This page documents the refereeing process that the publication FiorenzaMartinengo2012 went through.
Originally article submission, July 30th, 2011
(submission letter, FiorenzaMartinengo2012Submission.tex, FiorenzaMartinengo2012Submission.pdf)
Steering committee selects and contacts anonymous expert referee, August 5, 2011
Submission of slightly revised article, December 24th, 2011
(re-submission letter, FiorenzaMartinengo2012SubmissionB.tex FiorenzaMartinengo2012SubmissionB.pdf).
Referee is notified of revision, December 28th, 2011
(post)
May 2nd, 2012
Referee notifies steering committee that report will be further delayed due to circumstances beyond the referee’s control.
First referee report received June 28, 2012
Domenico Fiorenza, Elena Martinengo,A short note on infinity-groupoids and the period map for projective manifolds
In this note the authors explain how the modern homotopy-theoretic formulation of deformation theory sheds light on some classical results in the subject, particularly the period mapping relating deformations of complex structure on a projective manifold to variations of the Hodge structure on its cohomology.
The role of differential graded Lie algebras (dgla’s) in deformation theory was first emphasized by Nijenhuis and Richardson in the 1960’s, and was further elaborated upon by Deligne, Drinfeld, Kontsevich and others. Classically, deformation theory is given by a functor associating to a dgla a formal space. This formal space (more precisely, a functor from the category of Artinian algebras to Sets) describes the formal neighborhood of a point in the moduli space of structures encoded by the dgla as the solutions of the Maurer-Cartan equation. The modern approach extends this formulation in two directions:
(1) the deformation functor of a dgla should take values not in formal spaces but formal stacks (in higher-dimensional groupoids), i.e. functors from Artinian algebras to simplicial sets, rather than just sets. This captures the higher-order symmetry information encoded in the components of the dgla of low (negative) degrees.
(2) (not discussed in the present note) these formal stacks should be derived, i.e. their domain should be extended from Artinian algebras to their homotopical version, chain dg algebras (or, to go beyond characteristic 0, to simplicial or E-infinity algebras). This captures the information about the singularities of the Maurer-Cartan variety encoded in the high (positive) degree components of the dgla, and in particular sheds light on the obstruction theory.
In addition to capturing new information, this extended formulation often provides conceptual explanation of the classical phenomena which are recovered as the 0-truncation of the resulting homotopy types.
The authors approach is to first construct a certain map of sheaves of dgla’s on a manifold X arising in a simple way from the contraction of differential forms by vector fields (they first explain this in the abstract setting of “Cartan homotopies” of dgla’s). Taking the (extended) deformation functors turns this into a map of infinity-sheaves of formal stacks. Taking derived global sections and 0-truncating produces the classical period map.
Throughout the note the authors keep their presentation informal, steering clear of the technicalities of infinity-categories. Since they are not claiming any new results but merely offering a conceptual explanation of some old ones, this is acceptable. My only complaint is that the presentation sometimes gets a bit too informal, to the point of confusing the reader. I have provided some comments on the text below, which I feel should be addressed before the note is published.
the statement of the Theorem on p. 3 is incorrect. First, the category Art should be replaced by dgArt or equivalent, otherwise the degree $\gt 1$ part of the dgla will stay invisible. Second, the essential image of Def is not all formal stacks but only those that can arise from formal moduli problems, and there are conditions describing those. The precise statement can be found in DAG X (http://www.math.harvard.edu/~lurie/papers/DAG-X.pdf), Thm 0.0.13 (see also Warning 0.0.12). In fact, it seems this result, although generally of fundamental importance, plays no role in this note, as all the deformation problems considered explicitly arise from dgla’s.
On p.4 the authors introduce the “internal hom” dgla, Hom(g,h) and claim that MC(Hom(g,h) tensored with differential forms on simplices) is homotopy equivalent to Hom_oo(g,h). But since the authors do not exhibit any other model for Hom_oo besides MC(Hom(g,h)…), the statement seems meaningless. The authors should state explicitly what other model they are comparing MC(Hom..) with.
It is not clear what role is played by L-infinity morphisms in the note, in view of the fact that a Cartan homotopy is defined by conditions describing when a generally L-infinity morphism is actually a strict map of dgla’s. In fact, it is also unclear why one needs the “Lie derivative” l to be a strict map rather than L-infinity.
the authors occasionally apply constructions that generally only make sense for nilpotent dgla’s (such as exp,) to arbitrary dgla’s. The interpretation is probably that it is supposed to be applied not to a single dgla, but to the functor from Art to nilpotent dgla’s defined by this dgla. But it would be less confusing if this were spelled out explicitly.
Likewise, the authors often say “infinity-groupoid”, when they really mean a formal stack in infinity-groupoids, e.g. in footnote 4 on p.9.
It would help if the authors explained what they mean by “the differential of P” (since P is a map of formal spaces, its differential must be its value on the dual numbers modulo constants).
Reaction of the authors to the first referee report, June 29, 2012
Dear Referee,
Thanks a lot for your extremely careful report, and for the truly insightful remarks and suggestions it contains. We agree with all of them and are now planning to revise the note accordingly.
Below, we are sketching the kind of revision we have in mind for each of your comments, so that -should you enjoy having a look at it- you can eventually comment on this before we actually implement it into the new version of the note.
With our best regards,
Domenico and Elena
the statement of the Theorem on p. 3 is incorrect. First, the category Art should be replaced by dgArt or equivalent, otherwise the degree >1 part of the dgla will stay invisible. Second, the essential image of Def is not all formal stacks but only those that can arise from formal moduli problems, and there are conditions describing those. The precise statement can be found in DAG X (http://www.math.harvard.edu/~lurie/papers/DAG-X.pdf), Thm 0.0.13 (see also Warning 0.0.12).
We absolutely agree. Actually, the Artin algebras involved in the note are differential graded ones, but writing “(differential graded) local Artin algebra” instead of “differential graded local Artin algebra” on page 2 was definitely not a good idea, while denoting their category as Art instead of dgArt was so bad to be evil.
Concerning the statement of the Theorem on page 3, again, we absolutely agree. At the time we wrote that in the first arXiv version of the note we were not aware of the rigorous description of formal moduli problems. Then, after Lurie’s ICM talk, we added a pointer to that but did not upgarde the statement from its “so informal to be false” version to the correct one. Luckily, we will be able to do this now.
In fact, it seems this result, although generally of fundamental importance, plays no role in this note, as all the deformation problems considered explicitly arise from dgla’s.
Right. We are accordingly planning to deemphasize the role played by this result in the note. Its original role was to make it not surprising for the reader that the hom-space of (L_oo) morphisms between dglas could be realized as the simplicial set of Maurer-Cartan elements of a suitable dgla. But this is precisely one of thise points where our being informal ends up with being confusing, so we are now planning to state the equivalence between dglas and formal moduli problems in its correct way, as a result of fundamental importane on its own, but not as something palying an actual specific role in the note.
On p.4 the authors introduce the “internal hom” dgla, Hom(g,h) and claim that MC(Hom(g,h) tensored with differential forms on simplices) is homotopy equivalent to Hom_oo(g,h). But since the authors do not exhibit any other model for Hom_oo besides MC(Hom(g,h)…), the statement seems meaningless. The authors should state explicitly what other model they are comparing MC(Hom..) with.
Again we agree. The correct way to express what we had in mind is to define Hom_oo(g,h) via its simplicial model MC(Hom(g,h)…) with no reference to other models. Namely, we are planning to reduce the whole first part of Section 2 to a single sentence like “The hom-space Hom_oo(g,h) of morphisms between G and h in the (oo,1)-category of dglas is conveniently modelled as the simplicial set MC(Hom(g,h)…), where Hom(g,h) is the Chevalley-Eilenberg-type dgla defined as follows…”, and revise accordingly the rest of the section.
It is not clear what role is played by L-infinity morphisms in the note, in view of the fact that a Cartan homotopy is defined by conditions describing when a generally L-infinity morphism is actually a strict map of dgla’s. In fact, it is also unclear why one needs the “Lie derivative” l to be a strict map rather than L-infinity.
Indeed having so much space for L_oo morphisms in the note is a reminescence of a time when we were less uded to homotopy invariant constructions and were on the other hand used to deal with explict L_oo morphisms as tools for explicit computations. At the “transition towards homotopy” period the note was written it was nice to us to see the definition of Cartan homotopy as a condition describing when a certain L_oo morphism was actually strict. But now that we don’t look at strict L_oo morphisms as something cool anymore (and indeed there is nothing intrinsic in them) we are going to deemphasize this. Namely, we are now planning to give the plain definition of Cartan homotopy at the beginning of the section as something motivated by classical Cartan identities and to show directly (ore just say, since it is a one line computation) that the Lie derivative associated to a Cartan homotopy is a dgla morphism. And next just to observe the gauge equivalence e^{-i}0=l expressing the fact that l is a dgla morphism homotopy equivalent to zero.
the authors occasionally apply constructions that generally only make sense for nilpotent dgla’s (such as exp,) to arbitrary dgla’s. The interpretation is probably that it is supposed to be applied not to a single dgla, but to the functor from Art to nilpotent dgla’s defined by this dgla. But it would be less confusing if this were spelled out explicitly. -Likewise, the authors often say “infinity-groupoid”, when they really mean a formal stack in infinity-groupoids, e.g. in footnote 4 on p.9.
Again, absolutely right. We are going to spell these out explicitly.
It would help if the authors explained what they mean by “the differential of P” (since P is a map of formal spaces, its differential must be its value on the dual numbers modulo constants).
Here we could add a few lines to Section 1 as follows: after having mentioned formal moduli problems we could recall that the tangent space to a formal moduli problem P is P(k[epsilon]/(epsilon^2)) and that the differential of a morphism \phi: P –> P’ is \phi(k[epsilon]/(epsilon^2)): P(k[epsilon]/(epsilon^2)) –> P’(k[epsilon]/(epsilon^2)). By the way, we are working with unitary local Artin algebras but of such an algebra we actually take only the maximal ideal m_A: should we better work directly with nilpotent Artin algebras (and so define the tangent space to P as P((epsilon)/(epsilon^2)) instead?
Reaction of the referee, June 30, 2012
There’s not much to say, really, except I’m glad we’re on the same page. I’m sorry for not noticing the parenthesized “(differential graded)” on page 2 and assuming the authors meant discrete Artinian algebras throughout. There’s also a question of what dgArt should really mean: dg algebras whose underlying superalgebra is Artinian, or any infinity-category equivalent to that, eg. of dg algebras whose homology is Artinian (one could even take those algebras to be quasi-free). Regarding the last question, I guess it’s largely a matter of taste. I’m used to having all my algebras unital. But in this case they are also augmented, so it amounts to the same thing.
Submission of revised version by the authors, August 17, 2012: this is what is now the final version
Final referee reaction, August 23, 2012: The referee had no further comments and accepted the revised version.