Let $L$ be a holomorphic line bundle on a complex manifold $X$, and assume it is equipped with a singular hermitian metric $h$ with local weight $\varphi$. Then, one can show that the de Rham class of $\frac{i}{\pi}\partial \overline{\partial} \varphi$ coincides with the first Chern class $c\_1(L)$ of the line bundle.


Is there a generalization of this result to higher-rank vector bundles? 


For example, let $E$ be a holomorphic vector bundle of rank $r$ on a complex manifold $X$ and let $h$ be a singular hermitian metric on $E$. Can we describe the Chern clases $c\_1(E),\ldots,c\_r(E)$ in terms of $h$ and its curvature form? 



Chern classes and singular hermitian metrics on vector bundles
Given a complex hermitian vector bundle $E$ of complex rank n over a smooth manifold $M$, a representative of each Chern class $ck(E)$ of $E$ are given as the coefficients of the characteristic polynomial of the curvature form $Ω$ of $E$.


$$\det \left(\frac {it\Omega}{2\pi} +I\right) = \sum\_k c\_k(E) t^k $$


Each Chern class
$c\_k$
is a real cohomology class


this expression for the Chern form expands as


$$ \sum\_k c\_k(E) t^k = \left[ I + i \frac{\mathrm{tr}(\Omega)}{2\pi} t + \frac{\mathrm{tr}(\Omega^2)-\mathrm{tr}(\Omega)^2}{8\pi^2} t^2 + i \frac{-2\mathrm{tr}(\Omega^3)+3\mathrm{tr}(\Omega^2)\mathrm{tr}(\Omega)-\mathrm{tr}(\Omega)^3}{48\pi^3} t^3 + \cdots \right]. $$


Let $S\in H^0(X,E)$ be a non-zero holomorphic section 


Then the Poincaré-Lelong formula says that


$$dd^c\log ||S||=[Z\_S]-\frac{\sqrt{-1}}{2\pi}\Theta(E)$$


where $\Theta$ is the Chern curvature. See <http://www.mathunion.org/ICM/ICM1994.2/Main/icm1994.2.0817.0827.ocr.pdf>


and <http://arxiv.org/abs/alg-geom/9708003>


**In singular setting:**


On a holomorphic line bundle a hermitian metric $h$
is just a scalar-valued
function so that $\Theta=\bar\partial\partial \log h$ as long as $\log h\in L\_{loc}^1(X)$ But for holomorphic
vector bundles with rank
$E
≥
2 $ it is not clear what the appropriate notions
of connection and curvature associated with
$h$
are.


Let
$h$
be a singular hermitian metric on a holomorphic
vector bundle
$E$
, and assume that
$h$
is negatively curved in the sense of Griffiths. Then $\log \det
h$ is a plurisubharmonic function and if $\det h\neq 0$ then 
$\log\det h\in L\_{loc}^1(X)$ due to the proposition 1.1 of the nice paper <http://link.springer.com/article/10.1007%2Fs11512-015-0212-4>


