I just have two questions.

1. For Theorem 0.1, is it enough to prove it simply using the presheaf structure on B_\varepsilon defined in section 2, page 6 ?  It seems to me that, at least for the proof of (a), (b), the corresponding Proposition 2.5 depends only the presheaf structure. For (c), it might be possible to define the notion of Nakano positivity, using variation of log-norms of local holomorphic sections of B_\varepsilon (which is given by the presheaf structure).   I do not know whether the author has some other motivations (say, for future applications) for using the non-trivial real analytic Hilbert bundle structure.

2. For theorem 0.2, finite rank case, it seems that one may use Hormander estimate with singular weight to produce local frames in Ker D^{0,1} (to avoid the Nash-Moser regularity argument). My question is:

Is it possible to use the Hormander method for the general real analytic (smooth) Hilbert bundles case ? If not, could the author give more hints why Nash-Moser + integral representation fits better with the general case ?

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Answer to the questions raised in Report C

>> 1. For Theorem 0.1, is it enough to prove it simply using the presheaf structure on B_\varepsilon defined in section 2, page 6 ?  It seems to me that, at least for the proof of (a), (b), the corresponding Proposition 2.5 depends only the presheaf structure. For (c), it might be possible to define the notion of Nakano positivity, using variation of log-norms of local holomorphic sections of B_\varepsilon (which is given by the presheaf structure). I do not know whether the author has some other motivations (say, for future applications) for using the non-trivial real analytic Hilbert bundle structure.

The proof of parts (a), (b) of Theorem indeed only uses the (pre)sheaf structure of ${\cal B}_\varepsilon$. The referee would like to thank the referee for this observation .  A paragraph of explanation has been added after the statement of Theorem 0.1.

I have not answered the final question in the manuscript, but the curvature tensor can be obtained formally just by computing dbar and ddbar of the square of the norm of holomorphic sections. The concept of Griffiths positivity can be seen directly by asking the square norm of holomorphic sections of the dual bundle to be plurisubharmonic. However, the concept of Nakano positivity a priori requires some tensor algebra, and cannot be seen as easily from properties of squares of holomorphic sections. Our main motivations are described in section 4, although the application to the invariance of plurigenera is conclusive only in a very special case. Other applications are not too unlikely, but we do not have anything precise to say at this stage.

>> 2. For theorem 0.2, finite rank case, it seems that one may use Hormander estimate with singular weight to produce local frames in Ker D^{0,1} (to avoid the Nash-Moser regularity argument). My question is:

>> Is it possible to use the Hormander method for the general real analytic (smooth) Hilbert bundles case ? If not, could the author give more hints why Nash-Moser + integral representation fits better with the general case ?

The referee is right that several well known approaches work - starting from the "elementary" one given by Malgrange in 1958. We have added some additional explanations after the statement of Theorem 0.2, as well as a reference to L. Hörmander and to a related paper of S. Webster.

No other changes have been made - they are all located in the introduction and in the reference list.