Let $S$ be a complex manifold and let $p : E \to S$ be a holomorphic vector bundle. Is there any advantage to knowing that $E$ carries a flat Hermitian metric $h$, ie a smooth Hermitian metric with curvature form $i \Theta_{E,h} = 0$, over just knowing that $E$ carries a flat connection $\nabla$?

For an example, consider a family of compact Kahler manifolds $\pi : \mathcal X \to S$ over a smooth, connected base $S$. Then the Hodge bundle $E^k$ whose sheaf of sections is $$ \mathcal E^k := \mathcal R^k\pi_* \mathbb C \otimes_{\mathbb C} \mathcal O_S $$ carries a natural flat connection $\nabla$, called the Gauss-Manin connection. This flat structure on the Hodge bundles has been studied quite a bit. For one application, if the family $\pi : \mathcal X \to S$ is polarized, then this situation gives a seminegative Hermitian metric on the vector bundle associated to the sheaf $\mathcal R^0 \pi_* \Omega_{\mathcal X/S}^k \hookrightarrow \mathcal E^k$.

In this situation we can make a pretty trivial observation: If the family is polarized, then the Hodge inner product in cohomology defines a smooth Hermitian metric $h$ on $E^k$, whose Chern connection is the Gauss-Manin connection. Indeed, if $u,v$ are local sections of $E^k$ and the polarization is given by the family $s \mapsto \omega(s)$ of Kahler classes, then $h$ is defined by $$ h(u,v)(s) = \int_{X_s} u(s) \wedge *_s \overline{v(s)}, $$ where $*_s$ is the Hodge star operator on cohomology defined by $\omega_s$. By definition, the family being polarized means that $\nabla \omega = 0$. Let's write $v = \sum_j L^{k-j} v_j$ for the fiberwise primitive decomposition of the $k$-class $v$, where $Lv := v \wedge \omega$ is the fiberwise Lefschetz operator. Then $* v = \sum_j a_j L^{n-k-j} v_j$, where $a_j$ are constants that don't matter, so $\nabla(*v) = *(\nabla v)$. Applying this and basic properties of the Gauss-Manin connection gives that $$ d h(u,v) = h(\nabla u, v) + h(u, \nabla v), $$ so the Gauss-Manin connection is the Chern connection of $h$.

I think it's very unlikely that someone didn't notice this already (Griffiths, for one?) but I still haven't seen people mention this when talking about the Gauss-Manin connection. This flat metric is not even necessary to show seminegativity of the Weil-Peterson metrics, so I wonder if it's of any use at all?

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    $\begingroup$ My recollection is that Griffiths did notice this. I don't have time to check myself right now, but see his "Periods of integrals… III" $\endgroup$ – Donu Arapura Mar 26 '14 at 15:21
  • $\begingroup$ Huh. I was reading that the other day and didn't see it. But then again, it would be strange if Griffiths hadn't remarked this. I'll check again. $\endgroup$ – Gunnar Þór Magnússon Mar 26 '14 at 15:37
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    $\begingroup$ You have most definitely seen this mentioned in the literature: in the definition of polarised RVHS the polarisation is required to be a flat (horizontal) pairing wrt the connection. So, to second Donu, this must be somewhere in Griffiths' "Periods". $\endgroup$ – Peter Dalakov Mar 26 '14 at 15:44
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    $\begingroup$ Don't we just require that the intersection product $(u,v) \mapsto u \wedge v \wedge \omega^{n-2k}$ is flat in polarized VHS? That defines a Hermitian metric that's not positive-definite (we're not just talking about primitive classes), so to me it seems like we have to remark separately that the Hodge inner product is flat. $\endgroup$ – Gunnar Þór Magnússon Mar 26 '14 at 15:49
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    $\begingroup$ This is a bit subtle, the indefinite metric is flat, the Hodge metric is not. I may try to write something more later (have to run now) but check ch.13 of Carlson-Muellerstarch-Peters. $\endgroup$ – Peter Dalakov Mar 26 '14 at 16:00

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