1 Introduction↩︎

Let us start with a unified way of posing reasonable questions in Riemannian geometry. Given a compact \(n-\)manifold \(M\), define \(\mathcal{RM}(M)\) to be the set of Riemannian metrics \(g\) on \(M\). Consider a nonnegative functional \(\mathcal{F}:\mathcal{RM}(M)\rightarrow \mathbb{R}\) and look for its infimum \(\inf_{\mathcal{RM}(M)} \mathcal{F}\). According to Berger (see [1], p. 530), there are several types of basic questions:

1. Is \(\inf_{\mathcal{RM}(M)} \mathcal{F}\) zero or positive? Classify the manifolds \(M\) for which \(\inf_{\mathcal{RM}(M)} \mathcal{F}\) is positive and those for which it is zero.

2. If positive, then is it attained by some Riemannian metric \(g\) on \(M\)? When these best metrics exist, try to classify them.

3. Find the possible values \(\inf_{\mathcal{RM}(M)} \mathcal{F}\) when \(M\) runs through all compact manifolds of a given dimension. Or at least, we can ask if the set of these values is discrete, or if zero is an isolated point of this set.

Of course, these questions are almost impossible to answer for a general functional \(\mathcal{F}\), so one has to restrict to certain special, but crucial cases, e.g., minimal curvature integrals \[\inf_{g\in \mathcal{RM}(M)} \int_M |K(g)|^{n/2}dV_g,\;\; \inf_{g\in {\mathcal{RM}(M)}} \int_M |Ric(g)|^{n/2}dV_g,\;\; \inf_{g\in {\mathcal{RM}(M)}} \int_M |S(g)|^{n/2}dV_g,\] where \(K(g)\), \(Ric(g)\) and \(S(g)\) stand for sectional curvature, Ricci curvature and scalar curvature respectively. We consider the \(L^{n/2}\) integral instead of general \(L^p\) integrals since it is the only scale-invariant case. Another important example is Gromov’s minimal volume (cf. [2]) \[{\rm MinVol}(M):=\inf_{|K(g)|\le 1} {\rm vol}_g(M)=\inf_{g\in {\mathcal{RM}(M)}} {\rm vol}_g(M) \|K(g)\|_{L^\infty}^{n/2}.\] The Gauss-Bonnet theorem implies \({\rm MinVol}(M)\ge C_n |\chi(M)|\), where \(\chi(M)\) denotes the Euler characteristic of \(M\). Gromov proved the following highly-nontrivial inequality \[{\rm MinVol}(M)\ge \frac{\|M\|}{(n-1)^nn!},\] where \(\|M\|\) is the simplicial volume of \(M\), namely, \(\|M\|:=\inf_c \sum |a_i|\), \(c=\sum a_i\sigma_i\) runs through all representations of the fundamental class \([M]\in H_n(M,\mathbb{R})\). Thus the minimal volume is positive whenever the simplicial volume is. Positivity of the minimal volume provides an obstruction for the collapsing phenomenon in Gromov’s convergence theory. Gromov raised the famous Gap Conjecture: \[\exists\, \varepsilon_n>0\;\text{such that}\;{\rm MinVol}(M)\le \varepsilon_n\Rightarrow {\rm MinVol}(M)=0\] (see [3], [4] for partial answers). A beautiful rigidity theorem due to Besson, Courtois and Gallot [5] states that if \(g\) is a Riemannian metric on a hyperbolic \(n-\)manifold \((M,g_0)\) such that \(Ric(g)\ge -(n-1)g\), then \({\rm vol}_g(M)\ge {\rm vol}_{g_0}(M)\) with equality if and only if \(g\) is isometric to \(g_0\). Note \(|K(g)|\le 1\) implies \(Ric(g)\ge -(n-1)g\). Thus the best metric for \({\rm MinVol}(M)\) is the hyperbolic metric \(g_0\) in this case.

Among \(K(g)\), \(Ric(g)\) and \(S(g)\), the last is the weakest. In his lecture notes on scalar curvature (cf. [6], p. 38–39), Gromov conjectured there exists \(C_n>0\) such that \[\label{eq:Gromov95conj95int} \int_M |S^-(g)|^{n/2}dV_g \ge C_n \|M\|,\;\;\;S^-(g):=\max\{-S(g),0\}.\tag{1}\] Note that 1 implies \({\rm vol}_g(M)\ge C_n \|M\|\) when \(S(g)\ge -1\). This conjecture looks extremely hard since basic tools, such as Bishop-Gromov’s comparison theorem and Cheng-Yau’s gradient estimates, require a lower bound on the Ricci curvature. However, LeBrun [7] was able to show \[\label{eq:LeBrun0} \int_M |S(g)|^{2} dV_g \ge 32 \pi^2 K_M^2\tag{2}\] for minimal complex surfaces of general type, i.e., the canonical line bundle \(K_M\) is big and nef. Here, \(K_M^2 := \int_M c_1(K_M)^2\). Moreover, this inequality is sharp. LeBrun’s method is based on Seiberg-Witten theory, which is not available when the real dimension is greater than \(4\). Actually, every simply-connected compact complex \(n-\)manifolds \(M\) with ample \(K_M\) and \(n\ge 3\) satisfies \[\inf_{g\in \mathcal{RM}(M)} \int_M |S(g)|^{n} dV_g=0,\] in view of Theorem 1 in [8].

The goal of this paper is to consider analogous questions for compact Hermitian manifolds. Our hope is to find certain connections linking differential-geometric and algebraic-geometric points of view. To that end, let us consider a general compact complex \(n-\)manifold \(M\), equipped with a smooth Hermitian metric \(g\) whose Kähler form is given by \(\omega\). Let \(Ric(g)\) and \(Ric(\omega)\) denote the Riemannian Ricci curvature (for the underlying Riemannian metric) and Chern Ricci curvature form respectively. Let \(S_C(g)\) denote the Chern scalar curvature of \(g\). The canonical volume of \(M\) is given by \[\label{eq:CanVolDef} \mathrm{CanVol}(M):= \limsup_{m\rightarrow+\infty}\frac{P_m(M)}{m^n/n!},\tag{3}\] where \(P_m(M):=\dim\Gamma(M,mK_M)\) is the \(m-\)th plurigenus of \(M\). It is known from [9] that \[\mathrm{CanVol}(M)=K_M^n:=\int_Mc_1(K_M)^n,\] when \(M\) is a Kähler manifold and \(K_M\) is nef. Recall that \(M\) is of general type (i.e., \(K_M\) is big) iff \(\mathrm{CanVol}(M)>0\). Our first result is given as follows.

Let \(\mathcal{HM}(M)\) denote the set of Hermitian metrics on a compact complex \(n-\)manifold \(M\). It is reasonable to define \[\mathcal{I}_C(M):=\inf_{g\in \mathcal{HM}(M)} \int_M |S_C(g)|^{n} dV_g,\;\;\mathcal{I}_C^-(M):=\inf_{g\in \mathcal{HM}(M)} \int_M |S_C^-(g)|^{n} dV_g,\] \[\mathrm{MinVol}_C(M):=\inf_{S_C(g)\ge -1} \mathrm{vol}_g(M).\] We have known that under the hypothesis in Corollary 2, \(\mathrm{MinVol}_C(M)\) is not attained by a Kähler metric. It remains unknown whether or not it is achieved by some Hermitian metric.

As a direct consequence of Theorem 1, we have

It is natural to formulate a Gromov-type Gap Conjecture as follows: \[\label{eq:gap95conj} \exists\, \varepsilon_n>0\;\text{such that}\;{\rm MinVol}_C(M)\le \varepsilon_n\Rightarrow {\rm MinVol}_C(M)=0.\tag{4}\] In case \(M\) is a projective \(n-\)manifold, it suffices to verify \[{\rm CanVol}(M) =0 \Rightarrow {\rm MinVol}_C(M)=0.\] If there exists some Hermitian metric \(g\) with \(S_C(g)\ge 0\), for instance, \(M\) is a Fano manifold, then \(\mathrm{MinVol}_C(M)=0\), since \(S_C(tg)=t^{-1}S_C(g)\) and \(dV_{tg}=t^ndV_g\). A typical example is the Hopf manifold \(M_\alpha:=(\mathbb{C}^n\setminus \{0\})/\langle \gamma_\alpha\rangle\), \(\gamma_\alpha=\alpha z\), \(0<|\alpha|<1\). Note that the Hermitian metric \(g=|z|^{-2} \sum dz_j\otimes d\bar{z}_j\) on \(\mathbb{C}^n\setminus \{0\}\) is \(\langle \gamma_\alpha\rangle-\)invariant, so it induces a Hermitian metric on \(M_\alpha\). A direct calculation shows the Ricci form \(Ric(\omega) = 2n i\partial\bar{\partial}\log |z|\ge 0\), where \(\omega\) is the Kähler form of \(g\), so \(S_C(g)\ge 0\) and \(\mathrm{MinVol}_C(M_\alpha)=0\).

The following result is entirely new, in contrast to the Riemannian case.

To support this conjecture, we shall show

Our analysis relies heavily on asymptotic behaviour of the Bergman kernel of \(mK_M\) as \(m\rightarrow \infty\), as well as the theory of Kähler-Ricci flow and existence of the singular Kähler-Einstein metric (when \(K_M\) is big). More precisely, ?? follows directly from the arithmetic-geometric mean inequality and the well-known asymptotic upper bound of the Bergman kernel of \(mK_M\) (which follows easily from the mean-value inequality of plurisubharmonic functions (cf. [12]; see also [13])). If \(M\) is a projective manifold of general type, then there exists a singular Kähler-Einstein metric \(g_{KE}\), which is smooth on some Zariski open subset \(U\subset M\) (cf. [14], see also [15]). With the help of asymptotic lower bound of the Bergman kernel of \(mK_M\) with respect to \(g_{KE}\), ?? for the big case can be reduced to find a sequence \(\{g_j\}\) of smooth Hermitian metrics on \(M\) such that

1. \(g_j\rightarrow g_{KE}\) in \(C^\infty_{\mathrm{loc}}-\)topology on \(U\);

2. both \(dV_{g_j}\) and \(|S_C(g_j)|\) are uniformly bounded on \(M\);

3. \(S_C(g_j)\ge -n\).

In case \(K_M\) is big and nef, the Kähler-Ricci flow fits well with this purpose. It remains open whether the above sequence exists for any compact Kähler manifold of general type. The proof of Theorem 7 is based on a rather delicate gluing technique using the Burns-Simanca metric (compare [16] and [17]).

An even more interesting problem is to find the relations between the canonical volume and the simplicial volume. For instance, it is unknown whether there exists a constant \(C_n>0\) such that \[\label{eq:CV-SV} \mathrm{CanVol}(M)\geq C_n\|M\|\tag{5}\] for any compact complex \(n-\)manifold \(M\). Note that 5 would imply a conjecture of Zhang [18] that \(\|M\|=0\) provided \({\rm CanVol}(M)=0\). Gromov [2] showed that if the Ricci curvature of a Riemannian metric \(g\) is bounded below by \(-1\), then \[{\rm vol}_g (M)\ge \frac{\|M\|}{(n-1)^nn!}.\] This combined with the argument in § 6 would imply 5 for projective manifolds of general type, provided that there exists a sequence \(\{g_j\}\) of Kähler metrics on \(M\) such that

1. \(g_j\rightarrow g_{KE}\) in \(C^\infty_{\mathrm{loc}}-\)topology on \(U\);

2. \(dV_{g_j}\) is uniformly bounded on \(M\);

3. \(Ric(\omega_j)\ge -\omega_j\), where \(\omega_j\) is the Kähler form of \(g_j\).

Using a result of LeBrun [19], we will show in § 8 that for certain complex surfaces of general type \(g_{KE}\) cannot be approximated by Kähler metrics such that both the volume form and the Ricci form are uniformly bounded. We will return to this topic in a future paper.

Finally, recall that the Kodaira dimension of a compact complex \(n-\)manifold is defined to be \[\kappa (M):=\limsup_{m\rightarrow+\infty}\frac{\log P_m(M)}{\log{m}}.\] It is well known that there exists a constant \(C>1\) such that \[C^{-1} m^{\kappa(M)} \le P_m(M) \le C m^{\kappa(M)}\] (see [20]). Two interesting questions immediately arise. The first is to look for a geometric bound similar as scalar curvature integral or minimal volume for the quantity \[\limsup_{m\rightarrow+\infty}\frac{P_m(M)}{{m}^{\kappa(M)}}.\] The other is to find the relationship between \(\kappa(M)\) and the Ricci rank \(\mathcal{R}_g\) of a Hermitian metric \(g\), i.e., the maximum of the number of negative eigenvalues of \(Ric(\omega)\). Liu [21] proved \(\kappa(M)=\mathcal{R}_g\) when \(g\) is a Kähler metric with nonpositive bisectional curvature (see also Wu-Zheng [22] for real-analytic case). In general, we have the following

Proposition 8 turns out to be a fairly simple consequence of a very precise asymptotic upper bound for the Bergman kernel, which is given as follows. Consider a compact Hermitian \(n-\)manifold \((M,g)\) and a holomorphic line bundle \(L\) over \(M\). Given a (possibly singular) Hermitian metric \(h=e^{-\phi}\) of \(L\), define the Bergman space of \(mL=L^{\otimes m}\) to be \[A^2_{g,h}(M,mL):=\left\{s\in \Gamma(M,mL): \int_M |s|^2_{h^{\otimes m}} dV_g<\infty\right\},\] where \(\Gamma(M,mL)\) denotes the space of holomorphic sections of \(mL\) over \(M\). The corresponding Bergman kernel function is given by \[B_{g,h,mL} (x)=\sum |s_j(x)|^2_{h^{\otimes m}}\] for an/any orthonormal basis \(\{s_j\}\) of \(A^2_{g,h}(M,mL)\). In case \(L=K_M\) and \(h=(dV_g)^{-1}\), we simply write \(B_m\) instead of \(B_{g,(dV_g)^{-1},mK_M}\).

A large literature exists on the asymptotic behaviour of the Bergman kernel of \(mL\) as \(m\rightarrow \infty\) in case \(L\) is ample, among them the most famous is the so-called Tian-Yau-Catlin-Zelditch asymptotic expansion (see e.g., [23], [24], [25], [26]), which has deep applications in Kähler geometry. Although the proof of Theorem 9 is only a refinement of known methods, the theorem itself might be of independent interest. For instance, it gives a new proof of the following classical vanishing result.

Recently, Theorem 10 has been extended substantially by Yang in [28].

2 Preliminaries↩︎

2.1 Chern scalar curvature↩︎

Let \(M\) be a complex \(n-\)manifold. Let \(g\) be a Hermitian metric given locally by \(\sum^n_{j,k=1}g_{j\bar{k}}dz_j\otimes {d\bar{z}_k}.\) Then locally \(\omega=i\sum^n_{j,k=1}g_{j\bar{k}}dz_j\wedge d\bar{z}_k\) is the Kähler form of \(g\). The Chern Ricci curvature form of \(\omega\) is defined by \[Ric(\omega)=-i\partial\bar{\partial}\log\det(g_{j\bar{k}}).\] The trace of \(Ric(\omega)=: i \sum R_{j\bar{k}}dz_j\wedge d\bar{z}_k\) is called the Chern scalar curvature \(S_C(g)\) of \(g\), namely, \[S_C(g)=\mathrm{tr}_g(Ric(\omega))=\sum g^{j\bar{k}}R_{j\bar{k}},\] where \((g^{j\bar{k}})=(g_{j\bar{k}})^{-1}\). It is well-known that the Riemannian scalar curvature equals twice the Chern scalar curvature if \(g\) is a Kähler metric. In general, the relationship between them is much more complicated (compare [29]).

2.2 Singular Hermitian metrics↩︎

Let \(M\) be a complex \(n-\)manifold and \(L\) a holomorphic line bundle over \(M\). A singular Hermitian metric of \(L\) may be written as \[h=h_0 e^{-\varphi}\] where \(h_0\) is a smooth Hermitian metric of \(L\) and \(\varphi\in L^1_{\mathrm{loc}}(M)\). \(L\) is said to be pseudoeffective if there exists a singular Hermitian metric \(h\) of \(L\) such that \[\Theta_h = \Theta_{h_0} + i\partial\bar{\partial} \varphi\] is a closed positive current.

The multiplier ideal sheaf \(\mathcal{I}(h)\subset \mathcal{O}_M(L)\) of \((M,L)\) is defined by \[\Gamma(U,\mathcal{I}(h)):=\left\{f\in \Gamma(U,\mathcal{O}_U(L)): |f|^2_{h_0} e^{-\varphi}\in L^1_{\rm loc}(U)\right\}.\] Following Tsuji [30], \(h\) is called an analytic Zariski decomposition if

1. \(\Theta_h\) is a closed positive current,

2. for every \(m\ge 0\), the natural inclusion \[H^0\left(M,mL\otimes \mathcal{I}(h^{\otimes m})\right ) \rightarrow H^0(M,mL)\] is an isomorphism.

2.3 Kähler-Ricci flow↩︎

Here we collect some basic facts about the Kähler-Ricci flow by following the lecture notes of Song and Weinkove (cf. [31], p. 19, 20, 48). Let \((M,g_0)\) be a compact Kähler \(n-\)manifold. A Kähler-Ricci flow means the following equation \[\label{eq:KR1} \frac{\partial \omega_t}{\partial t}=-Ric(\omega_t),\;\;\;\omega_t|_{t=0}=\omega_0,\tag{6}\] and the normalized Kähler-Ricci flow is given by \[\label{eq:KR2} \frac{\partial \omega_t}{\partial t}=-Ric(\omega_t)-\omega_t,\;\;\;\omega_t|_{t=0}=\omega_0,\tag{7}\] where \(\omega_t\) is the Kähler form of \(g_t\). We shall focus on 7 . Then the following properties hold:

1. There exists a unique solution \(g_t\) of 7 on some maximal time interval \([0,T)\) with some \(0<T\le \infty\). If \(K_M\) is nef, then one can take \(T=\infty\).

2. Set \(C_0:=-\inf_M S_C(g_0)-n\). Then \[\label{eq:ScalarLower} S_C(g_t) \ge -n - C_0 e^{-t},\;\;\; t\in [0,T).\tag{8}\]

3. Let \(C_0\) be as above. Then \[\label{eq:volumeCompare} dV_{g_t} \le e^{C_0(1-e^{-t})} dV_{g_0},\;\;\; t\in [0,T).\tag{9}\] In particular, \(dV_{g_t}\) is uniformly bounded from above for \(t\in [0,T)\).

A deeper observation is the following theorem of Zhang [32] (for a generalization to the semi-ample case, see Song-Tian [33]).

2.4 Singular Kähler-Einstein metrics↩︎

The Kähler-Ricci flow was first used by Cao [34] to give an alternative proof of Yau’s theorem on the existence of Kähler-Einstein metric on compact Kähler manifolds with trivial or negative first Chern class (cf. Yau [35]; see also Aubin [36] for the negative case). Tsuji [37] (see also Tian-Zhang [38]) proved the following

One calls \(g_{KE}\) a singular Kähler-Einstein metric on \(M\). Tsuji’s work was extended by Eyssidieux-Guedj-Zeriahi [39], Boucksom et al. [15] and Song-Tian [14] to general projective manifolds of general type.

2.5 The Burns-Simanca metric↩︎

Denote by \(\mathrm{Bl}_0\mathbb{C}^n\) the blow-up of \(\mathbb{C}^n\) at the origin, i.e., \[\mathrm{Bl}_0\mathbb{C}^n:=\left\{(z,[\zeta])\in\mathbb{C}^n\times\mathbb{P}^{n-1},\;z_j\zeta_k=z_k\zeta_j,\;1\leq j,k\leq n\right\}.\] Here \([\zeta]=[\zeta_1:\cdots:\zeta_n]\) is the homogeneous coordinate on \(\mathbb{P}^{n-1}\). It is known that \(\mathrm{Bl}_0\mathbb{C}^n\) is the total space of the tautological line bundle over \(\mathbb{P}^{n-1}\) with a natural projection \(\pi:\mathrm{Bl}_0\mathbb{C}^n\rightarrow\mathbb{P}^{n-1}\). In particular, \(\mathrm{Bl}_0\mathbb{C}^n\) is a complex manifold. There is another natural holomorphic map \[\varpi:\mathrm{Bl}_0\mathbb{C}^n\rightarrow\mathbb{C}^n,\;\;\;(z,[\zeta])\mapsto z,\] which maps \(\mathrm{Bl}_0\mathbb{C}^n\setminus\varpi^{-1}(0)\) biholomorphically to \(\mathbb{C}^n\setminus\{0\}\). Recall that \(\varpi^{-1}(0)\) is called the exceptional locus, which is a submanifold of \(\mathrm{Bl}_0\mathbb{C}^n\) (biholomorphic to \(\mathbb{P}^{n-1}\)).

The Burns-Simanca metric \(g_{BS}\) is a Kähler metric on \(\mathrm{Bl}_0\mathbb{C}^n\) with zero scalar curvature (cf. [40], [41]). On the complement of the exceptional divisor, its Kähler form can be written as \[\omega_{BS}=i\partial\bar{\partial}\left(\frac{|z|^2}{2}+\psi(z)\right),\] where \[\label{eq:BS95zero} \psi(z)=a\log|z|^2+\eta(|z|^2)\tag{10}\] for some smooth function \(\eta\) on \([0,+\infty)\) and \(a>0\). When \(n=2\), one can take \(a=1\) and \(\eta\equiv0\); when \(n\geq3\), \(\eta\) cannot be written explicitly, but we have the asymptotic expansion \[\label{eq:BS95infty} \psi(z)=-|z|^{4-2n}+O(|z|^{2-2n}),\;\;\;|z|\to \infty\tag{11}\] (see [17]).

The formula 10 enables us to extend \(\omega_{BS}\) through the exceptional locus. Indeed, if \((z,[\zeta])\in\mathrm{Bl}_0\mathbb{C}^n\), then \[\log|z|^2=\log\left(\sum^n_{j=1}\frac{|\zeta_j|^2|z_k|^2}{|\zeta_k|^2}\right)=\log|\zeta|^2+\log|z_k|^2-\log|\zeta_k|^2,\] where \(z_k,\zeta_k\neq0\), so that \[i\partial\bar{\partial}\log|z|^2=i\partial\bar{\partial}\log|\zeta|^2.\] As a consequence, we have \[\label{eq:BS95FS} \omega_{BS}=a\cdot \pi^*\omega_{FS}+\varpi^*\left(i\partial\bar{\partial}\left(\frac{|z|^2}{2}+\eta(|z|^2)\right)\right)\tag{12}\] outside the exceptional locus, where \(\omega_{FS}\) is the Fubini-Study metric on \(\mathbb{P}^{n-1}\).

2.6 An \(L^2\) estimate for \(\bar{\partial}\)↩︎

We present the following refinement of Demailly-Nadel’s vanishing theorem which might be useful for other purposes.

3 Upper asymptotic bounds of the Bergman kernel↩︎

4 Lower asymptotic bounds of the Bergman kernel↩︎

With the help of Theorem 14, Tian’s method of peak functions (cf. [25]) applies to the following

5 Proof of Theorem 1↩︎

Recall that \(M_0\) is the open subset of \(M\) where \(\lambda_j<0\) for all \(1\leq{j}\leq{n}\). By 14 , ?? , ?? and Fatou’s lemma, we have

\[\begin{align} \label{eq:LeBrun95M950} \mathrm{CanVol}(M) & = & \limsup_{m\rightarrow+\infty}\frac{P_m(M)}{m^n/n!}\leq \int_M\limsup_{m\rightarrow+\infty}\frac{B_m}{m^n/n!}dV_g\nonumber\\ &\leq& \frac{(-1)^nn!}{\pi^n}\int_{M_0}\lambda_1\cdots\lambda_n{dV_g}\nonumber\\ &\leq& \frac{n!}{\pi^n}\int_{M_0}\left(\frac{-\lambda_1-\cdots-\lambda_n}{n}\right)^n{dV_g}\nonumber\\ &\leq& \frac{n!}{(n\pi)^n}\int_M|S^-_{C}|^ndV_g, \end{align}\tag{25}\] i.e., ?? holds, which implies ?? since \[{\rm vol}_g (M)\ge \int_M|S^-_{C}(g)|^ndV_g\] if \(S_C(g)\ge -1\).

Suppose there exists a Hermitian metric \(g\) such that \(S_C(g)\ge -1\) and \[{\rm vol}_g(M) = \frac{(n\pi)^n}{n!} {\rm CanVol}(M).\] In particular, \({\rm CanVol}(M)>0\), i.e., \(M\) is of general type. On the other hand, 25 implies \[{\rm vol}_g(M_0) \ge \frac{(n\pi)^n}{n!} {\rm CanVol}(M).\] Thus \[{\rm vol}_g(M\setminus M_0)=0\] and equalities in 25 hold. It follows that \(\lambda_1=\cdots=\lambda_n=:\lambda\) on \(M_0\), hence on \(M\) by continuity. Note that \[\lambda=\frac{1}{n}\mathrm{tr}_g(\lambda\omega)=\frac{1}{n}\mathrm{tr}_g(Ric(\omega))=\frac{S_C(g)}{n}.\] In case \(g\) is Kähler, the Chern connection coincides with the Levi-Civita connection, and hence \(g\) is also a Riemannian Einstein metric. Since the real dimension of \(M\) is greater than \(2\), we conclude that \(\lambda\) is a constant (cf. Berger [1]). Moreover, \(\lambda<0\) for \(M\) is of general type.

6 Proofs of Theorem 4 and Corollary 6↩︎

6.1 Proof of Theorem 4 when \(K_M\) is big and nef↩︎

First of all, we have \[\label{eq:MinVol95S} \mathrm{MinVol}_C(M)\geq\mathcal{I}_C^-(M).\tag{26}\] Indeed, given any Hermitian metric \(g\) on \(M\), the scalar curvature \(S_C(g)\) is bounded from below on \(M\). Since \(S_C(tg)=t^{-1}S_C(g)\) and \(dV_{tg}=t^ndV_g\), we see that the integral \(\int_M|S_C^-(g)|^ndV_g\) is invariant under scaling. Then we may assume that \(S_C(g)\geq-1\), so that \[\int_M|S_C^-(g)|^ndV_g\leq\int_M1dV_g\leq\mathrm{vol}_g(M),\] from which 26 immediately follows.

By ?? and 26 , it suffices to verify the follow inequalities: \[\label{eq:MinVol95CanVol} \mathrm{MinVol}_C(M) \leq \frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)\tag{27}\] \[\label{eq:big95nef952} \mathcal{I}_C(M)\leq\frac{(n\pi)^n}{n!}\mathrm{CanVol}(M).\tag{28}\]

Note that a compact Kähler manifold with a big line bundle is always algebraic, in view of a celebrated result of Siu [42]. Thus by Theorem 12, we know that there exists an analytic subset \(E\subset M\) such that the solution \(g_t\) of the normalized Kähler-Ricci flow 7 converges in \(C^\infty_{\rm loc}(M\setminus E)\) to the singular Kähler-Einstein metric \(g_{KE}\). Moreover, the Kähler form \(\omega_{KE}\) of \(g_{KE}\) extends to a closed positive current on \(M\). Consider the Bergman space of \(mK_M\) associated to \(dV_{KE}\), i.e., \[A^2_{g_0,(dV_{KE})^{-1}}(M,mK_M)\] and the corresponding Bergman kernel \[B_{g_0,(dV_{KE})^{-1},mK_M}(x).\] Since \[Ric(\omega_{KE})=-\omega_{KE},\qquad \text{on}\;M\setminus E,\] it follows from ?? that \[\label{eq:limit95Bergman95M95E} \liminf_{m\rightarrow+\infty}\frac{B_{g_0,(dV_{KE})^{-1},mK_M}(x)}{m^n}\ge \frac{1}{\pi^n}\cdot \frac{dV_{KE}}{dV_{g_0}}(x),\;\;\;\forall\;x\in M\setminus E.\tag{29}\] Then we have \[\begin{align} \label{eq:lower95CanVol951} && \liminf_{m\rightarrow+\infty}\frac{\dim A^2_{g_0,(dV_{KE})^{-1}}(M,mK_M)}{m^n/n!}\nonumber\\ &=& n!\liminf_{m\rightarrow+\infty}\int_{M}\frac{B_{g_0,(dV_{KE})^{-1},mK_M}(x)}{m^n}dV_{g_0}(x)\notag\\ &\geq& \frac{n!}{\pi^n}\mathrm{vol}_{KE}(M\setminus E)\;\;\;(\text{Fatou's lemma and\;}\eqref{eq:limit95Bergman95M95E}). \end{align}\tag{30}\] Thus \[\begin{align} \label{eq:lower95CanVol952} \mathrm{CanVol}(M) = \limsup_{m\rightarrow+\infty}\frac{P_m(M)}{m^n/n!} & \geq & \limsup_{m\rightarrow+\infty}\frac{\dim A^2_{g_0,(dV_{KE})^{-1}}(M,mK_M)}{m^n/n!}\nonumber\\ & \geq & \frac{n!}{\pi^n}\mathrm{vol}_{KE}(M\setminus E). \end{align}\tag{31}\]

On the other side, it follows from 9 that there exists for each \(\delta>0\), a neighbourhood \(E_\delta\) of \(E\) such that \(\mathrm{vol}_{g_t}(E_\delta)<\delta\) for \(t\in[0,\infty)\). Since \(g_t\rightarrow g_{KE}\) in \(C^\infty(M\setminus E_\delta)\), we have \(\mathrm{vol}_{g_t} (M\setminus E_\delta)\leq\mathrm{vol}_{KE}(M\setminus E_\delta)+\delta\) when \(t\gg1\), so that \[\label{eq:lower95CanVol953} \mathrm{vol}_{g_t} (M) \leq \mathrm{vol}_{g_t}(M\setminus E_\delta)+\delta \leq \mathrm{vol}_{KE}(M\setminus E_\delta)+2\delta \leq \frac{\pi^n}{n!}\mathrm{CanVol}(M)+2\delta,\tag{32}\] in view of 31 .

By 8 , the Kähler metric \((n+C_0 e^{-t})g_t\) on \(M\) satisfies \(S_C\geq-1\). Thus \[\label{eq:MinVolUpper} \mathrm{MinVol}_C(M) \leq (n+C_0e^{-t})^n\mathrm{vol}_{g_t}(M).\tag{33}\] This combined with 32 yields \[\mathrm{MinVol}_C(M) \leq (n+C_0 e^{-t})^n\left(\frac{\pi^n}{n!}\mathrm{CanVol}(M)+2\delta\right),\] when \(t\gg1\) and \(0<\delta\ll1\). Letting \(t\rightarrow\infty\) and \(\delta\rightarrow0\), we get 27 .

For 28 , we infer from Theorem 11 that there exists a constant \(C>0\) such that \[\label{eq:scalar95bdd95upper} |S_C(g_t)|\leq C,\;\;\;\forall\;t\in[0,\infty),\tag{34}\] so that \[\label{eq:I95M95small} \int_{E_\delta}|S_C(g_t)|^ndV_{g_t} \leq C^n\mathrm{vol}_{g_t} (E_\delta)<C^n\delta.\tag{35}\] Moreover, \(S_C(g_t)\) converges uniformly on \(M\setminus E_\delta\) to \(S_C(g_{KE})=-n\), so that \[\label{eq:I95M95big} \lim_{t\rightarrow\infty}\int_{M\setminus E_\delta}|S_C(g_t)|^ndV_{g_t} = n^n\mathrm{vol}_{KE}(M\setminus E_\delta)\leq \frac{(n\pi)^n}{n!}\mathrm{CanVol}(M),\tag{36}\] in view of 31 . By 35 and 36 , we have \[\label{eq:I95C95limit} \mathcal{I}_C (M)\leq\limsup_{t\rightarrow\infty}\int_M|S_C(g_t)|^ndV_{g_t} \leq \frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)+C^n\delta,\tag{37}\] from which 28 immediately follows.

6.2 Proof of Theorem 4 when \(K_M\) is nef but not big↩︎

Recall that the numerical dimension of the nef line bundle \(K_M\) is defined by \[\label{eq:nd} \mathrm{nd}(M)=\mathrm{nd}(K_M)=\max\left\{k:\;1\leq k\leq n, c_1(K_M)^k\neq0\;\text{in}\;H^{k,k}(M,\mathbb{R})\right\}.\tag{38}\] Take a smooth \((1,1)\) form \(\gamma\) with \(c_1(K_M)=[\gamma]\) and a smooth volume form \(\Omega\) on \(M\) such that \(\gamma=i\partial\bar{\partial}\log\Omega\) and \(\int_M\Omega=1\). Let \(g_t\) be the solution to the normalized Kähler-Ricci flow 7 and \(\omega_t\) the associated Kähler form. Set \[\widehat{\omega}_t:=(1-e^{-t})\gamma+e^{-t}\omega_0.\] It follows from 7 that \[\frac{d}{dt}[\omega_t]=[-\mathrm{Ric}(\omega_t)-\omega_t]=c_1(K_M)-[\omega_t],\] i.e., \[\frac{d}{dt}(e^t[\omega_t])=c_1(K_M)e^t.\] Solving this ODE in \(H^{1,1}(M,\mathbb{R})\), we obtain \[[\omega_t]=(1-e^{-t})[c_1(K_M)]+e^{-t}[\omega_0]=[\widehat{\omega}_t].\] In particular, \[\label{eq:volume95int95cohomology} \mathrm{vol}_{g_t}(M)=\int_M\widehat{\omega}_t^n.\tag{39}\] By 8 , we have \[\label{eq:vol95nef95not95big} \mathrm{MinVol}_C(M)\leq \left(n+C_0e^{-t}\right)^n\mathrm{vol}_{g_t}(M)=\left(n+C_0e^{-t}\right)^n\int_M\widehat{\omega}_t^n.\tag{40}\] Thus it remains to verify \(\int_M\widehat{\omega}_t^n\rightarrow0\) as \(t\rightarrow+\infty\). This follows immediately from [43]. For the sake of completeness, we shall include the very simple proof as follows.

Since \([\gamma]^k=c_1(K_M)^k=0\) in \(H^{k,k}(M,\mathbb{R})\) for \(k>\mathrm{nd}(M)\), so \[\begin{align} \int_M\widehat{\omega}_t^n &=& \int_M\left(\gamma + e^{-t}(\omega_0-\gamma)\right)^n\\ &=& \sum^n_{l=0}\binom{n}{l}e^{-(n-l)t}\int_M\gamma^l\wedge(\omega_0-\gamma)^{n-l}\\ &=& e^{-(n-\mathrm{nd}(M))t}\binom{n}{\mathrm{nd}(M)}\int_M\gamma^{\mathrm{nd}(M)}\wedge\left(\omega_0-\gamma\right)^{n-\mathrm{nd}(M)}+o(e^{-(n-\mathrm{nd}(M))t}),\;\;\;t\rightarrow+\infty. \end{align}\] In particular, we get \[\label{eq:volume95numerical95dim951} C^{-1}e^{-(n-\mathrm{nd}(M))t}\mathrm{vol}_{g_0}(M) \leq \mathrm{vol}_{g_t}(M) \leq Ce^{-(n-\mathrm{nd}(M))t}\mathrm{vol}_{g_0}(M).\tag{41}\] for some constant \(C\) independent of \(t\). As \(\mathrm{Kod}(M)<n\) implies \(\mathrm{nd}(M)<n\) (cf. [9]), it follows that \(\mathrm{vol}_{g_t}(M)\rightarrow0\) as \(t\rightarrow+\infty\).

Finally, we will show \(\mathcal{I}_C(M)=0\). Consider the average Chern scalar curvature of \(g_t\) \[\label{eq:average95sc} \overline{S}_t :=\frac{1}{\mathrm{vol}_{g_t}(M)}\int_M S_C(g_t)\,dV_{g_t}.\tag{42}\] We claim that the function \(t\mapsto\overline{S}_t\) is bounded on \([0,\infty)\). Since \(g_t\) is a Kähler metric, we have the standard identity \[S_C(g_t)\,\omega_t^n = n\,Ric(\omega_t)\wedge \omega_t^{n-1},\] so that \[\int_M S_C(g_t)\,dV_{g_t} =n\int_M Ric(\omega_t)\wedge \omega_t^{n-1}.\] Recall that \([Ric(\omega_t)]=-c_1(K_M)=-[\gamma]\), so \[\label{eq:scalar95integral} \int_M S_C(g_t)\,dV_{g_t} = -n\int_M\gamma\wedge\omega_t^{n-1} = -n\int_M\gamma\wedge\widehat{\omega}_t^{n-1}.\tag{43}\] A similar argument as above yields \[\begin{align} \int_M\gamma\wedge\widehat{\omega}_t^{n-1} &=& \int_M\gamma\wedge\left(\gamma + e^{-t}(\omega_0-\gamma)\right)^{n-1}\\ &=& \sum^{n-1}_{l=0}\binom{n-1}{l}e^{-(n-1-l)t}\int_M\gamma^{l+1}\wedge(\omega_0-\gamma)^{n-1-l}. \end{align}\] Note that \([\gamma]^{l+1}=c_1(K_M)^{l+1}=0\) in \(H^{l+1,l+1}(M,\mathrm{R})\) when \(l\geq\mathrm{nd}(M)\). We get \[\left|\int_M\gamma\wedge\widehat{\omega}_t^{n-1}\right|=O(e^{-(n-\mathrm{nd}(M))t}).\] This combined with 42 , 41 and 43 gives \[\label{eq:average95sc95bdd} |\overline{S}_t|\leq C, \qquad \forall\,t\geq 0.\tag{44}\] Here \(C\) is independent of \(t\).

We will proceed the argument through suitable conformal deformation. Set \[h_t=e^{u_t/n}g_t\] for a real-valued smooth function \(u_t\) on \(M\). The volume form is given by \[\label{eq:volume95form95conformal95t} dV_{h_t}=e^{u_t}dV_{g_t},\tag{45}\] and a straightforward calculation shows \[\label{eq:sc95conformal95t} S_C(h_t)=e^{-u_t/n}\left(S_C(g_t)-\Delta_{g_t} u_t\right),\tag{46}\] where \(\Delta_{g_t}\) denotes the complex Laplacian with respect to the metric \(g_t\), i.e., \[\Delta_{g_t}\varphi=\mathrm{tr}_{g_t}(i\partial\bar{\partial}\varphi),\;\;\;\forall\,\varphi\in C^\infty(M).\] For each \(t\geq 0\), the Poisson equation \[\label{eq:Poisson} \Delta_{g_t}u_t = S_C(g_t)-\overline{S}_t\tag{47}\] admits a smooth solution \(u_t\) since the right-hand side has integral zero (see [44]). By 46 and 47 , we get \[S_C(h_t)=e^{-u_t/n}\overline{S}_t.\] This together with 45 and 44 yield \[\int_M|S_C(h_t)|^ndV_{h_t}=\int_M|\overline{S}_t|^ndV_{g_t}\leq C^n\mathrm{vol}_{g_t}(M).\] Since \(\mathrm{nd}(M)<n\), we conclude \[\int_M|S_C(h_t)|^ndV_{h_t}\rightarrow0,\;\;\;t\rightarrow+\infty,\] in view of 41 . This implies \(\mathcal{I}_C(M)=0\).

6.3 Proof of Corollary 6↩︎

The pull-back \(F^*K_M\) of \(K_M\) is a line bundle over \(M'\), whose volume can be defined in a similar way: \[\mathrm{vol}(F^*K_M):= \limsup_{m\rightarrow+\infty}\frac{\dim\Gamma(M,F^*K_M)}{m^n/n!}.\] It follows from [45] (see also [46]) that \[\label{eq:deg95vol95pullback} \mathrm{vol}(F^*K_M)=\deg(F)\cdot\mathrm{CanVol}(M).\tag{48}\] Let \(R_F\) denote the line bundle associated to the effective divisor locally given by the zeros of the complex Jacobian determinant of \(F\). We have \[K_{M'}=F^*K_M+ R_F.\] By [47], we know that \[\label{eq:volume95increasing95effective} \mathrm{CanVol}(M')=\mathrm{vol}(F^*K_M+ R_F)\geq\mathrm{vol}(F^*K_M).\tag{49}\] Thus \[\mathrm{CanVol}(M')\geq \deg(F)\cdot\mathrm{CanVol}(M).\] This together with Corollary 3 and Theorem 4 yield ?? ?? . Moreover, if \(F\) is a covering map, then \(R_F=0\) and \(K_{M'}\) is also nef, so that \[\mathrm{CanVol}(M')=\mathrm{vol}(F^*K_M)=\deg(F)\cdot\mathrm{CanVol}(M).\] Thus equalities in ?? ?? hold, in view of Theorem 4.

7 Proof of Theorem 7↩︎

Since the canonical volume is a bimeromorphic invariant (see, e.g., [47], [20]), it follows that \[\label{eq:canvol95bimeromorphic} \mathrm{CanVol}(\widehat M)=\mathrm{CanVol}(M).\tag{50}\] By 50 and Theorem 4, it suffices to prove that \(\mathrm{MinVol}_C\) and \(\mathcal{I}_C^-\) (resp. \(\mathcal{I}_C\)) are invariant under the blow-up at finite points when \(K_M\) is nef (resp. big and nef). For simplicity, we will only treat the case of blowing-up at one point and leave the general case to interested readers. Let \[\varpi:\widehat{M}=\mathrm{Bl}_pM\rightarrow M\] be the blow-up at \(p\in M\).

From the proof of Theorem 4, we learn that if \(K_M\) is nef, then for any \(\varepsilon>0\), there exists a Kähler metric \(g\) on \(M\) such that \(S_C(g)\geq-1\) and \[\label{eq:approximating95metric95I95minus} \int_M|S_C^-(g)|^ndV\leq \frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)+\varepsilon,\tag{51}\] \[\label{eq:approximating95metric95I} \int_M|S_C(g)|^ndV\leq \frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)+\varepsilon,\tag{52}\] \[\label{eq:approximating95metric95vol} \mathrm{vol}_g(M)\leq \frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)+\varepsilon.\tag{53}\]

The main idea is to modify \(g\) to obtain certain metric on \(\widehat{M}\) with analogous properties. More precisely, we will divide the proof into four steps:

• Glue any given metric with the Burns-Simanca metric near the exceptional divisor;

• Derive quantitative estimates of the scalar curvature and the volume for the glued metric;

• Deduce the equalities for \(\mathcal{I}_C\) and \(\mathcal{I}_C^-\);

• Apply a conformal correction to achieve the sharp lower bound \(S_C\ge -1\), and conclude the equality for \(\mathrm{MinVol}_C\).

Step 1: Construction of the glued metric. We will glue the background metric \(g\) on \(M\) with a rescaled Burns-Simanca metric near the exceptional divisor, following LeBrun [16] and Székelyhidi [17].

Since \(g\) is Kähler, the Kähler form \(\omega\) of \(g\) can be written as \[\omega=i\partial\bar{\partial}\left(\frac{|z|^2}{2}+\varphi(z)\right)\] in a small coordinate ball \(B_\rho\) around \(p\), where \(\varphi(z)=O(|z|^4)\) as \(z\rightarrow0\). Let \(\delta>0\) be a small parameter. Consider the rescaled Burns-Simanca metric (see § 2.5) \[\omega^\delta_{BS} :=\varpi^*i\partial\bar{\partial}\left(\frac{|z|^2}{2}+\delta^2\psi\left(\frac{z}{\delta}\right)\right)\] on \(\varpi^{-1}(B_\rho)\), where \(z\in B_\rho\setminus\{0\}\). We will define a modified metric \(g_\delta\) on \(\widehat{M}\) by gluing \(g\) and \(g_{BS}^\delta\) as follows. Let \(\chi:[0,\infty)\to[0,1]\) be a smooth cut-off function with \(\chi\equiv 0\) on \([0,1]\) and \(\chi\equiv 1\) on \([2,\infty)\). Take \[r_\delta:=\delta^\alpha,\qquad 0<\alpha<{1}/{2}.\] Clearly, \(B_{2r_\delta}\subset\subset B_\rho\) for \(\delta\ll 1\). On \(B_\rho\), we define \[\label{eq:Phi95delta} \Phi_\delta(z) :=\frac{|z|^2}{2} +\chi\left(\frac{|z|}{r_\delta}\right)\varphi(z) +\left(1-\chi\left(\frac{|z|}{r_\delta}\right)\right)\delta^2\psi\left(\frac{z}{\delta}\right).\tag{54}\] Note that \(\varpi^*i\partial\bar{\partial} \Phi_\delta\) is a real \((1,1)\)-form on \(\widehat{M}\), which coincides with \(\omega^\delta_{BS}\) (resp. \(\varpi^*\omega\)) on \(\varpi^{-1}(B_{r_\delta})\) (resp. \(\varpi^{-1}(B_\rho\setminus B_{2r_\delta})\)). The glued metric \(g_\delta\) is defined through \[\label{eq:glue95Kahler} \omega_\delta:=\begin{cases} \varpi^*i\partial\bar{\partial} \Phi_\delta,\;\;\;&\text{on }\varpi^{-1}(B_\rho),\\ \varpi^*\omega,\;\;\;&\text{on }\widehat{M}\setminus\varpi^{-1}(B_\rho), \end{cases}\tag{55}\] provided that \(\omega_\delta\) is positive on the gluing annulus \[A_\delta:=\varpi^{-1}(B_{2r_\delta}\setminus B_{r_\delta}).\] To see this, let us use rescaled coordinates \(w=z/r_\delta\). Consider the holomorphic map \[\iota:\{0<|w|\leq\rho/r_\delta\}\xrightarrow{z=r_\delta w}\left\{0<|z|\leq \rho\right\}\xrightarrow{\varpi^{-1}}\widehat{M}\] and define the \((1,1)\)-tensor \[\widetilde{g}_\delta:=r_\delta^{-2}\iota^*g_\delta\] on \(\{0<|w|\leq\rho/r_\delta\}\). With respect to the coordinate \(w\), the Kähler form \(\widetilde{\omega}_\delta\) of \(\widetilde{g}_\delta\) can be written as \[\widetilde{\omega}_\delta=i\partial\bar{\partial}\,\widetilde{\Phi}_\delta,\] where \[\widetilde{\Phi}_\delta(w)=\frac{|w|^2}{2} +\chi\left(|w|\right)r_\delta^{-2}\varphi(r_\delta w) +\bigl(1-\chi\left(|w|\right)\bigr)r_\delta^{-2}\delta^2\psi\left(\frac{r_\delta w}{\delta}\right),\;\;\;0<|w|\leq\rho/r_\delta.\] Denote by \(g_{\mathrm{eucl}}\) the Euclidean metric on \(\mathbb{C}^n\) (whose Kähler form is given by \(i\partial\bar{\partial}(|w|^2/2)\)).

As a consequence, \(\omega_\delta=r_\delta^2(\iota^{-1})^*\widetilde{\omega}_\delta\) is positive on \(A_\delta\) for all sufficiently small \(\delta\). Hence \(g_\delta\) is a genuine Kähler metric on \(\widehat M\).

Step 2: Gluing estimates.

Let us first show the following

Now we give the crucial estimates.

Step 3: Equalities for \(\mathcal{I}_C\) and \(\mathcal{I}_C^-\).

Suppose that \(K_M\) is nef. Recall that \(g\) satisfies 51 . By ?? , we get \[\int_{\widehat{M}}|S_C^-(g_\delta)|^n dV_{g_\delta} \leq \int_M|S_C^-(g)|^n dV_g+O(r_\delta^2) \leq \frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)+\varepsilon+O(r_\delta^2),\] so that \[\mathcal{I}^-_C(\widehat{M})\leq \frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)+\varepsilon+O(r_\delta^2).\] Letting \(\delta\rightarrow0\) and \(\varepsilon\rightarrow0\), we obtain \[\mathcal{I}^-_C(\widehat{M})\leq \frac{(n\pi)^n}{n!}\mathrm{CanVol}(M).\] This combined with Theorem 1 gives \[\mathcal{I}^-_C(\widehat{M})= \frac{(n\pi)^n}{n!}\mathrm{CanVol}(M).\] Using 52 and ?? instead of 51 and ?? , we obtain the equality for \(\mathcal{I}_C(M)\).

Step 4: Equality for \(\mathrm{MinVol}_C\).

This part is the most involved. Suppose \(K_M\) is nef. It is known from 53 that for any \(\varepsilon>0\) there exists a Kähler metric \(g\) with \(S_C(g)\ge -1\) and \[\mathrm{vol}_g(M)\le \frac{(n\pi)^n}{n!}\,\mathrm{CanVol}(M)+\varepsilon.\] Fix \(\tau\in(0,1)\). Replacing \(g\) by \(\lambda g\) with \(\lambda=(1-\tau)^{-1}>1\), we get \[S_C(g)\ge -1+\tau\] and \[\mathrm{vol}_g(M)\le (1-\tau)^{-n}\left(\frac{(n\pi)^n}{n!}\,\mathrm{CanVol}(M)+\varepsilon\right).\] If \(\tau\) is sufficiently small, then \[\mathrm{vol}_g(M)\le \frac{(n\pi)^n}{n!}\,\mathrm{CanVol}(M)+2\varepsilon.\] This combined with ?? implies \[\label{eq:volume95canvol95epsilon} \mathrm{vol}_{g_\delta}(\widehat{M})\leq \frac{(n\pi)^n}{n!}\,\mathrm{CanVol}(M)+3\varepsilon\tag{64}\] when \(\delta\) is sufficiently small. However, this is not enough to verify the equality for \(\mathrm{MinVol}_C\), for we merely have \[\label{eq:sc95glue} S_C(g_\delta)\geq \begin{cases} 0\;\;\;\;\;&\text{on}\;\varpi^{-1}(B_{r_\delta}),\\ -C_0\;\;\;\;\;&\text{on}\;A_\delta=\varpi^{-1}(B_{2r_\delta}\setminus B_{r_\delta}),\\ -1+\tau\;\;\;\;\;&\text{on}\;\widehat{M}\setminus \varpi^{-1}(B_{2r_\delta}), \end{cases}\tag{65}\] in view of ?? , ?? and ?? . It is unclear whether \(C_0\leq1\).

To overcome this difficulty, we will use again the technique of conformal deformation. Set \(h_\delta=e^{u_\delta/n}g_\delta\) for a real-valued smooth function \(u_\delta\) on \(\widehat M\). Recall that \[\label{eq:sc95conformal} S_C(h_\delta)=e^{-u_\delta/n}\left(S_C(g_\delta)-\Delta_{g_\delta} u_\delta\right),\tag{66}\] where \(\Delta_{g_\delta}\) denotes the complex Laplacian with respect to the metric \(g_\delta\). We are going to prove the following

We first show the following

Pick a smooth radial function \(f\in C_0^\infty(\mathbb{C}^n)\) such that \[\begin{align} & & -C_0\le f\le 0 \quad \text{on } \left\{\frac{1}{2}\le |w|\le 3\right\},\\ & & f\equiv -C_0 \quad \text{on } \{1\le |w|\le 2\},\\ & & \mathrm{supp}\,f\subset \left\{\frac{1}{2}\le |w|\le 3\right\}. \end{align}\] Let \(N(w)=-c_n|w|^{2-2n}\) be the Newtonian kernel in real dimension \(2n\ge 4\), and define \[v:=N*f.\] Then \(v\) is smooth and radial on \(\mathbb{C}^n\), satisfying \(v\geq0\) and \[\Delta_{\mathrm{eucl}}v:=2\sum^n_{j=1}\frac{\partial^2v}{\partial w_j\partial \bar{w}_j}=f.\] We also need the following elementary fact.

By Proposition 19 and 64 , we obtain \[\mathrm{MinVol}_C(M)\leq \mathrm{vol}_{h_\delta}(\widehat{M})\leq \frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)+4\varepsilon\] when \(0<\delta\ll\rho\ll1\), while Theorem 1 gives \[\mathrm{MinVol}_C(M)\geq \frac{(n\pi)^n}{n!}\mathrm{CanVol}(M).\] The proof of Theorem 7 is complete.

8 Failure of regularization of \(g_{KE}\) with Ricci curvatures bounded↩︎

It is also natural to consider \[\mathcal{I}_R(M):=\inf_{g\in\mathcal{RM}(M)}\int_M|Ric(g)|^{n/2}dV_g\] for any compact Riemannian \(n-\)manifold \(M\) (see LeBrun [19]). By [19], if \(M\) is a projective surface with \(K_M\) big and nef and \(M_k\) is the blow-up of \(M\) at \(k\) points, then \[\label{eq:Ricci95integral} \mathcal{I}_R(M_k)\geq 8\pi^2\mathrm{CanVol}(M)+k\;\left(>8\pi^2\mathrm{CanVol}(M_k)\right)\tag{75}\] holds.

Since \(K_{M_k}\) is big, there exists a singular Kähler-Einstein metric \(g_{KE}\) which is smooth on a Zariski open subset \(U\) of \(M_k\). Based on 75 , we have

Acknowledgements. The first author is supported by National Natural Science Foundation of China, No. 12271101; the third author is supported by National Natural Science Foundation of China, No. 12471079.

References↩︎