Nakano and dual-Nakano positivity of the adjoint bundle $S^2E\otimes\det E\otimes K_S$ for ample vector bundles of rank at least the base dimension

For a compact Kähler manifold $S$ of dimension $n$ and an ample holomorphic vector bundle $E\to S$ of rank $r$, Liu–Sun–Yang proved sharp Nakano- and dual-Nakano-positivity statements for the adjoint bundles $S^kE\otimes(\det E)^2\otimes K_S$ ($k\ge\max\{n-r,0\}$) and $E\otimes(\det E)^k\otimes K_S$ ($k\ge\max\{n+1-r,2\}$), where $K_S$ is the canonical line bundle. In Section 5 (Remark 5.4) of that paper they pose the boundary problem left between these ranges: is $S^2E\otimes\det E\otimes K_S$ Nakano-positive and dual-Nakano-positive when $E$ is ample and $r\ge n$? They note it would follow from ampleness of $S^{n+2}E\otimes K_S$, itself unknown. No resolution or refutation of this boundary case is known in the literature to date.

Background

Let $S$ be a compact Kähler manifold of complex dimension $n$ and let $E\to S$ be a holomorphic vector bundle of rank $r$ equipped with a smooth Hermitian metric $h$. The Chern curvature tensor $\Theta=(R_{i\bar j\alpha\bar\beta})$ of $(E,h)$, indexed by base coordinates $i,j$ and fiber coordinates $\alpha,\beta$, defines several nested notions of positivity. The bundle $(E,h)$ is Griffiths-positive if the quadratic form $\sum R_{i\bar j\alpha\bar\beta}u^i\bar u^jv^\alpha\bar v^\beta$ is positive on all nonzero decomposable tensors $u\otimes v$; Nakano-positive if $\sum R_{i\bar j\alpha\bar\beta}u^{i\alpha}\overline{u^{j\beta}}>0$ for every nonzero element $u^{i\alpha}$ of $T^{1,0}S\otimes E$; and dual-Nakano-positive if the same expression with fiber and base indices transposed, $\sum R_{i\bar j\alpha\bar\beta}u^{i\beta}\overline{u^{j\alpha}}$, is positive, which is equivalent to the dual bundle $(E^*,h^*)$ being Nakano-negative. Nakano positivity implies Griffiths positivity, but Nakano and dual-Nakano positivity are logically independent conditions, and all of them make sense as properties of the bundle $E$ itself by demanding the existence of some metric with the stated curvature sign.

On the algebraic side, $E$ is called ample if the tautological line bundle $\mathcal O_{\mathbb P(E^*)}(1)$ is ample on the projectivization $\mathbb P(E^*)$ of the dual bundle. For line bundles, ampleness is equivalent to the existence of a positively curved metric by the Kodaira embedding theorem (a positively curved line bundle has enough holomorphic sections to embed the base into projective space); Griffiths conjectured in the late 1960s that the same equivalence holds for vector bundles, and this conjecture is settled only for base curves (Campana–Flenner) and remains open in dimension at least two. Producing positive metrics on bundles constructed from an ample $E$ is therefore a central difficulty. Here $S^kE$ denotes the $k$-th symmetric power, $\det E$ the determinant line bundle, and $K_S$ the canonical line bundle, so $S^2E\otimes\det E\otimes K_S$ is an adjoint-type bundle of rank $r(r+1)/2$.

Two classical precursors frame the question: the Demailly–Skoda theorem that $(E,h)$ Griffiths-positive with $r>1$ forces $E\otimes\det E$ to be Nakano-positive, and Berndtsson's theorem (via positivity of direct images in holomorphic fibrations) that $S^kE\otimes\det E$ is Nakano-positive for every $k\ge 0$ whenever $E$ is ample. Neither addresses twists by the canonical bundle, which is typically not positive.

Liu–Sun–Yang built an ampleness-to-positivity mechanism — if $S^{r+k}E\otimes\det E^*\otimes F$ is ample then $S^kE\otimes F$ is both Nakano- and dual-Nakano-positive — and combined it with a Fujita–Ye–Zhang ampleness lemma ($\det E\otimes K_S$ is ample for $r\ge n+1$, apart from the single exception $(\mathbb P^n,\mathcal O(1)^{\oplus(n+1)})$) to prove the sharp statements that $S^kE\otimes(\det E)^2\otimes K_S$ is Nakano- and dual-Nakano-positive for $k\ge\max\{n-r,0\}$ and that $E\otimes(\det E)^k\otimes K_S$ is Nakano- and dual-Nakano-positive for $k\ge\max\{n+1-r,2\}$, with both lower bounds sharp (witnessed by $(\mathbb P^4,\mathcal O(1)^{\oplus 2})$-type examples where the bundles fail to be Nakano-positive below the bound). They verified the special case $S^2T\mathbb P^n\otimes\det T\mathbb P^n\otimes K_{\mathbb P^n}$, using the Fubini–Study metric and the Veronese embedding. In that source they observe that the boundary case $S^2E\otimes\det E\otimes K_S$ for $r\ge n$ would follow from ampleness of $S^{n+2}E\otimes K_S$, equivalently of $\mathcal O_{\mathbb P(E^*)}(n+2)\otimes\pi^*K_S$ under the projection $\pi:\mathbb P(E^*)\to S$, via the ampleness-to-positivity mechanism above, but this sharp Fujita-type ampleness at the boundary rank $r=n$ is itself unknown.

The motivation is twofold: Nakano positivity forces cohomology vanishing by the Nakano vanishing theorem (a Nakano-positive bundle $F$ satisfies $H^q(S,\Omega_S^n\otimes F)=0$ for $q\ge 1$), so metric positivity yields new $H^{n,q}$-type vanishing theorems; and the sharp examples show the question sits exactly at the boundary of what ampleness hypotheses can currently deliver.

Problem Statement

Let $S$ be a compact Kähler manifold of complex dimension $n$ and let $E\to S$ be an ample holomorphic vector bundle of rank $r\ge n$. Is the adjoint vector bundle

$S^2 E \otimes \det E \otimes K_S$

Nakano-positive and dual-Nakano-positive — that is, does it carry a smooth Hermitian metric with Nakano-positive Chern curvature and a smooth Hermitian metric with dual-Nakano-positive Chern curvature? (As usual, these are properties of the bundle itself; a single metric satisfying both conditions at once would also give an affirmative answer.) This is the problem posed in Section 5 (Remark 5.4) of Liu–Sun–Yang's 'Positivity and vanishing theorems for ample vector bundles'. A negative answer would consist of an explicit ample bundle $E$ of rank $r\ge n$ for which $S^2E\otimes\det E\otimes K_S$ admits no Nakano-positive metric, or none that is dual-Nakano-positive.

References

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