Scalar curvature, Kodaira dimension and ${{\widehat{A}}}$-genus

Abstract

Let (X, g) be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure J compatible with g, then the Kodaira dimension of (X, J) is equal to $-\infty$ and the canonical bundle $K_X$ is not pseudo-effective. We also introduce the complex Yamabe number $\lambda _c(X)$ for compact complex manifold X, and show that if $\lambda _c(X)$ is greater than 0, then $\kappa (X)$ is equal to $-\infty$; moreover, if X is also spin, then the Hirzebruch A-hat genus ${{\widehat{A}}}(X)$ is zero.

Appendix: The scalar curvature relation on compact complex manifolds

Let’s recall some elementary settings (e.g. [24, Section 2]). Let $(M, g, \nabla )$ be a 2n-dimensional Riemannian manifold with the Levi–Civita connection $\nabla$. The tangent bundle of M is also denoted by $T_{{\mathbb {R}}}M$. The Riemannian curvature tensor of $(M,g,\nabla )$ is

$$\begin{aligned} R(X,Y,Z,W)=g\left( \nabla _X\nabla _YZ-\nabla _Y\nabla _XZ-\nabla _{[X,Y]}Z,W\right) \end{aligned}$$

for tangent vectors $X,Y,Z,W\in T_{{\mathbb {R}}}M$. Let $T_{{\mathbb {C}}}M=T_{{\mathbb {R}}}M\otimes {{\mathbb {C}}}$ be the complexification. We can extend the metric g and the Levi–Civita connection $\nabla$ to $T_{{{\mathbb {C}}}}M$ in the ${{\mathbb {C}}}$-linear way. Hence for any $a,b,c,d\in {{\mathbb {C}}}$ and $X,Y,Z,W\in T_{{\mathbb {C}}}M$, we have

$$\begin{aligned} R(aX,bY,cZ, dW)=abcd\cdot R(X,Y,Z,W).\end{aligned}$$

Let (M, g, J) be an almost Hermitian manifold, i.e., $J:T_{{\mathbb {R}}}M\rightarrow T_{{\mathbb {R}}}M$ with $J^2=-1$, and for any $X,Y\in T_{{\mathbb {R}}}M$, $g(JX,JY)=g(X,Y)$. The Nijenhuis tensor $N_J:\Gamma (M,T_{{\mathbb {R}}}M)\times \Gamma (M,T_{{\mathbb {R}}}M)\rightarrow \Gamma (M,T_{{\mathbb {R}}}M)$ is defined as

$$\begin{aligned} N_J(X,Y)=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY].\end{aligned}$$

The almost complex structure J is called integrable if $N_J\equiv 0$ and then we call (M, g, J) a Hermitian manifold. We can also extend J to $T_{{\mathbb {C}}}M$ in the ${{\mathbb {C}}}$-linear way. Hence for any $X,Y\in T_{{\mathbb {C}}}M$, we still have $g(JX,JY)=g(X,Y).$ By Newlander–Nirenberg’s theorem, there exists a real coordinate system $\{x^i,x^I\}$ such that $z^i=x^i+\sqrt{-1}x^I$ are local holomorphic coordinates on M. Let’s define a Hermitian form $h:T_{{\mathbb {C}}}M\times T_{{\mathbb {C}}}M\rightarrow {{\mathbb {C}}}$ by

$$\begin{aligned} h(X,Y):= g(X,Y), \ \ \ \ \ X,Y\in T_{{\mathbb {C}}}M. \end{aligned}$$

(6.1)

By J-invariant property of g,

$$\begin{aligned} h_{ij}:=h\left( \frac{\partial }{\partial z^i},\frac{\partial }{\partial z^j}\right) =0, \quad \text{ and }\quad h_{{{\overline{i}}}{{\overline{j}}}}:=h\left( \frac{\partial }{\partial \overline{z}^i},\frac{\partial }{\partial {{\overline{z}}}^j}\right) =0 \end{aligned}$$

(6.2)

and

$$\begin{aligned} h_{i{{\overline{j}}}}:=h\left( \frac{\partial }{\partial z^i},\frac{\partial }{\partial {{\overline{z}}}^j}\right) =\frac{1}{2}\left( g_{ij}+\sqrt{-1}g_{iJ}\right) . \end{aligned}$$

(6.3)

It is obvious that $(h_{i{{\overline{j}}}})$ is a positive Hermitian matrix. Let $\omega$ be the fundamental two-form associated to the J-invariant metric g:

$$\begin{aligned} \omega (X,Y)=g(JX,Y). \end{aligned}$$

(6.4)

In local complex coordinates,

$$\begin{aligned} \omega =\sqrt{-1}h_{i{{\overline{j}}}} dz^i\wedge d{{\overline{z}}}^j. \end{aligned}$$

(6.5)

In the local holomorphic coordinates $\{z^1,\ldots , z^n\}$ on M, the complexified Christoffel symbols are given by

$$\begin{aligned} \Gamma _{AB}^C= & {} \sum _{E}\frac{1}{2}g^{CE}\big (\frac{\partial g_{AE}}{\partial z^B}+\frac{\partial g_{BE}}{\partial z^A}-\frac{\partial g_{AB}}{\partial z^E}\big )\nonumber \\= & {} \sum _{E}\frac{1}{2}h^{CE}\big (\frac{\partial h_{AE}}{\partial z^B}+\frac{\partial h_{BE}}{\partial z^A}-\frac{\partial h_{AB}}{\partial z^E}\big ) \end{aligned}$$

(6.6)

where $A,B,C,E\in \{1,\ldots ,n,{\overline{1}},\ldots ,{\overline{n}}\}$ and $z^{A}=z^{i}$ if $A=i$, $z^{A}={\overline{z}}^{i}$ if $A={\overline{i}}$. For example

$$\begin{aligned} \Gamma _{ij}^k=\frac{1}{2}h^{k\overline{\ell }}\left( \frac{\partial h_{j{{\overline{\ell }}}}}{\partial z^i}+\frac{\partial h_{i\overline{\ell }}}{\partial z^j}\right) ,\ \Gamma _{{{\overline{i}}}j}^k=\frac{1}{2}h^{k{\overline{\ell }}}\left( \frac{\partial h_{j{{\overline{\ell }}}}}{\partial {{\overline{z}}}^i}-\frac{\partial h_{j{\overline{i}}}}{\partial {{\overline{z}}}^\ell }\right) . \end{aligned}$$

(6.7)

We also have $\Gamma _{{{\overline{i}}}{{\overline{j}}}}^k=\Gamma _{ij}^{{{\overline{k}}}}=0$ by the Hermitian property $h_{pq}=h_{{{\overline{i}}}{{\overline{j}}}}=0$. The complexified curvature components are

$$\begin{aligned} R_{ABC}^D=\sum _ER_{ABCE}h^{ED}=-\left( \frac{\partial \Gamma _{AC}^D}{\partial z^B}-\frac{\partial \Gamma _{BC}^D}{\partial z^A}+\Gamma _{AC}^F\Gamma _{FB}^D-\Gamma _{BC}^F\Gamma _{AF}^D\right) . \end{aligned}$$

(6.8)

By the Hermitian property again, we have

$$\begin{aligned} R_{i{{\overline{j}}}k}^{l}=-\left( \frac{\partial \Gamma ^{l}_{ik}}{\partial {\overline{z}}^j}-\frac{\partial \Gamma ^{l}_{{{\overline{j}}}k}}{\partial z^i}+\Gamma _{ ik}^{s}\Gamma ^{l}_{{{\overline{j}}}s}-\Gamma _{ {{\overline{j}}}k}^{s}\Gamma ^{l}_{ is}-{\Gamma _{{{\overline{j}}} k}^{{{\overline{s}}}}\Gamma _{i{\overline{s}}}^l}\right) . \end{aligned}$$

(6.9)

It is computed in [24, Lemma 7.1] that

Lemma 6.1

On the Hermitian manifold (M, h), the Riemannian Ricci curvature of the Riemannian manifold (M, g) satisfies

$$\begin{aligned} Ric(X,Y)=h^{i{{\overline{\ell }}}}\left[ R\left( \frac{\partial }{\partial z^i}, X,Y, \frac{\partial }{\partial {{\overline{z}}}^\ell }\right) +R\left( \frac{\partial }{\partial z^i}, Y,X, \frac{\partial }{\partial {{\overline{z}}}^\ell }\right) \right] \end{aligned}$$

(6.10)

for any $X,Y\in T_{{\mathbb {R}}}M$. The Riemannian scalar curvature is

$$\begin{aligned} s=2h^{i{{\overline{j}}}}h^{k{{\overline{\ell }}}}\left( 2R_{i{{\overline{\ell }}} k{{\overline{j}}}}-R_{i{{\overline{j}}} k{{\overline{\ell }}}}\right) . \end{aligned}$$

(6.11)

The following result is established in [24, Corollary 4.2] (see also some different versions in [11]). For readers’ convenience we include a straightforward proof without using “normal coordinates”.

Lemma 6.2

On a compact Hermitian manifold $(M,\omega )$, the Riemannian scalar curvature s and the Chern scalar curvature $s_{\mathrm C}$ are related by

$$\begin{aligned} s=2s_{\mathrm C}+\left( \langle \partial \partial ^*\omega +{\overline{\partial }}{\overline{\partial }}^*\omega ,\omega \rangle -2|\partial ^*\omega |^2\right) -\frac{1}{2}|T|^2, \end{aligned}$$

(6.12)

where T is the torsion tensor with

$$\begin{aligned} T_{ij}^k=h^{k{{\overline{\ell }}}}\left( \frac{\partial h_{j{{\overline{\ell }}}}}{\partial z^i}-\frac{\partial h_{i{{\overline{\ell }}}}}{\partial z^j}\right) . \end{aligned}$$

Proof

For simplicity, we denote by

$$\begin{aligned} s_\text {R}=h^{i{{\overline{j}}}}h^{k{{\overline{\ell }}}}R_{i{{\overline{\ell }}} k{{\overline{j}}}} \quad \text{ and }\quad s_\text {H}=h^{i{{\overline{j}}}}h^{k{{\overline{\ell }}}}R_{i{{\overline{j}}} k{{\overline{\ell }}}}. \end{aligned}$$

Then, by formula (6.11), we have $s=4s_\text {R}-2s_\text {H}.$ In the following, we shall show

$$\begin{aligned} s_{\text {H}}=s_{\text {C}}-\frac{1}{2}\langle \partial \partial ^*\omega +{\overline{\partial }}{\overline{\partial }}^*\omega ,\omega \rangle -\frac{1}{4}|T|^2 \end{aligned}$$

(6.13)

and

$$\begin{aligned} s_\text {R}=s_\text {C}-\frac{1}{2}|\partial ^*\omega |^2-\frac{1}{4}|T|^2. \end{aligned}$$

(6.14)

It is easy to show that

$$\begin{aligned} {\overline{\partial }}^*\omega =2\sqrt{-1}\overline{\Gamma _{{\overline{i}}k}^k}dz^i \end{aligned}$$

(6.15)

and so

$$\begin{aligned} -\frac{\partial \partial ^*\omega +{\overline{\partial }}{\overline{\partial }}^*\omega }{2}=\sqrt{-1}\left( \frac{\partial \Gamma _{{\overline{j}} k}^k}{\partial z^i}+\frac{\partial \overline{\Gamma _{{\overline{i}}k}^k}}{\partial {\overline{z}}^j}\right) dz^i\wedge d{{\overline{z}}}^j. \end{aligned}$$

(6.16)

On the other hand, by formula (6.9), we have

$$\begin{aligned} {R}_{i{{\overline{j}}}k}^k=-\frac{\partial \Gamma ^{k}_{ik}}{\partial {{\overline{z}}}^j}+\frac{\partial \Gamma ^{k}_{{{\overline{j}}}k}}{\partial z^i}+{\Gamma _{{{\overline{j}}} k}^{{\overline{s}}}\Gamma _{i{{\overline{s}}}}^k}. \end{aligned}$$

(6.17)

A straightforward calculation shows

$$\begin{aligned} h^{i{{\overline{j}}}} {\Gamma _{{{\overline{j}}} k}^{{{\overline{s}}}}\Gamma _{i{\overline{s}}}^k}=-\frac{1}{4}|T|^2. \end{aligned}$$

Moreover, we have

$$\begin{aligned} \left( -\frac{\partial \Gamma ^{k}_{ik}}{\partial {{\overline{z}}}^j}+\frac{\partial \Gamma ^{k}_{{{\overline{j}}}k}}{\partial z^i}\right) -\left( \frac{\partial \Gamma _{{{\overline{j}}} k}^k}{\partial z^i}+\frac{\partial \overline{\Gamma _{{\overline{i}}k}^k}}{\partial {\overline{z}}^j}\right) =-\frac{\partial \Gamma ^{k}_{ik}}{\partial {\overline{z}}^j}-\frac{\partial \overline{\Gamma _{{\overline{i}}k}^k}}{\partial {\overline{z}}^j}=-\frac{\partial ^2\log \det (g)}{\partial z^i\partial {{\overline{z}}}^j}\nonumber \\ \end{aligned}$$

(6.18)

where the last identity follows from (6.7). Indeed, we have

$$\begin{aligned} \Gamma ^{k}_{ik}+\overline{\Gamma _{{\overline{i}}k}^k}=h^{k{{\overline{\ell }}}}\frac{\partial h_{k{{\overline{\ell }}}}}{\partial z^i}=\frac{\partial \log \det (g)}{\partial z^i}. \end{aligned}$$

Hence, we obtain

$$\begin{aligned} s_\text {H}+\left\langle \frac{\partial \partial ^*\omega +{\overline{\partial }}{\overline{\partial }}^*\omega }{2}, \omega \right\rangle =s_{\text {C}}-\frac{1}{4}|T|^2 \end{aligned}$$

which proves (6.13). Similarly, one can show (6.14). $\square$