric, taken as the initial metric of the Chern–Ricci flow, and suppose that the flow has a finite maximal time $\tau $. With the Ricci convention of corollary 12.34.7, the limiting background $\theta _\tau =\omega _0-\tau \operatorname {Ric}(\omega _0)$ is nef. To identify the region where the flow converges smoothly as the complement of a null locus, one first needs bigness of $\theta _\tau $. The dual currents detect the failure of strict positivity, but the degeneration of their determinants does not by itself locate an analytic exceptional set.
For a big smooth background $\theta $, the non-Hermitian locus $E_{\mathrm{nH}}(\theta )$ is the intersection of the singular sets of the strict currents $\theta +\mathrm{dd}^{\mathrm{c}}u$, those bounded below by a positive multiple of a Hermitian form, whose potentials have almost analytic singularities; one such current attains the intersection. Here a potential has almost analytic singularities if it is locally a positive multiple of the logarithm of the norm of an ideal up to a bounded remainder, and this remainder becomes smooth after subtracting the divisorial logarithms on a suitable log resolution. This is the class of singularities used in the theorem stated in section 25.22.
For nef and big $\theta $, define
\[ \operatorname {Null}(\theta ) =\bigcup _{\substack {V\subset X\text{ irreducible analytic}\\ \dim V>0,\ \theta |_ V\text{ not big}}}V, \]
where bigness on a singular subvariety has its intrinsic analytic meaning. By the theorem recalled in section 25.22, the equality $E_{\mathrm{nH}}(\theta )=\operatorname {Null}(\theta )$ holds for nef and big smooth backgrounds on arbitrary compact complex manifolds. The numerical criterion recalled there has additional hypotheses: bounded mass on $X$ and positive lower volume on its positive-dimensional submanifolds.
The flow theorem recalled in section 25.22 identifies the finite-time singularities of the Chern–Ricci flow with the null locus under positive limiting lower volume, for arbitrary initial metrics in dimension two and for Guan–Li initial metrics in higher dimensions, that is, those with $\mathrm{dd}^{\mathrm{c}}\omega _0=0$ and $\mathrm{d}\omega _0\wedge \mathrm d^{\mathrm c}\omega _0=0$. Positivity of $\mathcal V_\mu $ cannot replace these hypotheses, as the example below shows.
The calculation with currents in corollary 12.34.7 applies to arbitrary initial metrics. When $\tau <\infty $, there are normalized Gauduchon currents $T$ annihilating $\theta _\tau $:
\[ \int _ X\theta _\tau \wedge T=0. \]
If $\mathcal V_\mu (\theta _\tau )>0$, the determinant-weighted measures of all such currents vanish, by proposition 12.34.2. This is a precise statement of degeneracy, but it does not imply that their traces concentrate on analytic sets, since a smooth positive semidefinite matrix can have deficient rank on a whole open set.
Theorem 12.34.10 shows that positivity of the variational volume does not imply bigness at the end of the flow: in every dimension at least three, there is a Chern–Ricci flow with finite maximal time whose limiting background is nef, not big, and has positive $\mathcal V_\mu $. Thus being the endpoint of a flow does not exclude the radial obstruction. In this model, proposition 12.34.12 shows that every normalized annihilating current has its trace supported on the degenerate fiber, with a vertical coefficient matrix of rank one. This support follows from the explicit semipositive representative, not from the vanishing of determinants alone.
If $\theta _\tau $ is big, whether by assumption or under additional hypotheses, the parabolic theorem recalled in section 25.22 gives convergence as currents and smooth local convergence off $E_{\mathrm{nH}}(\theta _\tau )$. Combined with nefness and the null-locus theorem, this identifies the analytic exceptional set. The example above does not contradict this conditional theorem, but any further numerical or geometric criterion must impose enough structure to exclude its nonbig endpoint. The local continuity of envelopes and the homogeneous replacement arguments of theorem 15.8.1 provide an elliptic foundation away from the poles of a strict barrier; the parabolic estimates are a separate input.
Completeness of the limiting metric requires an additional estimate. In theorem 10.24.1, the metric defined by the unique finite-energy solution of a divisorial exponential equation in a big closed class is incomplete on every nonempty connected open subset of its ample locus, because its minimal divisorial singularity cancels the prescribed pole. Nefness alone does not help: theorem 10.24.7 keeps a nonintegrable pole with smooth positive coefficient and a complete cusp, but has a disjoint null curve at finite distance; its solution has zero Lelong numbers and finite energy, but is not of minimal ordinary type. The construction of theorem 10.24.16 with compatible initial data also gives an incomplete auxiliary metric, with the exact identities for the resolution and the prescribed volume, but its class is not nef. Simultaneous nefness and compatibility remain a separate question, and none of these elliptic examples decides whether the limits of an evolving flow are complete; see section 25.19.
For integrable poles weakened by logarithmic factors, corollary 10.25.5 gives a sufficient condition for completeness in terms of the density, the potential and the tangential determinant; the proof controls normal lengths and applies in arbitrary Hermitian backgrounds. The torus models of example 10.25.11, example 10.25.12 attain its critical exponent and show that it is sharp, even for a Kähler background and full mass. Their pole coefficients need not extend smoothly and positively across the divisor, so the geometric smoothing question remains separate.
The regularity of the residual also matters. By theorem 5.39.1, a Hölder residual on a compact manifold that defines a smooth positive metric off its analytic singular set puts that set at finite distance. Continuity alone does not suffice: the prescribed radial densities of theorem 10.22.7 have bounded continuous minimal remainders and complete regular ends for the range $n<d\leq 2n$ of the exponent $d$ in that theorem. These examples concern the specified densities; they say nothing about compatibility with an evolving Chern–Ricci flow.
24.4.16 Balanced and pluriclosed metricsB8BA04F9415A
The balanced–pluriclosed conjecture asks whether a compact complex manifold carrying both a balanced metric $\beta $ and a pluriclosed metric $\omega $ must be Kähler. The two metrics may differ; a single metric satisfying both conditions is already Kähler by lemma 24.4.19. The source of the conjecture and its formulation through a flow are given in section 25.22.
A manifold in Fujiki's class $\mathcal C$ that carries a pluriclosed metric is Kähler, by the result recalled there. It would therefore suffice to construct a closed Kähler current from the two metrics. The scalar potential equation alone cannot produce the required closedness.
Indeed, for every distributional potential $u$,
\[ \mathrm{d}(\omega +\mathrm{dd}^{\mathrm{c}}u)=\mathrm{d}\omega . \]
Thus if the pluriclosed metric is not closed, no potential in its background makes it Kähler; the finite-volume equation, including its smooth solution, does not change this differential datum. Similarly, the Gauduchon metric in a conformal class need not be balanced. Positivity and a scalar normalization cannot replace either missing closedness condition.
The closed form $\beta ^{n-1}$ pairs trivially with exact real two-forms, so it can give a conserved trace along a flow. When $c_1(X)=0$ in real cohomology, the flow conjecture recalled in section 25.22 proposes that, in the presence of a balanced metric, the pluriclosed flow exists for all time and converges to a Kähler metric. The invariant examples cited there use the conserved trace. For general metrics, the analytic difficulty is to turn this integral into a pointwise bound.
Parallel torsion gives a setting in which the integral constraint of a balanced trace becomes pointwise. Suppose $n\geq 2$, write $J$ for the complex structure of $X$, let $g$ be a Hermitian metric on $X$, and let $D$ be a real connection on $TX$ that preserves $g$ and $J$ and has parallel torsion $T^ D$:
\[ Dg=0,\qquad DJ=0,\qquad DT^ D=0, \qquad T^ D(U,V)=D_ UV-D_ VU-[U,V]. \]
The balanced and pluriclosed metrics considered below need not equal $g$. Write
\[ E_ D^ k=\{ \alpha \in \Omega ^ k(X,\mathbb R):D\alpha =0\} \]
for the space of $D$-parallel real $k$-forms. Since $X$ is connected, evaluation at one point is injective on this space, so it is finite-dimensional. Its complexification inherits the decomposition by bidegree because $DJ=0$.
Lemma 24.4.17.2066C3EEC6F6 There is a linear projection $S:\Omega ^\bullet (X,\mathbb R)\to E_ D^\bullet $ with $S\mathrm d=\mathrm dS$. Its complexification preserves bidegree and commutes with $\partial $ and $\bar\partial $. It preserves strict positivity in bidegrees $(1,1)$ and $(n-1,n-1)$.
Proof. Step 1: the projection. For a parallel form $\alpha $, the formula for exterior differentiation in terms of a connection expresses $\mathrm d\alpha $ as an algebraic contraction of $T^ D$ with $\alpha $. Since $DT^ D=0$, it follows that $\mathrm dE_ D^ k\subseteq E_ D^{k+1}$. The Hodge star of $g$ is also parallel and maps $E_ D^ k$ onto $E_ D^{2n-k}$. Consequently the pairing
\[ E_ D^ k\times E_ D^{2n-k}\longrightarrow \mathbb R, \qquad (\alpha ,\beta )\longmapsto \int _ X\alpha \wedge \beta \]
is nondegenerate: pairing a nonzero $\alpha $ with $*_ g\alpha $ gives its squared $L^2$ norm, which is positive. Hence every smooth real $k$-form $\alpha $ determines a unique $S_ k\alpha \in E_ D^ k$ such that
3EA53204CDA3
24.4.18
\begin{equation} \int _ X S_ k\alpha \wedge \beta =\int _ X\alpha \wedge \beta \qquad (\beta \in E_ D^{2n-k}). \end{equation}
This defines a projection onto $E_ D^ k$. For $\gamma \in E_ D^{2n-k-1}$ we have $\mathrm d\gamma \in E_ D^{2n-k}$, so Stokes' theorem gives
\[ \int _ X\mathrm dS_ k\alpha \wedge \gamma =(-1)^{k+1}\int _ X\alpha \wedge \mathrm d\gamma =\int _ X S_{k+1}(\mathrm d\alpha )\wedge \gamma . \]
Both forms being compared are parallel, so nondegeneracy proves $\mathrm dS_ k\alpha =S_{k+1}\mathrm d\alpha $.
Step 2: positivity. Fix $p\in X$. Let $H$ be the closure of the full $D$-holonomy group at $p$, a compact subgroup of the unitary group, with probability Haar measure $m_ H$. For a path from $p$ to $x$, write $P_ x:T_ pX\to T_ xX$ for parallel transport along it. The average
\[ R_\alpha (x)=\int _ H h^*(P_ x^*\alpha _ x)\, \mathrm dm_ H(h) \]
is independent of the path, since a change of path acts by an element of $H$. It is an $H$-invariant tensor at $p$ depending smoothly on $x$, because paths can be chosen smoothly on each coordinate neighborhood. Average it over $X$:
\[ a_ p=\frac1{\operatorname {Vol}_ g(X)} \int _ X R_\alpha (x)\, \mathrm dV_ g(x). \]
Being $H$-invariant, the tensor $a_ p$ extends uniquely to a global parallel form $\widetilde S_ k\alpha $. For parallel $\beta $, write $\alpha \wedge \beta =f\, \mathrm dV_ g$. Since parallel transport is unitary and $\beta $ is invariant under holonomy, averaging gives
\[ R_\alpha (x)\wedge \beta _ p=f(x)\, \mathrm dV_{g,p}. \]
Integrating this equality shows that $\widetilde S_ k\alpha $ satisfies (24.4.18); hence $\widetilde S_ k=S_ k$.
Every parallel transport and every holonomy element preserves reality, bidegree and the positive cones. A smooth strictly positive $(q,q)$-form, for $q=1$ or $n-1$, is bounded below by a positive multiple of the $q$th power of the fundamental form of $g$, uniformly on the compact manifold, and both probability averages preserve that lower bound. Therefore $S$ preserves strict positivity and bidegree. Comparing bidegrees in $S\mathrm d=\mathrm dS$ proves that $S$ commutes with $\partial $ and $\bar\partial $. $\square$
Lemma 24.4.19.4865DD012C9C If a Hermitian form $\zeta $ satisfies $\mathrm d\zeta ^{n-1}=0$ and $\mathrm{dd}^{\mathrm{c}}\zeta =0$, then $\mathrm d\zeta =0$.
Proof. For $n=2$, the balanced condition is the assertion $\mathrm d\zeta =0$. Suppose $n\geq 3$ and set $\alpha =\partial \zeta $. Since $\zeta $ is balanced, $\alpha \wedge \zeta ^{n-2}=0$, so $\alpha $ is a primitive $(2,1)$-form. Applying $\partial \bar\partial $ to the balanced power gives
\[ 0=\partial \bar\partial \zeta ^{n-1} =(n-1)\partial \bar\partial \zeta \wedge \zeta ^{n-2} +(n-1)(n-2)\alpha \wedge \bar\alpha \wedge \zeta ^{n-3}. \]
The first term vanishes because $\zeta $ is pluriclosed. For a primitive $(2,1)$-form, pointwise Hermitian linear algebra gives
\[ -\mathrm i\, \alpha \wedge \bar\alpha \wedge \frac{\zeta ^{n-3}}{(n-3)!} =|\alpha |_\zeta ^2\frac{\zeta ^ n}{n!}. \]
Thus $\alpha =0$, and since $\zeta $ is real, also $\bar\partial \zeta =0$, so $\mathrm d\zeta =0$. $\square$
Theorem 24.4.20.781DAEC1B4CE Let $D$ be a connection on $TX$ as above, preserving $g$ and $J$ and with parallel torsion. If $X$ admits a balanced metric and a pluriclosed metric, then it admits a $D$-parallel Kähler metric.
Proof. Let $\beta $ be a balanced metric and $\omega $ a pluriclosed metric. By lemma 24.4.17, $S(\beta ^{n-1})$ is strictly positive, parallel and closed. It has a unique smooth positive $(n-1)$-st root $B$. Indeed, after the fixed normalization of wedge powers, the corresponding map on matrices is $G\mapsto C=(\det G)G^{-1}$, whose inverse is
\[ G=(\det C)^{1/(n-1)}C^{-1}. \]
By uniqueness the root is invariant under parallel transport. Thus $DB=0$ and $\mathrm dB^{n-1}=0$. Similarly, $P=S\omega $ is a parallel positive pluriclosed form.
Consider the finite-dimensional space of real parallel directions of exact type
\[ \mathcal A_ D=\{ (\mathrm d\xi )^{1,1}:\xi \in E_ D^1\} . \]
Every such direction is $\partial \bar\partial $-closed, since
\[ (\mathrm d\xi )^{1,1}=\partial \xi ^{0,1}+\bar\partial \xi ^{1,0}. \]
For a parallel positive form $Q$, the volume functional has derivative
B755FCE3BC15
24.4.21
\begin{equation} \left.\frac{\mathrm d}{\mathrm dt}\right|_{t=0} \frac1{n!}\int _ X(Q+t(\mathrm d\xi )^{1,1})^ n =\frac1{(n-1)!}\int _ X\xi \wedge \mathrm dQ^{n-1}, \end{equation}
because the $(2,0)$ and $(0,2)$ parts of $\mathrm d\xi $ pair to zero with $Q^{n-1}$. Since $\mathrm dQ^{n-1}$ is parallel, nondegeneracy of the parallel pairing shows that all these derivatives vanish exactly when $Q$ is balanced.
Apply this to the balanced form $B$. For $a\in \mathcal A_ D$, the function $\operatorname {tr}_ B a$ is parallel and hence constant. Its integral against $B^ n$ vanishes by (24.4.21), so it vanishes identically. Therefore every form in the positive affine slice
\[ \mathcal P=\{ Q\in P+\mathcal A_ D:Q>0\} \]
has the same pointwise trace $\operatorname {tr}_ B Q=\operatorname {tr}_ B P=:C>0$.
Evaluate the parallel forms at one point. The positive eigenvalues of $Q\in \mathcal P$ relative to $B$ sum to $C$, so they are bounded above by $C$. Thus $\mathcal P$ is bounded in its finite-dimensional affine space. Its closure is
\[ \overline{\mathcal P}=\{ Q\in P+\mathcal A_ D:Q\geq 0\} , \]
because $(1-t)Q+tP$ is positive for $0<t\leq 1$ whenever $Q$ is semipositive; in particular, this closure is compact. Since both forms are parallel, the relative eigenvalues and the relative determinant are constant on $X$. Hence
\[ \frac1{n!}\int _ XQ^ n =\det (B^{-1}Q)\frac1{n!}\int _ XB^ n. \]
At a relative boundary point of $\overline{\mathcal P}$ the semipositive matrix is singular, so this volume is zero, whereas it is positive at $P$. A maximum of the volume on the compact set $\overline{\mathcal P}$ therefore lies in $\mathcal P$.
At this interior maximum all the derivatives in (24.4.21) vanish, so the maximizing form is balanced; it is pluriclosed because it lies in $P+\mathcal A_ D$. By lemma 24.4.19 it is closed. Since it is positive and $D$-parallel, this proves the theorem. $\square$
The trace constraint in this argument is pointwise because the metrics and the directions of variation are parallel. This additional structure makes the positive slice compact and the volume vanish on its boundary; for a general family, the conserved balanced trace gives no such control.
Problem 24.4.22.58C2B4B7107E Suppose that a compact complex manifold carries a balanced metric and a pluriclosed metric. Use the two metrics to construct closed positive data, such as a closed Kähler current, or to control a geometric flow that can change the torsion. Determine which estimate, beyond the finite balanced trace, prevents degeneration of the metrics. Can determinant estimates for Gauduchon currents contribute to that estimate?
The complete determinant profile also need not control ellipticity. Under the lower-power $\mathrm{dd}^{\mathrm{c}}$-closed condition, theorem 12.18.8 computes its exact value on every $L^ n$ density root. The nonclosed pluriclosed family of theorem 12.18.15 keeps this entire profile, the bounded-potential mass extrema and a balanced trace fixed while its least relative eigenvalue tends to zero. Its torsion is fixed and nonzero. The underlying torus is already Kähler; the example shows which scalar data an additional estimate must go beyond, without constructing a pluriclosed-flow trajectory.
24.4.23 Pluriclosed metrics under deformation39D58BE832C0
Pluriclosed metrics persist under small deformations of the complex structure when the Aeppli number $h_ A^{1,1}$ stays constant. Let $M$ be a compact smooth manifold of real dimension $2n$, $n\geq 1$, and let $J_ t$, $t\in U\subset \mathbb R^\ell $, be a smooth family of integrable complex structures, where $U$ is an open neighborhood of $0$. Write $\Omega _ t^{p,q}$ for the smooth complex forms of type $(p,q)$ for $J_ t$. A pluriclosed form belongs to the kernel of $\partial _ t\bar\partial _ t$, and the corresponding Aeppli group is
\[ H_ A^{1,1}(M,J_ t) =\frac{\ker (\partial _ t\bar\partial _ t:\Omega _ t^{1,1}\to \Omega _ t^{2,2})}{\partial _ t\Omega _ t^{0,1}+\bar\partial _ t\Omega _ t^{1,0}}, \qquad h_ A^{1,1}(M,J_ t)=\dim _{\mathbb C}H_ A^{1,1}(M,J_ t). \]
Constancy of this dimension gives a smooth family of projections onto the forms satisfying the closedness constraint, and positivity then persists near a given strictly positive form.
Lemma 24.4.24.90818FEADA4E The number $h_ A^{1,1}(M,J_ t)$ is finite. If it is constant near $0$, then, after shrinking $U$, the orthogonal projections
\[ \mathcal P_ t:\Omega _ t^{1,1}\longrightarrow \ker (\partial _ t\bar\partial _ t) \]
for any smooth family of auxiliary Hermitian metrics vary smoothly with $t$ on smooth forms and preserve reality.
Proof. Step 1: an elliptic operator. Fix $t$ and suppress it from the notation; the auxiliary metric defines the adjoints. Put $C=\partial \bar\partial $ and $E=\bar\partial \partial ^*$. The nonnegative self-adjoint operator
A0AA509F3862
24.4.25
\begin{equation} \mathcal L=C^*C+CC^*+E^*E+EE^* +\partial \partial ^*+\bar\partial \bar\partial ^* \end{equation}
is elliptic of order four. Indeed, at a nonzero real covector $\xi $, the principal symbols of $\partial $ and $\bar\partial $ are, up to the common factor $\mathrm i$, exterior multiplication by the $(1,0)$ and $(0,1)$ parts of $\xi $. These parts are orthogonal and have the same norm $r>0$. Split the exterior algebra according to which of these two directions are present. The four terms of order four act on the four resulting summands respectively, and the sum of their symbols is $r^4\operatorname {Id}$. The terms of lower order do not affect ellipticity.
The quadratic form of $\mathcal L$ is a sum of squared norms, which gives the inclusion $\subseteq $ in
\[ \ker \mathcal L=\ker C\cap \ker \partial ^*\cap \ker \bar\partial ^*. \]
For the reverse inclusion, note that $C^*=\bar\partial ^*\partial ^*$, $E=\bar\partial \partial ^*$ and $E^*=\partial \bar\partial ^*$ all vanish under these three conditions. Elliptic theory on the compact manifold gives a finite-dimensional space of smooth forms $\ker \mathcal L$, the orthogonal projection $\mathcal H$ onto it and the Green operator $\mathcal G$, with $\mathcal L\mathcal G=\operatorname {Id}-\mathcal H$.
Step 2: the projection. Set $R=\mathcal L-C^*C$. Each term of $R$ has image in $\operatorname {im}\partial +\operatorname {im}\bar\partial $. Thus every smooth $(1,1)$-form $\alpha $ has the decomposition
321E95CFB730
24.4.26
\begin{equation} \alpha =\mathcal H\alpha +R\mathcal G\alpha +C^*C\mathcal G\alpha . \end{equation}
The first two terms lie in $\ker C$, since $C\partial =C\bar\partial =0$, and the last is orthogonal to $\ker C$. Consequently
is the orthogonal projection onto $\ker C$. If $C\alpha =0$, the last term in (24.4.26) vanishes, so the Aeppli class of $\alpha $ is represented by $\mathcal H\alpha $. A harmonic form in $\operatorname {im}\partial +\operatorname {im}\bar\partial $ is zero, by orthogonality. This identifies $H_ A^{1,1}$ with $\ker \mathcal L$ and proves finiteness.
Step 3: dependence on $t$. Now identify the varying type bundles and their $L^2$ spaces smoothly with those at $t=0$, using a unitary identification for the auxiliary metrics. The coefficients of $\mathcal L_ t$ then vary smoothly. Choose a circle $|z|=\varepsilon $ enclosing no nonzero eigenvalue of $\mathcal L_0$. The resolvents of $\mathcal L_0$ on this circle are bounded from $H^ s$ to $H^{s+4}$. The Neumann series for their perturbations gives smooth resolvents on the circle for small $t$, and the contour projections have constant rank. Constancy of $h_ A^{1,1}$ makes that rank equal to $\dim \ker \mathcal L_ t$. Hence the contour projections are exactly $\mathcal H_ t$, and $\mathcal L_ t$ has no additional small eigenvalues.
The operators
\[ \mathcal G_ t=(\mathcal L_ t+\mathcal H_ t)^{-1}-\mathcal H_ t \]
vary smoothly on every Sobolev scale, by the same inverse argument and elliptic regularity. Formula (24.4.27) therefore gives the asserted smooth family $\mathcal P_ t$. Complex conjugation sends $C_ t$ to $-C_ t$, exchanges $E_ t$ with $E_ t^*$, and exchanges the last two terms of (24.4.25). Thus $\mathcal L_ t$, $\mathcal H_ t$, $\mathcal G_ t$ and $\mathcal P_ t$ commute with conjugation. $\square$
Theorem 24.4.28.5C15639B8E53 Suppose that $(M,J_0)$ carries a pluriclosed Hermitian form $\omega _0$ and that $h_ A^{1,1}(M,J_ t)$ is constant near $0$. Then for small $t$ there is a smooth family of pluriclosed Hermitian forms $\omega _ t$ on $(M,J_ t)$ with the prescribed value $\omega _0$ at $t=0$.
Proof. Let $\pi _ t^{1,1}$ denote the projection onto forms of type $(1,1)$ for $J_ t$, and use the operators of lemma 24.4.24. Define
\[ \omega _ t=\mathcal P_ t\bigl (\pi _ t^{1,1}\omega _0\bigr ). \]
These are smooth real $(1,1)$-forms satisfying $\partial _ t\bar\partial _ t\omega _ t=0$. At $t=0$ the projection fixes $\omega _0$, since $\omega _0$ already satisfies this equation. The tensors $\omega _ t(\cdot ,J_ t\cdot )$ converge uniformly to the positive definite tensor $\omega _0(\cdot ,J_0\cdot )$, which has a uniform positive lower bound by compactness. Hence they remain positive definite after shrinking $U$. $\square$
The construction projects onto $\partial _ t\bar\partial _ t$-closed forms and obtains positivity by continuity near $\omega _0$; closedness and conditions on the curvature would impose further equations. Thus the theorem provides pluriclosed metrics along a given deformation, while the balanced metric required in 24.4.22 remains a separate geometric condition.
24.4.29 Morse inequalities and the choice of volumeB20929459DA6
In the closed Kähler theory, subtracting a nef class has a controlled cost in volume. For closed nef classes $\alpha ,\beta $ on a compact Kähler manifold, the transcendental Morse inequality states that
\[ \operatorname {Vol}(\alpha -\beta ) \geq \int _ X\alpha ^ n-n\int _ X\alpha ^{n-1}\wedge \beta , \]
where $\operatorname {Vol}$ is the usual volume of a class. Its source, and the differentiability of the volume and the duality of cones that follow from it, are recalled in section 25.22. To formulate an analogue on an arbitrary compact complex manifold, one must first choose the volume to be estimated.
For nonclosed backgrounds, the intersection numbers on the right-hand side are in general not invariant under a change of representatives. Moreover, the upper volume of a class and $\mathcal V_\mu $ have different definitions and can behave differently with respect to finiteness. A Hermitian analogue must therefore specify which invariant it estimates.
The first-variation formula gives a finite quantity to study. Fix smooth real $(1,1)$-forms $\theta ,\eta $ and $t>0$. Suppose that $\theta _ s=\theta -s\eta $ has a smooth positive representative for $0\leq s\leq t$, and let $\gamma _ s$ be a minimizing Gauduchon metric for its smooth equation. By proposition 12.3.2 and the fundamental theorem of calculus,
C29E513B11B7
24.4.30
\begin{equation} \lambda _\mu (\theta -t\eta ) =\lambda _\mu (\theta ) -\int _0^ t \frac{\int _ X\eta \wedge \gamma _ s^{n-1}}{D_\mu (\gamma _ s)}\, \mathrm{d}s. \end{equation}
The quotient in the integrand does not change when the minimizer is rescaled. This identity holds on the positive segment $0\leq s\leq t$ and uses no bound on the upper Monge–Ampère mass. A uniform upper bound for its integrand gives a lower bound for the remaining root volume. On a compact Kähler manifold, theorem 12.12.2 extends the identity up to a nef endpoint that has no smooth positive representative, and bounds the resulting improper integral, when a closed nef core of positive volume is supplemented by a semipositive form $B$. Its test metrics keep the prescribed density, and (12.12.5) isolates the correction involving $\mathrm{dd}^{\mathrm{c}}B$. There the endpoint is big because of the closed core; example 12.12.7 gives a nonclosed example with an explicit strict current at the endpoint. Positive volume at the endpoint alone cannot provide such a current, by theorem 12.14.6, so a criterion going beyond the closed core must exclude that obstruction.
For the warped products of section 12.13, theorem 12.13.1 computes the finite improper cost for every choice of minimizing metric, and proposition 12.13.4 gives its rate from the codimension and vanishing order of the fiber coefficient. These paths can end with positive root volume but no big representative, and proposition 12.13.6 excludes every closed nef positive-volume core at a zero of that coefficient. Thus exact control of the subtraction cost in this product family does not supply a general bigness criterion.
Problem 24.4.31.EF3C6D763307 In the setting of (24.4.30), find geometric hypotheses that control the integral on the right-hand side, the cost of the subtraction, up to a nef endpoint, beyond the regime of a closed nef core treated in theorem 12.12.2. Determine when positive root volume forces bigness in the absence of such a core. For closed backgrounds, compare the resulting estimate with the classical Morse inequalities; for nonclosed backgrounds, the estimate should keep track of the differential data and of the dependence on $\mu $.
The singularity and divisorial constructions of these constructions become relevant here: they measure how much positivity can be spent on imposed poles while the residual Gauduchon volume stays finite. They do not yet give the moving intersection product or the birational volume formula of the closed theory. For the nonanalytic ruled type of example 12.35.13, the improper subtraction cost remains finite as the variational volume vanishes, while the upper ordinary-type volume has a positive limit. This family makes the choice of invariant explicit even for an imposed type with infinite first moment. Recovering a general moving product or birational formula remains a further goal.
24.4.32 Oeljeklaus–Toma manifolds68A7CB3104FE
Oeljeklaus–Toma manifolds are not Kähler, yet they carry a closed semipositive form $\theta $ whose potentials are all continuous (theorem 10.29.4). On a threefold of type $(2,1)$, the natural Gauduchon metric $\omega =\theta +y_1y_2\, \mathrm{dd}^{\mathrm{c}}|z|^2$ has an exactly quadratic mass functional, indefinite in sign, and its masses on fiber-invariant potentials are bounded above and below by positive constants (proposition 10.29.10). The known curvature criteria for finite upper and positive lower mass in dimension three require a Hermitian metric whose $\mathrm{dd}^{\mathrm{c}}$ has a sign, and these threefolds admit no such metric (proposition 10.29.14; see section 25.18 for the criteria). By (10.29.12), the remaining difficulty lies entirely in the fluctuation of a potential around its fiber average.
Problem 24.4.33.7E8975FF8CBB Let $X$ be an Oeljeklaus–Toma threefold of type $(2,1)$, with the Hermitian form $\omega $ of proposition 10.29.10. Does $X$ have the bounded mass property, that is, is $v_+(\omega )<+\infty $? Is the lower mass $v_-(\omega )$ positive? Equivalently, writing $F(u)$ for the fluctuation term in (10.29.12), is $F$ bounded above on smooth $\omega $-plurisubharmonic functions $u$, and is $\inf _ u\bigl (\mathrm M_\omega (Au)+F(u)\bigr )>0$? The same questions can be asked for Oeljeklaus–Toma manifolds of any type $(s,t)$ with $s\geq 2$.
By proposition 13.14.15, the answer to the first question does not depend on the choice of Hermitian form. A negative answer would require potentials whose fluctuations make $\operatorname {Re}(\partial _{w_1}v\, \overline{\partial _{w_2}v})$ large, while keeping $\omega +\mathrm{dd}^{\mathrm{c}}u\geq 0$. By lemma 10.29.3, every nonzero Fourier mode on the fiber tori depends on $z$, so for a single mode $c(y)\cos (\cdots )$ the positivity of the $z$-coefficient of $\omega +\mathrm{dd}^{\mathrm{c}}u$ bounds $|c|$ by a multiple of $y_1y_2$ divided by the square of the $z$-frequency; the frequencies in the $x$-directions are not controlled pointwise. Whether compactness and the automorphy under the units bound the fluctuation term is the content of the problem. Unlike the Hopf threefold of section 23.1, the manifold contains no curves along which heat potentials could be concentrated, by the theorem of Ornea and Verbitsky recalled before corollary 10.29.7.
The rigidity in theorem 10.29.4 comes from two features of the null foliation of $\theta $: its leaves carry no nonconstant plurisubharmonic functions bounded above, and their closures are real tori on which translations act transitively. For the pullback of a Kähler form under a holomorphic fibration, the null leaves are the compact fibers, and $\theta $-plurisubharmonic functions may have poles.
Problem 24.4.34.A65ADE22C9ED Let $\theta $ be a smooth closed semipositive real $(1,1)$-form on a compact complex manifold $X$ whose kernel has constant rank. Suppose that every leaf of the null foliation of $\theta $ is Liouville, in the sense that plurisubharmonic functions bounded above on it are constant, and that the closure of every leaf is a compact real submanifold whose real codimension equals the complex rank of $\theta $. Is every $\theta $-plurisubharmonic function continuous? If not, describe the possible singularity types in $\mathrm{PSH}(X,\theta )$ in terms of the closures of the leaves.
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