1 Introduction

Since the early 2010s, the study of positive scalar curvature metrics has been revived with a new focus on quantitative results (in contrast with the topological results obtained in the 1980s). The usual metric invariants (volume and diameter) cannot be controlled by the scalar curvature alone as the example of $\mathbb {S}^2\times \mathbb {R}$ shows. Thus, one needs to investigate more subtle invariants. For instance, Bray–Brendle–Neves in dimension 3 [Reference Bray, Brendle and Neves3] and Zhu in dimension at most 7 [Reference Zhu12] have shown the following result.

Note that, for $n=2,$ this follows immediately from the Gauss–Bonnet theorem since, in this case, the scalar curvature is twice the Gauss curvature.

The proof of this result uses stable minimal hypersurfaces. The theorem is actually slightly stronger as it applies to Riemannian manifolds $(M^n,g)$ with a nonzero degree map $M^n\to \mathbb {S}^2\times \mathbb {T}^{n-2}$ .

Following Berger [Reference Berger1], the systole $\mathrm {sys}_h(g)$ of a homology class $h\in H_k(M)$ with $(M^n,g)$ a Riemannian manifold is the infimum of the areas of smooth representatives of h. Since $\pi _2(\mathbb {S}^2\times \mathbb {T}^{n-2})\simeq H_2(\mathbb {S}^2)\subset H_2(\mathbb {S}^2\times \mathbb {T}^{n-2})$ by the Hurewicz theorem, the conclusion of Theorem 1.1 can then be restated as

$$ \begin{align*}\mathrm{sys}_{[\mathbb{S}^2\times\{\ast\}]}\leq 4\pi\end{align*} $$

with the equality case covered.

This suggests that lower bounds on the scalar curvature can under appropriate topological assumptions provide an upper bound on the systole of some classes $h\in H_2(M)$ or for $\mathrm {sys}_2(g)=\inf _{h\in H_2(M)\backslash \{0\}}\mathrm {sys}_h(g)$ . Replacing $\mathbb {S}^2\times \mathbb {T}^{n-2}$ with other manifolds in Theorem 1.1 has remained elusive for a long time, although some progress on the case of $\mathbb {S}^2\times \mathbb {S}^2$ was made in [Reference Richard7].

More recently, the case of Kähler manifolds was investigated by Sha in [Reference Sha8, Reference Sha9] who showed an optimal upper bound for $\mathrm {sys}_2(\omega )$ for a closed Kähler surface $(X^2,\omega )$ with a positive average scalar curvature. This relies on the cohomological nature of the 2-systoles and the average scalar curvature for Kähler manifolds.

A few months later, Cecchini et al. [Reference Cecchini, Hirsch and Zeidler4] as well as Stryker [Reference Stryker10] used Dirac operator methods to prove upper bounds on the slightly weaker stable 2-systole of Riemannian positive scalar curvature metrics on various manifolds, including $\mathbb {CP}^n$ .

Even more recently, Tsiamis used both complex geometric and index theoretic tools in [Reference Tsiamis11] to obtain optimal upper bounds for $\mathrm {sys}_2(\omega )$ , where $(X,\omega )$ is any compact Kähler manifold with a positive scalar curvature as well as refined stable 2-systole bounds for various Riemannian manifolds $(M^n,g)$ , including, for instance, the case where $M^n$ is diffeomorphic to a Fano manifold.

This short note describes an inequality which instead of seeking an upper bound on $\mathrm {sys}_2(\omega )$ for some Kähler manifolds $(X,\omega )$ relates the systoles $\mathrm {sys}_{h_i}(\omega )$ for suitably chosen $h_1,\dots ,h_N\in H_2(X)$ in the presence of a product structure and positive scalar curvature.

To state our main result, we need a definition.

Definition 1.1 A compact complex manifold of complex dimension $n (X,J)$ is said to have an unconditionally nonpositive total scalar curvature if, for any $\eta $ in $\mathcal {K}(X),$ the Kähler cone of X (i.e., the open convex cone in $H^{1,1}(X)$ of Kähler classes, i.e., classes represented by some Kähler form),

$$ \begin{align*}\int_X c_1(X)\wedge\eta^{n-1}\leq 0.\end{align*} $$

We will focus on compact Kähler manifolds of the form $X=X_0\times \Pi _{k=1}^N\mathbb {CP}^{n_k}$ and denote by $L_k\subset \mathbb {CP}^{n_k}$ a generic complex line, and by $\omega _k$ the Fubini–Study metric on $\mathbb {CP}^{n_k}$ with a sectional curvature between $\tfrac {1}{4}$ and $1$ . We will show, for instance, the following theorem.

Theorem 1.3 Let $X_0$ be a compact Kähler manifold of complex dimension $n_0\geq 0$ with an unconditionally nonpositive total scalar curvature; assume further that $X_0$ is connected if $n_0=1$ .

Let $X=X_0\times \Pi _{k=1}^N\mathbb {CP}^{n_k}$ . Let $\omega $ be any Kähler metric on X such that:

$$ \begin{align*}\int_X\mathrm{scal}_\omega\omega^n\geq S \int_X\omega^n , \end{align*} $$

where $S=\sum _{k=1}^N {n_k(n_k+1)}{}$ .

Then,

$$ \begin{align*}\sum_{k=1}^N\frac{n_k(n_k+1)}{\mathrm{sys}_{[L_k]}(\omega)}\geq \sum_{k=1}^N \frac{n_k(n_k+1)}{4\pi}. \end{align*} $$

Moreover, if equality is achieved, then $\int _X\mathrm {scal}_\omega \omega ^n= S \int _X\omega ^n$ and $c_1(X_0)$ vanishes.

Note that the constant $\sum _{k=1}^N {n_k(n_k+1)}{}$ is the scalar curvature of $\Pi _{k=1}^N\mathbb {CP}^{n_k}$ endowed with the product of the Fubini–Study metrics on each factor. Furthermore, due to the simple connectedness of the complex projective spaces and Knneth formula, any Kähler class on X decomposes as $\pi _0^*\eta _0+\sum _{k=1}^N a_k\pi _k^*[\omega _k]$ with $\eta _0$ Kähler class on $X_0$ , $a_k>0$ , $\omega _k$ Fubini–Study metric on $\mathbb {CP}^{n_k}$ , and $\pi _k$ the projection on $X_0$ if $k=0$ , $\mathbb {CP}^{n_k}$ if $k>0$ .

The key ingredient is that the integral of the scalar curvature and the systoles of complex submanifolds can be recovered from purely cohomological data for Kähler manifolds, an idea which is also exploited in [Reference Sha8, Reference Sha9, Reference Tsiamis11].

This can be easily turned into an optimal upper bound for the $2$ -systole of X.

Corollary 1.5 If $(X,\omega )$ satisfies the same assumptions as in Theorem 1.3,

(1.1) $$ \begin{align} \min_k \mathrm{sys}_{[L_k]}(\omega) \leq 4\pi. \end{align} $$

In particular, $\mathrm {sys}_2(\omega )\leq 4\pi $ .

Moreover, if equality is achieved, then $\omega $ is in the Kähler class of $\omega _0+\sum _{k=1}^N\omega _k$ , where $\omega _0$ is a Ricci flat Kähler metric on $X_0$ .

If we moreover assume a pointwise lower bound on the scalar curvature, we can improve the rigidity part.

Corollary 1.6 If $(X,\omega )$ satisfies the same assumptions as in Theorem 1.3, and in addition,

$$ \begin{align*}\mathrm{scal}_\omega\geq S\end{align*} $$

and

$$ \begin{align*}\min_k \mathrm{sys}_{[L_k]}(\omega) = 4\pi.\end{align*} $$

then $\omega $ is biholomorphically isometric to $\omega _0+\sum _{k=1}^N\omega _k$ , where $\omega _0$ is a Ricci flat Kähler metric on $X_0$ .

If we assume $X_0$ carries some definite amount of negative curvature, we can get a scale-invariant inequality between various 2-systoles under a nonnegativity assumption on the scalar curvature. To illustrate this phenomenon, we prove the following proposition.

Proposition 1.7 Let $X=\Sigma _g\times \mathbb {CP}^n $ , where $\Sigma _g$ is a smooth projective curve of genus $g\geq 2$ , and let $\omega $ be a Kähler metric on X such that $\int _X\mathrm {scal}_\omega \omega ^{n+1}\geq 0$ ; then,

$$ \begin{align*}\mathrm{sys}_{[L]}(\omega)\leq \frac{n(n+1)}{2(g-1)}\mathrm{sys}_{[\Sigma_g]}(\omega),\end{align*} $$

where L is a generic projective line in $\mathbb {CP}^n$ .

Equality is achieved if and only if $\omega $ is in a positive multiple of the Kähler class of the product metric $\frac {n(n+1)}{2}\omega _{FS}+\omega _h$ , where $\omega _{FS}$ is the Fubini–Study metric on $\mathbb {CP}^n$ and $\omega _h$ is the hyperbolic metric on $\Sigma _g$ .

If equality is achieved and moreover $\mathrm {scal}_\omega \geq 0$ , then $\mathrm {scal}_\omega =0$ and $\omega $ is holomorphically isometric to a positive multiple of $\frac {n(n+1)}{2}\omega _{FS}+\omega _h$ .