Curvature, macroscopic dimensions, and symmetric products of surfaces

Alexander Dranishnikov; Ekansh Jauhari; Luca F. Di Cerbo

arxiv: 2503.01779 · v6 · submitted 2025-03-03 · 🧮 math.GT · math.AG· math.DG

Curvature, macroscopic dimensions, and symmetric products of surfaces

Luca F. Di Cerbo , Alexander Dranishnikov , Ekansh Jauhari This is my paper

Pith reviewed 2026-05-23 01:22 UTC · model grok-4.3

classification 🧮 math.GT math.AGmath.DG

keywords symmetric productssurfacesmacroscopic dimensionpositive scalar curvaturesymplectic asphericityKaehler manifoldsminimal model theorypositivity in algebraic geometry

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The pith

Symmetric products of surfaces sharply distinguish between two notions of macroscopic dimension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the curvature and symplectic asphericity properties of symmetric products of surfaces. It establishes that these spaces serve as examples capable of differentiating two distinct notions of macroscopic dimension for closed Riemannian manifolds with positive scalar curvature. The work extends this to address conjectures on the existence of positive scalar curvature metrics in the Kaehler projective setting. It draws connections between minimal model theory, positivity in algebraic geometry, and macroscopic dimensions.

Core claim

Symmetric products of surfaces possess curvature and symplectic asphericity properties that allow them to sharply distinguish between two distinct notions of macroscopic dimension, providing a means to investigate questions about positive scalar curvature on manifolds while linking these geometric ideas to the minimal model theory and positivity properties from algebraic geometry.

What carries the argument

Symmetric products of surfaces with their curvature and symplectic asphericity properties, which function as distinguishing examples for macroscopic dimension notions.

If this is right

Symmetric products of surfaces answer nuanced questions about closed Riemannian manifolds with positive scalar curvature.
These spaces sharply distinguish two distinct notions of macroscopic dimension.
The approach addresses conjectures on positive scalar curvature metrics in the Kaehler projective setting.
Connections emerge between minimal model theory, positivity in algebraic geometry, and macroscopic dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Symmetric products might serve as test cases for probing other geometric invariants beyond the two dimension notions.
Product constructions in higher dimensions could be checked for similar distinguishing power on dimension notions.
The linkage between algebraic positivity and macroscopic dimension may produce new invariants for manifold classification.

Load-bearing premise

Symmetric products of surfaces must have the specific curvature and symplectic asphericity properties required to serve as distinguishing examples between the two notions of macroscopic dimension.

What would settle it

A direct computation or example showing that the two notions of macroscopic dimension assign the same value to symmetric products of surfaces would demonstrate that the distinction fails for these spaces.

read the original abstract

We present a detailed study of the curvature and symplectic asphericity properties of symmetric products of surfaces. We show that these spaces can be used to answer nuanced questions arising in the study of closed Riemannian manifolds with positive scalar curvature. For example, we prove that symmetric products of surfaces sharply distinguish between two distinct notions of macroscopic dimension introduced by Gromov and the second-named author. As a natural generalization of this circle of ideas, we address the Gromov--Lawson and Gromov conjectures in the Kaehler projective setting and draw new connections between the theories of the minimal model, positivity in algebraic geometry, and macroscopic dimensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Symmetric products of surfaces separate two macroscopic dimension notions and the paper uses them to address Kaehler versions of the Gromov-Lawson conjecture.

read the letter

The main takeaway is that symmetric products of surfaces give concrete examples separating two notions of macroscopic dimension from Gromov and Dranishnikov, and the authors apply this to the Gromov-Lawson and Gromov conjectures in the Kaehler projective setting while linking to minimal models and positivity in algebraic geometry. The paper studies curvature and symplectic asphericity properties of these spaces in detail and claims they answer nuanced questions about positive scalar curvature manifolds. This distinction is presented as a theorem, which would be useful if the calculations check out. What looks new is the sharp separation via symmetric products and the extension into the Kaehler case. The authors build directly on existing work on macroscopic dimensions, which is appropriate here. The soft spots are limited. The abstract states the results clearly but gives no derivations or explicit checks that the spaces satisfy one notion but not the other, so the strength rests on whether those verifications hold in the full text. The algebraic geometry connections read more as pointers than fully developed arguments. No circularity or internal gaps show up in the stated claims. This work is aimed at researchers in geometric topology and differential geometry who track macroscopic invariants or positive curvature questions. A reader focused on examples that distinguish related notions or on Kaehler extensions would find it relevant. It deserves a serious referee to examine the technical steps. I would send it out for peer review.

Referee Report

0 major / 2 minor

Summary. The paper studies the curvature and symplectic asphericity properties of symmetric products of surfaces. It proves that these spaces sharply distinguish two distinct notions of macroscopic dimension (due to Gromov and the second-named author) and uses them to address the Gromov--Lawson and Gromov conjectures in the Kähler projective setting, while drawing connections to minimal model theory and positivity in algebraic geometry.

Significance. If the central claims hold, the work supplies concrete geometric examples that separate the two macroscopic dimension notions, which bears directly on questions about positive scalar curvature manifolds. The extension to the Kähler setting and the links to algebraic geometry provide a potentially useful bridge between geometric topology and complex geometry.

minor comments (2)

[Abstract / Introduction] The abstract refers to 'two distinct notions of macroscopic dimension' without naming them; the introduction should state the precise definitions (e.g., the Gromov and second-author versions) before the distinction is claimed.
[Introduction] The phrase 'second-named author' is acceptable in the abstract but the introduction should identify the author explicitly when the notions are attributed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No specific major comments were provided in the report, so we have no points to address point-by-point at this stage. We will incorporate any minor suggestions during the revision process.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central claim is a proof that symmetric products of surfaces distinguish two macroscopic dimension notions (one introduced by Gromov and the second author) via their curvature and symplectic asphericity properties. The abstract presents this as a theorem addressing external conjectures (Gromov-Lawson, Gromov) without any reduction of the result to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The derivation uses the spaces as distinguishing examples with independent mathematical content, and no quoted step equates the output to its inputs by construction. This is the normal case of a self-contained proof paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no information is given on free parameters, background axioms, or new postulated entities.

pith-pipeline@v0.9.0 · 5639 in / 992 out tokens · 43741 ms · 2026-05-23T01:22:18.000676+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Sharp systolic inequalities for K\"ahler manifolds
math.DG 2026-05 unverdicted novelty 7.0

Sharp systolic inequalities for Kähler manifolds with positive scalar curvature attain equality on CP^n with Fubini-Study metric and imply Gromov's rational-essentialness conjecture.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper · 2 internal anchors

[1]

Abramovich, Subvarieties of semiabelian varieties

[Abr94] D. Abramovich, Subvarieties of semiabelian varieties. Compositio Math. 90 (1994), no. 1, 37–52. [ACGH85] E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of algebraic curves. Vol. I. Grundlehren Math. Wiss., 267, Springer-Verlag, New York,

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[Bab93] I. K. Babenko, Asymptotic invariants of smooth manifolds. Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 1–38. [BCHM09] C. Birkar, P. Cascini, C. D. Hacon, J. McKernan, Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2009), no. 2, 405–468. [Bis13] I. Biswas, On K¨ ahler structures over symmetric products of ...

work page internal anchor Pith review Pith/arXiv arXiv 1993
[3]

Cheeger, D

[CG71] J. Cheeger, D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differential Geom. 6 (1971), 119–129. [CG72] J. Cheeger, D. Gromoll, On the Structure of Complete Manifolds of Nonnegative Curvature. Ann. of Math. 96 (1972), no. 3, 413–443. CURVATURE, MACROSCOPIC DIMENSIONS, AND SYMMETRIC PRODUCTS 39 [DD22] M. Daher, A....

work page 1971
[4]

Di Cerbo, L

[DD15] G. Di Cerbo, L. F. Di Cerbo, Positivity in K¨ ahler–Einstein theory. Math. Proc. Cam- bridge Philos. Soc. 159 (2015), no. 2, 321–338. [DCP24] L. F. Di Cerbo, R. Pardini, On the Hopf problem and a conjecture of Liu–Maxim– Wang. Expo. Math. 42 (2024), 125543, pp

work page 2015
[5]

Dold, Homology of symmetric products and other functors of complexes

[Dol58] A. Dold, Homology of symmetric products and other functors of complexes. Ann. of Math. 68 (1958), no. 2, 54–80. [Dol62] A. Dold, Decomposition theorems for S(n)-complexes. Ann. of Math. 75 (1962), no. 1, 8–16. [DT58] A. Dold, R. Thom, Quasifaserungen und unendliche symmetrische Produkte. Ann. of Math. 67 (1958), no. 2, 239–281. [Dra11a] A. Dranish...

work page 1958
[6]

[GK67] S. I. Goldberg, S. Kobayashi, Holomorphic bisectional curvature. J. Differential Geom. 1 (1967), 225–233. [Gom98] R. E. Gompf, Symplectially aspherical manifolds with non-trivial π2. Math. Res. Lett. 5 (1998), 599–603. [GH78] P. Griffiths, J. Harris, Principles of Algebraic Geometry. Pure Appl. Math., Wiley- Interscience, New York,

work page 1967
[7]

Gromov, B

[GrH24] M. Gromov, B. Hanke, Torsion Obstructions to Positive Scalar Curvature, SIGMA 20 (2024), 069, 22p. [Gro83] M. Gromov, Filling Riemannian manifolds. J. Differential Geom. 18 (1983), no. 1, 1–147. [Gro91] M. Gromov, K¨ ahler hyperbolicity and L2-Hodge theory. J. Differential Geom. 33 (1991), 263–292. [Gro93] M. Gromov, Asymptotic Invariants of Infin...

work page 2024
[8]

Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher sig- natures

[Gro96] M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher sig- natures. In Functional analysis on the eve of the 21st century. Vol. II , ed. S. Gindikin et al., Progr. Math., 132, Birkh¨ auser Boston, Inc., Boston, 1996, pp. 1–213. [Gro07] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces . Mod. Birkh¨ ause...

work page 1996
[9]

Gromov, H

[GL80] M. Gromov, H. B. Lawson, The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), no.3, 423-434. [GL83] M. Gromov, H. B. Lawson, Positive scalar curvature and the Dirac operator on com- plete Riemannian manifolds. Publ. Math. I.H.E.S. 58 (1983), 295–408. [HM03] C. D. Hacon J. McKernan, On Shokurov...

work page 1980
[10]

Heier, B

[HW12] G. Heier, B. Wong, Scalar curvature and uniruledness on projective manifolds. Comm. Anal. Geom. 20 (2012), no. 4, 751–764. [Hit74] N. Hitchin, On compact four-dimensional Einstein manifolds. J. Differential Geom. 9 (1974), 435–442. [Jau25] E. Jauhari, LS-category and sequential topological complexity of symmetric products. Preprint, arXiv:2503.0453...

work page arXiv 2012
[11]

Some Remarks on Symmetric Products of Curves

[JS00] M. Joachim, T. Schick, Positive and negative results concerning the Gromov–Lawson– Rosenberg conjecture. In Geometry and topology: Aarhus (1998) , ed. K. Grove et al., Contemp. Math., 258, AMS, Providence, 2000, pp. 213–226. [Kal98] S. Kallel, Divisor spaces on punctured Riemann surfaces. Trans. Amer. Math. Soc. 350 (1998), no. 1, 135–164. [Kal04] ...

work page internal anchor Pith review Pith/arXiv arXiv 1998
[12]

Kallel, P

[KS06] S. Kallel, P. Salvatore, Symmetric products of two dimensional complexes. In Recent developments in algebraic topology , ed. A. ´Adem et al., Contemp. Math., 407, AMS, Providence, 2006, pp. 147–161. [KR06] M. Katz, Y. B. Rudyak, Lusternik–Schnirelmann category and systolic category of low-dimensional manifolds. Comm. Pure Appl. Math. 59 (2006), no....

work page 2006
[13]

Kotschick, C

[KN13] D. Kotschick, C. Neofytidis, On three-manifolds dominated by circle bundles. Math. Z. 274 (2013), no. 1-2, 21–32. [Kut12] S. Kutsak, Essential manifolds with extra structures. Topology Appl. 159 (2012), 2635–2641. [LM89] H. B. Lawson, Jr., M.-L. Michelsohn, Spin Geometry . Princeton Math. Ser., 38, Princeton University Press, Princeton,

work page 2013
[14]

LeBrun, Kodaira dimension and the Yamabe problem

[LeB99] C. LeBrun, Kodaira dimension and the Yamabe problem. Comm. Anal. Geom. 7 (1999), 133–156. [LMW21] Y. Liu, L. Maxim, B. Wang, Aspherical manifolds, Mellin transformation and a ques- tion of Bobadilla–Koll´ ar.J. Reine Angew. Math. 781 (2021), 1–18. [LK08] J. Lott, B. Kleiner, Notes on Perelman’s papers. Geom. Topol. 12 (2008), no. 5, 2587–2855. [LS...

work page 1999
[15]

[Mac62] I. G. Macdonald, Symmetric products of an algebraic curve. Topology 1 (1962), 319–

work page 1962
[16]

Mattuck, Picard bundles

[Mat61a] A. Mattuck, Picard bundles. Illinois J. Math. 5 (1961), 550–564. [Mat61b] A. Mattuck, Symmetric products and Jacobians. Amer. J. Math. 83 (1961), 189–206. [Mil69] R. James Milgram, The homology of symmetric products. Trans. Amer. Mat. Soc. 138 (1969), 251–265. [Nak57] M. Nakaoka, Cohomology of symmetric products. J. Inst. Polytech. Osaka City Uni...

work page 1961
[17]

Rosenberg, S

[RS01] J. Rosenberg, S. Stolz, Metrics of positive scalar curvature and connection with surgery. In Survey on Surgery Theory, Vol. 2 , ed. S. Cappell et al. , Ann. of Math. Stud., 149, Princeton University Press, Princeton, 2001, pp. 353–386. [RO99] Y. B. Rudyak, J. Oprea, On the Lusternik–Schnirelmann category of symplectic man- ifolds and the Arnold con...

work page 2001

[1] [1]

Abramovich, Subvarieties of semiabelian varieties

[Abr94] D. Abramovich, Subvarieties of semiabelian varieties. Compositio Math. 90 (1994), no. 1, 37–52. [ACGH85] E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of algebraic curves. Vol. I. Grundlehren Math. Wiss., 267, Springer-Verlag, New York,

work page 1994

[2] [2]

[Bab93] I. K. Babenko, Asymptotic invariants of smooth manifolds. Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 1–38. [BCHM09] C. Birkar, P. Cascini, C. D. Hacon, J. McKernan, Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2009), no. 2, 405–468. [Bis13] I. Biswas, On K¨ ahler structures over symmetric products of ...

work page internal anchor Pith review Pith/arXiv arXiv 1993

[3] [3]

Cheeger, D

[CG71] J. Cheeger, D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differential Geom. 6 (1971), 119–129. [CG72] J. Cheeger, D. Gromoll, On the Structure of Complete Manifolds of Nonnegative Curvature. Ann. of Math. 96 (1972), no. 3, 413–443. CURVATURE, MACROSCOPIC DIMENSIONS, AND SYMMETRIC PRODUCTS 39 [DD22] M. Daher, A....

work page 1971

[4] [4]

Di Cerbo, L

[DD15] G. Di Cerbo, L. F. Di Cerbo, Positivity in K¨ ahler–Einstein theory. Math. Proc. Cam- bridge Philos. Soc. 159 (2015), no. 2, 321–338. [DCP24] L. F. Di Cerbo, R. Pardini, On the Hopf problem and a conjecture of Liu–Maxim– Wang. Expo. Math. 42 (2024), 125543, pp

work page 2015

[5] [5]

Dold, Homology of symmetric products and other functors of complexes

[Dol58] A. Dold, Homology of symmetric products and other functors of complexes. Ann. of Math. 68 (1958), no. 2, 54–80. [Dol62] A. Dold, Decomposition theorems for S(n)-complexes. Ann. of Math. 75 (1962), no. 1, 8–16. [DT58] A. Dold, R. Thom, Quasifaserungen und unendliche symmetrische Produkte. Ann. of Math. 67 (1958), no. 2, 239–281. [Dra11a] A. Dranish...

work page 1958

[6] [6]

[GK67] S. I. Goldberg, S. Kobayashi, Holomorphic bisectional curvature. J. Differential Geom. 1 (1967), 225–233. [Gom98] R. E. Gompf, Symplectially aspherical manifolds with non-trivial π2. Math. Res. Lett. 5 (1998), 599–603. [GH78] P. Griffiths, J. Harris, Principles of Algebraic Geometry. Pure Appl. Math., Wiley- Interscience, New York,

work page 1967

[7] [7]

Gromov, B

[GrH24] M. Gromov, B. Hanke, Torsion Obstructions to Positive Scalar Curvature, SIGMA 20 (2024), 069, 22p. [Gro83] M. Gromov, Filling Riemannian manifolds. J. Differential Geom. 18 (1983), no. 1, 1–147. [Gro91] M. Gromov, K¨ ahler hyperbolicity and L2-Hodge theory. J. Differential Geom. 33 (1991), 263–292. [Gro93] M. Gromov, Asymptotic Invariants of Infin...

work page 2024

[8] [8]

Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher sig- natures

[Gro96] M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher sig- natures. In Functional analysis on the eve of the 21st century. Vol. II , ed. S. Gindikin et al., Progr. Math., 132, Birkh¨ auser Boston, Inc., Boston, 1996, pp. 1–213. [Gro07] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces . Mod. Birkh¨ ause...

work page 1996

[9] [9]

Gromov, H

[GL80] M. Gromov, H. B. Lawson, The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), no.3, 423-434. [GL83] M. Gromov, H. B. Lawson, Positive scalar curvature and the Dirac operator on com- plete Riemannian manifolds. Publ. Math. I.H.E.S. 58 (1983), 295–408. [HM03] C. D. Hacon J. McKernan, On Shokurov...

work page 1980

[10] [10]

Heier, B

[HW12] G. Heier, B. Wong, Scalar curvature and uniruledness on projective manifolds. Comm. Anal. Geom. 20 (2012), no. 4, 751–764. [Hit74] N. Hitchin, On compact four-dimensional Einstein manifolds. J. Differential Geom. 9 (1974), 435–442. [Jau25] E. Jauhari, LS-category and sequential topological complexity of symmetric products. Preprint, arXiv:2503.0453...

work page arXiv 2012

[11] [11]

Some Remarks on Symmetric Products of Curves

[JS00] M. Joachim, T. Schick, Positive and negative results concerning the Gromov–Lawson– Rosenberg conjecture. In Geometry and topology: Aarhus (1998) , ed. K. Grove et al., Contemp. Math., 258, AMS, Providence, 2000, pp. 213–226. [Kal98] S. Kallel, Divisor spaces on punctured Riemann surfaces. Trans. Amer. Math. Soc. 350 (1998), no. 1, 135–164. [Kal04] ...

work page internal anchor Pith review Pith/arXiv arXiv 1998

[12] [12]

Kallel, P

[KS06] S. Kallel, P. Salvatore, Symmetric products of two dimensional complexes. In Recent developments in algebraic topology , ed. A. ´Adem et al., Contemp. Math., 407, AMS, Providence, 2006, pp. 147–161. [KR06] M. Katz, Y. B. Rudyak, Lusternik–Schnirelmann category and systolic category of low-dimensional manifolds. Comm. Pure Appl. Math. 59 (2006), no....

work page 2006

[13] [13]

Kotschick, C

[KN13] D. Kotschick, C. Neofytidis, On three-manifolds dominated by circle bundles. Math. Z. 274 (2013), no. 1-2, 21–32. [Kut12] S. Kutsak, Essential manifolds with extra structures. Topology Appl. 159 (2012), 2635–2641. [LM89] H. B. Lawson, Jr., M.-L. Michelsohn, Spin Geometry . Princeton Math. Ser., 38, Princeton University Press, Princeton,

work page 2013

[14] [14]

LeBrun, Kodaira dimension and the Yamabe problem

[LeB99] C. LeBrun, Kodaira dimension and the Yamabe problem. Comm. Anal. Geom. 7 (1999), 133–156. [LMW21] Y. Liu, L. Maxim, B. Wang, Aspherical manifolds, Mellin transformation and a ques- tion of Bobadilla–Koll´ ar.J. Reine Angew. Math. 781 (2021), 1–18. [LK08] J. Lott, B. Kleiner, Notes on Perelman’s papers. Geom. Topol. 12 (2008), no. 5, 2587–2855. [LS...

work page 1999

[15] [15]

[Mac62] I. G. Macdonald, Symmetric products of an algebraic curve. Topology 1 (1962), 319–

work page 1962

[16] [16]

Mattuck, Picard bundles

[Mat61a] A. Mattuck, Picard bundles. Illinois J. Math. 5 (1961), 550–564. [Mat61b] A. Mattuck, Symmetric products and Jacobians. Amer. J. Math. 83 (1961), 189–206. [Mil69] R. James Milgram, The homology of symmetric products. Trans. Amer. Mat. Soc. 138 (1969), 251–265. [Nak57] M. Nakaoka, Cohomology of symmetric products. J. Inst. Polytech. Osaka City Uni...

work page 1961

[17] [17]

Rosenberg, S

[RS01] J. Rosenberg, S. Stolz, Metrics of positive scalar curvature and connection with surgery. In Survey on Surgery Theory, Vol. 2 , ed. S. Cappell et al. , Ann. of Math. Stud., 149, Princeton University Press, Princeton, 2001, pp. 353–386. [RO99] Y. B. Rudyak, J. Oprea, On the Lusternik–Schnirelmann category of symplectic man- ifolds and the Arnold con...

work page 2001