Let $\mathcal E\to X$ be a stable vector bundle over a polarized projective manifold $(X,\omega)$. It is well-known, that in this case $\mathcal E$ admits Hermitian-Einstein metric, i.e., a metric $h$, such that (may be up to some constants)

$$ {\rm tr}_\omega F(\mathcal E, h)=\lambda \rm Id, $$ where

- $F(\mathcal E, h)\in \Lambda^{1,1}(T^*M)\otimes \rm End(\mathcal E)$ is the curvature of Chern connection on $(\mathcal E, h)$
- ${\rm tr}_\omega\colon \Lambda^{1,1}(T^*M)\otimes \rm End(\mathcal E)\to \rm End(\mathcal E)$ is a contraction with $\omega$
- $\lambda=\int_X c_1(\mathcal E)\wedge\omega^{n-1}/\int_X \omega^n$.

Assume additionally that $\mathcal E$ admits a metric $h_0$ such that the corresponding curvature $F(\mathcal E, h_0)$ is Griffiths non-negative, i.e., for any $v\in T_x^{1,0}X$, $e\in \mathcal E_x$, $h_0\bigl(F(\mathcal E, h_0)(v,\bar v)e,\bar e\bigr)\ge 0$.

**Question:** Is it true that under this additional non-negativity assumption the Hermitian-Einstein metric $h$ is also Griffiths non-negative?

**Upd Oct 5.** In the present form the question might seem not well-motivated, so I would like to make two points.

1) Why one should expect this to be true?

Hermitian-Einstein metric can be constructed as limits of a heat-type flow (see, e.g., Yum-Tong Siu, *Lectures on Hermitian-Einstein metrics for stable bundles and Kahler-Einstein metrics*, 1986). On the other hand, various geometric flows are known to preserve certain positivity conditions. For example, Ricci flow preserves positivity of curvature operator. So the question can be resolved by studying which positivity conditions are preserved by the relevant heat flow.

2) Why one might care?

Existence of Hermitian-Einstein metric implies Kobayashi-Lubke inequality relating $c_2(\mathcal E)$, $c_1(\mathcal E)$ and $\omega$. At the same time, for a Griffiths non-negative vector bundle there is a whole family of Fulton-Lazarsfeld inequalities, involving higher Chern classes. It is reasonable to expect, that once $\mathcal E$ admits a metric that is both Hermitian-Einstein and positive, then stronger inequalities hold. This in turn can help in various computations involving Hirzebruch-Riemann-Roch formula.