The Poincaré Index on Singular Varieties

Round 1

Reviewer 1 Report

Summary of the Work

The aim of the work is to compute the indices of vector fields on singular varieties.  The author showed some simple methods of computation of the homological indices of various kinds given on complex varieties with singularities of different types.

Main Results Obtained

- The author computed the index in the case of Gorenstein curves and graded surfaces, monomial varieties and non-normal surfaces.

- The proposed method does not require the use of computers, integration, deformations or spectral sequences.

 

General Remarks

- This manuscript belongs to a series of works developed by the author over the past few years. In particular, this paper is a continuation of a work on the Poincaré index recently published by the author in the journal Universe-MDPI.

- The first four Sections, up to page 8/19, introduce well known concepts and definitions on indices of vector fields. However, I agree with the author that it is essential to spend an effort to establish the nomenclature. This makes reading work faster. Furthermore, I appreciated his lucid and synthetic synthesis. The novelty of the manuscript can be found in the last six Sections. The reader specialised in the field, can “jump” directly to the last six Sections.

- The "Conclusions" section is missing. The inclusion of this Section is not a mere exercise as it is useful both because it summarises the main results obtained in a few lines and because it outlines the limits of the present work while illustrating the perspectives of the works to be accomplished in the future.

- I did not find mathematical mistakes (even if a detailed check on the mathematical structure, coherence of the definitions etc. would take time, much longer than the time allowed for the evaluation of the work).

- Some relevant works in the field, recently appeared in the literature, have not been quoted in this work. The author is invited to complete the list of the works cited in the Sections “References” (see some suggestions below).

 

Suggestions

This work is vulnerable in some respects. Some of them are mentioned below.

1) In the first fourth Sections of the manuscript the authors provide a brief overview of all some indices. However, the relations among them are not clearly established, and their relations with the various generalisations to complex analytic singular varieties of the Chern classes of complex manifolds is not shown. For completeness, the author is invited to fill this gap by inserting a brief Section in this regard.

2) To proceed on to the next point, it is first of all important to answer, in the most exhaustive way, the following question: “The author has investigated the Poincare' index on singular varieties, but what are the main applications from the physical point of view?” In physics, the singularities are referred to as topological defects. To avoid repeating the conclusions of the work already published in the Universe-MDPI journal, let us analyse the very important application of the Poincare'–Hopf theorem for line fields. Indeed, nematic fields are of interest in soft-matter physics, where they are used to model nematic liquid crystals.

2a) To attract the interest of physicists more, may the author cite some concrete examples of systems showing point defects in two and three dimensions, and disclinations (i.e., lines of singularities) in three dimensions? How to extend the approach proposed by the author to calculate the value of the indices in these cases?

2b) By the way, according to the author, the Poincaré'-Hopf theorem can also be applied to biaxial nematics?

3) The author did not mention what the drawbacks of using the Poincare index are. For instance, Singular points are extracted using Poincare' index. However, the procedure indicated by the authors requires accuracy in locating singular points. Even though, Poincare\ index may be modified to improve accuracy, more computation due to modified Poincare' definition is required. Another drawback is the following. As known, Poincare' index can be used to detect core points. However, false core points cannot be eliminated completely (ref. to the work of B. Cho et al. 2000). Similarly, for clarity, the author is asked to include a short table illustrating the main advantages and disadvantages of using the Poincaré index and related indices.

4) We come now to an important point. May the author's approach be extended to manifolds with boundary? If yes, how?

5) (it is not mandatory to answer this question)

Do we focus our attention on singularity lines? According to the author, the topology of M3 can dictate which nodes and links can occur?

 

Conclusions

In my opinion the work deserves to be published. However, some gaps need to be filled. It is recommended that the author takes the above suggestions into account. In particular, the author is invited to insert a final section that illustrates the perspectives of his work and its potential applications in physics and engineering, indicating the main drawbacks of the Poincaré index and related indices.

Author Response

See PDF-file

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper needs minor revision. Please see the attached file.

Comments for author File: Comments.pdf

Author Response

See PDF-file

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The author answered (more or less) the questions raised in my first report. Many of questions have not received an answer and the author has preferred to deal with them in forthcoming works (perhaps because, at the moment, he is unable to answer). There is still a lot of work to do in this research field ! Anyhow, I can say that in general I am satisfied with this work and I confirm the fact that the manuscript deserves to be published.