Convex Algebraic Geometry of Curvature Operators

Mario Kummer; Renato G. Bettiol; Ricardo A. E. Mendes

Convex Algebraic Geometry of Curvature Operators

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Pith's one-line read For curvature operators, semidefinite descriptions exist only in low dimensions.

desk verdict New curvature-operator counterexamples to Helton-Nie; the proof is sound, with one terse step that deserves expansion. read the letter →

arxiv 1908.03713 v2 pith:INUGMJKP submitted 2019-08-10 math.DG math.AG

Renato G. Bettiol , Mario Kummer , Ricardo A. E. Mendes This is my paper · ORCID

classification math.DGmath.AG MSC 14P1053B2053C2190C22

keywords curvatureoperatorssectionalboundsspectrahedraspectrahedralshadowssemidefiniteprogrammingGrassmanniansumsofsquaresHelton-Nieconjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper determines exactly which dimensions admit a semidefinite description of the set of algebraic curvature operators satisfying a sectional curvature bound. In dimensions at most three these sets are spectrahedra; in dimension four they are projections of spectrahedra but not spectrahedra themselves; and in dimensions five and higher they are neither, which supplies a new family of counterexamples to the Helton–Nie conjecture that every convex semialgebraic set is a spectrahedral shadow. The paper also gives efficient membership tests for dimension four via Sturm's root counting, and for higher dimensions it constructs nested inner and outer semidefinite approximations that converge to the true set.

What carries the argument

The machinery turns curvature operators into quadratic forms on the Grassmannian $\mathrm{Gr}_2(n)$: an algebraic curvature operator $R$ is identified with the quadratic form $q_R$ on Plücker coordinates, and the condition $\mathrm{sec}_R\ge 0$ becomes nonnegativity on the real Grassmannian. Under this identification, $\mathcal{R}_{\mathrm{sec}\ge 0}(n)$ is exactly the cone $P_{\mathrm{Gr}_2(n)}$ of nonnegative quadratic forms, while sums of squares in the homogeneous coordinate ring correspond to strongly nonnegative curvature. Two named objects carry the argument: the Finsler–Thorpe trick says that in dimension four, $\mathrm{sec}_R\ge 0$ iff $R+x*$ is positive semidefinite for some real $x$, exhibiting $\mathcal{R}_{\mathrm{sec}\ge 0}(4)$ as a spectrahedral shadow; and the discriminant $p_k(R)=\mathrm{disc}_x(\det(R-k\,\mathrm{Id}+x*))$ gives the minimal defining polynomial of the algebraic interior in dimension four. For $n\ge 5$, the proof applies a criterion that detects when such a cone is not a spectrahedral shadow, using a quadratic form on $\mathrm{Gr}_2(5)$ that is nonnegative but not a sum of squares, and a translation-invariance property of its homogenization.

What would settle it

Take the specific nonnegative non-sum-of-squares quadratic form on $\mathrm{Gr}_2(5)$ that the paper uses, write its homogenization in the affine chart, and check whether every coefficient of the shifted polynomial $f^h(t, x-y)$ belongs to the subspace $L$ spanned by the affine-chart images of quadratic forms; a single violation would invalidate the $n\ge 5$ theorem. Alternatively, produce an explicit spectrahedral shadow description of $\mathcal{R}_{\mathrm{sec}\ge 0}(5)$; if one exists, the theorem's first part is false.

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Extended reading notes

Core claim

The central discovery, stated as Theorem A, is a complete dimensional dichotomy. For every real k and every n, the convex semialgebraic set $\mathcal{R}_{\mathrm{sec}\ge k}(n)$ of algebraic curvature operators with $\mathrm{sec}\ge k$ (and its mirror $\mathcal{R}_{\mathrm{sec}\le k}(n)$) is a spectrahedron when $n\le 3$, a spectrahedral shadow but not a spectrahedron when $n=4$, and not even a spectrahedral shadow when $n\ge 5$. The proof for $n=4$ rests on the Finsler–Thorpe trick: $\mathrm{sec}_R\ge 0$ in dimension four is equivalent to $R+x*$ being positive semidefinite for some real $x$, where $*$ is the Hodge star, so $\mathcal{R}_{\mathrm{sec}\ge 0}(4)$ is the linear projection of a spectrahedron. The proof for $n\ge 5$ identifies $\mathcal{R}_{\mathrm{sec}\ge 0}(n)$ with the cone of nonnegative quadratic forms on the Grassmannian $\mathrm{Gr}_2(n)$, extracts a criterion from recent work on spectrahedral shadows to show such a cone is not a spectrahedral shadow, and exhibits the required non-sum-of-squares input using a known explicit quadratic form. Theorem C adds that in dimension four the set is an algebraic interior whose minimal defining polynomial is the discriminant of $\det(R+x*)$, and Theorem B provides nested inner approximations by spectrahedral shadows and outer approximations by spectrahedra whose unions and intersections exhaust the set.

Load-bearing premise

For dimensions five and up, the whole non-representability argument rests on a single technical criterion and on one explicitly supplied polynomial that must satisfy a certain translation-coefficient property; if that criterion does not apply here, or the polynomial fails that property, the paper's counterexample claim for $n\ge 5$ has no proof.

Editorial extensions

If this is right

In dimension 4, membership in $\mathcal{R}_{\mathrm{sec}\ge 0}(4)$ (and strict variants) can be decided exactly by Sturm's root-counting algorithm, without semidefinite programming.
For $n\ge 5$, Algorithm 1 terminates correctly on every input outside a measure-zero bad set, so sectional curvature bounds can be tested in practice up to numerical precision.
The inner approximations $I_m$ and outer approximations $O_m$ are $O(n)$-invariant and geometric, so they define coordinate-free curvature conditions; in particular $I_0$ is strongly nonnegative curvature and $O_0$ is nonnegative Ricci curvature.
Because the approximations do not stabilize for $n\ge 5$, no finite truncation of the Lasserre-type hierarchy or of the Weitzenböck-formula hierarchy can exactly capture $\mathrm{sec}\ge 0$ in high dimensions.
The same dimensional dichotomy holds for semi-Riemannian curvature operators with the natural analogue of sectional curvature bounds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Since $\mathcal{R}_{\mathrm{sec}\ge 0}(n)$ for $n\ge 5$ is not a spectrahedral shadow, any exact semidefinite representation would have to use an infinite-dimensional or non-polynomial lifting; a concrete test would be to see whether the boundary of the cone has positive curvature in the sense of convex algebraic geometry that obstructs such lifts.
The construction of the bad example suggests a general recipe: any Grassmannian $\mathrm{Gr}_k(n)$ with $2\le k\le n-2$ and $n\ge 5$ yields a non-spectrahedral-shadow cone of nonnegative quadratic forms, so the phenomenon is not special to $\mathrm{sec}\ge 0$ but is shared by positivity conditions defined by other $O(n)$-representations.
A natural extension would be to replace the Grassmannian by other homogeneous varieties and ask whether the corresponding nonnegative quadratic forms are spectrahedral shadows exactly when sums of squares coincide; the paper's criterion might be the right tool to test this.
The algorithms' numerical robustness could be benchmarked against random curvature operators in dimensions 5 and 6; if the bad set, though of measure zero, is approached by typical inputs, the stopping criterion might require many iterations in practice.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here is my take.

The paper is serious and mostly rigorous. Its real contribution is Theorem A: the sectional-curvature bound sets are spectrahedra for n≤3, spectrahedral shadows but not spectrahedra for n=4, and not even spectrahedral shadows for n≥5. The last part is a new family of Helton-Nie counterexamples, and the authors correctly note the conjecture was already dead via Scheiderer and Fawzi. The point is that these are natural convex bodies from Riemannian geometry. The n=4 minimal defining polynomial in Theorem C and the O(n)-invariant inner/outer approximations in Theorem B are also new and useful.

I want to address the specific worry about Proposition 4.1. The stress-test concern is that the coefficient-in-L condition is asserted without proof, and if it failed, the n≥5 non-shadow conclusion would lose its only proof. I read the algebra carefully and the condition holds. Once you write the Plücker coordinates of the two-row matrix Γ(t) as t^k times the six affine coordinates or the 2×2 minors of Γ′, the translated polynomial f^h(t,x-y) expands by multilinearity into linear combinations, with y-coefficients, of products of these same functions. Those products span L by definition. So Scheiderer's criterion applies. The step is compressed—it deserves a few explicit lines or a footnote—but it is not an error.

Two minor caveats. Proposition 5.7 uses [BM, Thm. A] for the outer approximations without stating the exact statement; that is normal use of prior work, but the dependency should be explicit. And the SDP-based membership algorithms assume exact feasibility, while interior-point methods return numerical approximations; the algorithms are rigorous in an oracle model, not as floating-point routines. The authors are honest that the n≥5 algorithm may fail to halt on a measure-zero bad set.

Who is this for? People in convex algebraic geometry, and differential geometers who care about curvature conditions as semialgebraic sets. It deserves serious peer review. The referee should push for a few more lines in Proposition 4.1 and a sentence about the SDP exactness caveat; otherwise it is ready.

Referee Report

0 major / 5 minor

Summary. This paper studies, from the viewpoint of convex algebraic geometry, the convex semialgebraic sets Rsec≥k(n) of algebraic curvature operators whose sectional curvature is bounded below (or above) by k. The main result, Theorem A, asserts a complete classification: for n≥5 these sets are not spectrahedral shadows; for n=4 they are spectrahedral shadows but not spectrahedra; for n≤3 they are spectrahedra. The proof uses the identification R[Gr_2(n)]_2 ≅ Sym^2_b(∧^2R^n), the Blekherman–Smith–Velasco/Zoltek non-SOS quadratic form, and a criterion extracted from Scheiderer's work. Theorem B provides O(n)-invariant nested inner approximations by spectrahedral shadows and outer approximations by spectrahedra; Theorem C characterizes Rsec≥0(4) as an algebraic interior whose minimal defining polynomial is the discriminant disc_x(det(R+x*)). The paper also gives SDP-based algorithms for n≥5 and Sturm-based algorithms for n=4, plus a semi-Riemannian analogue in Appendix A.

Significance. The results, if correct, are significant. They supply a new geometric family of counterexamples to the Helton–Nie conjecture, give a concrete and reusable form of Scheiderer's non-shadow criterion, and turn the dimension-four case into an explicit algebraic-interior description with efficient membership tests. The proof structure is sound: Propositions 3.2 and 3.5 establish the algebraic-interior structure; Appendix B proves the needed irreducibility of the discriminant of symmetric matrices; and the n≥5 argument reduces via Corollary 4.2 to Scheiderer's criterion. I found no circularity: Theorem A(1) does not depend on the authors' earlier work, and the only use of [BM] is for the outer approximations in Section 5, which is external. The paper also includes checkable algorithmic consequences and a reproducible Macaulay2 verification in Example 5.3.

minor comments (5)

[Theorem C, final sentence] The stated component identification is false as written: for every k, p_k((k+1)Id) = disc_x(det(Id+x*)) = disc_x((1+x)^3(1-x)^3) = 0, so (k+1)Id cannot belong to the set {p_k>0}, let alone to C_k. The intended condition is presumably that (k+1)Id lies in the closure of C_k, or equivalently that C_k is the component whose closure is Rsec≥k(4). Please correct this sentence.
[Section 4, Proposition 4.1] The verification of the coefficient-in-L hypothesis of Theorem 2.14 is compressed into the sentence 'This and multilinearity of the determinant imply...'. The claim is correct, but because this is the load-bearing connection between the Zoltek form and Scheiderer's criterion, please spell out the argument: write Γ_t(x-y)=Γ_t(x)-Γ_t(y), expand each 2×2 minor by row multilinearity, and observe that every t-coefficient of the resulting product is a y-dependent linear combination of products of two t-coefficients of minors of Γ_t(x), each of which lies in L by definition of ψ.
[Section 4, Corollary 4.2] The isomorphism in the induction step should read Gr_k(n+1) ≅ Gr_{n+1-k}(n+1), not Gr_{n-k}(n+1). The subsequent range reduction remains valid after this correction.
[Section 2.6, Theorem 2.14] The proof of Theorem 2.14 delegates the key implication to 'the exact same reasoning as in [Sch18b, Ex. 4.20, Rem. 4.21]'. Since the theorem is advertised as a conveniently applicable criterion, a fuller derivation, or an explicit statement of the cited results, would make the paper more self-contained and easier to verify.
[Section 5.4, Algorithm 3] In line 5, the condition that σ_i has a root in (a_j,a_{j+1}) is tested for each i; the implementation via Sturm's root-isolating partitions should be described explicitly so that the decidability of this test is transparent.

Circularity Check

0 steps flagged · score 2.0 of 10

No constructional circularity; Theorem A rests on Scheiderer's external criterion and the Zoltek/BSV16 non-SOS form, with only a minor same-author citation in the outer approximations.

full rationale

Walking the derivation chain, no claim reduces by construction to its own input. Theorem A(1) is obtained by identifying P_{Gr2(n)} with Rsec≥0(n) and applying Scheiderer's external criterion (Theorem 2.14) to Zoltek's nonnegative non-SOS form P. The Claim in Proposition 4.1 proves f=ψ(P) is nonnegative and not a sum of squares, while the additional coefficient condition on f^h(t,x−y) is asserted in one sentence ('This and multilinearity of the determinant imply...'). That sentence is a possible unverified technical step, but it is not a circular reduction of the conclusion to the inputs. Theorem C and Theorem A(2) are self-contained from Finsler's lemma, discriminant identities, and the irreducibility result in Appendix B. Theorem B's outer approximation uses [BM, Thm. A] for the nontrivial inclusion in Proposition 5.7; this is a same-author citation, but it is a parameter-free prior theorem about Weitzenböck curvature terms and it does not assume the spectrahedral-shadow conclusion, so it is independent support rather than circularity. The algorithms contain no fitted parameters, and no predicate is renamed as a prediction. The only mild same-author dependency is the citation of [BM, Thm. A] in the outer relaxations, which justifies the low nonzero score rather than a circularity finding.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No fitted constants and no invented objects appear in this paper. The proofs rely on standard algebraic geometry theorems and one same-author external theorem, [BM, Thm. A], used for the outer approximations. The polynomial p_k is constructed from known discriminant operations, not fitted to data.

assumptions (7)

standard math Tarski-Seidenberg theorem: projections of semialgebraic sets are semialgebraic; quantifier elimination for polynomial sentences exists.
Used in Section 1 to assert Rsec>=k(n) is semialgebraic from the quantified sectional-curvature condition.
standard math Blekherman-Smith-Velasco theorem: P_X=Sigma_X for a real projective variety X iff X has minimal degree; for Gr2(n) this holds iff n<=4.
Used in Example 2.11 to identify Sigma_Gr2(n) with Rsec>=0(n) for n<=4 and to obtain P in P_Gr2(5) minus Sigma_Gr2(5) for n>=5.
standard math Scheiderer's criterion: certain nonnegative non-SOS quadratic forms yield convex sets whose dual cones are not spectrahedral shadows, extracted as Theorem 2.14.
This is the engine behind Theorem A(1); the paper gives a proof sketch referencing [Sch18b, Lem. 4.17, Prop. 4.18, Prop. 4.19] rather than a full independent proof.
standard math Finsler's lemma (Lemma 2.10): for real symmetric A,B, nonnegativity of A on the kernel of B is equivalent to existence of x with A+xB positive semidefinite.
Underlies Thorpe's trick in Proposition 3.1 and therefore the n=4 classification and Algorithms 2 and 3.
domain assumption [BM, Thm. A]: Weitzenbock nonnegativity for all traceless symmetric p-tensors characterizes sec R>=0.
Cited in Proposition 5.7 to prove that the outer approximations O_m intersect to Rsec>=0(n); authored by two of the three present authors and not reproved in this preprint.
standard math Scheiderer's Positivstellensatz for projective real varieties, [Sch12, Cor. 4.2].
Used in Proposition 5.2 to show every curvature operator with sec R>0 is eventually contained in an inner approximation I_m.
domain assumption Identification of algebraic curvature operators with quadratic forms on the Grassmannian Gr2(n) via the Plucker relations.
This identification in Section 2.2 converts sectional-curvature bounds into nonnegativity of quadratic forms on Gr2(n), connecting the geometry to convex algebraic geometry.

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Pith. "Pith review of Convex Algebraic Geometry of Curvature Operators." pith.science (2026). https://pith.science/paper/INUGMJKP

@misc{pith2026190803713,
  author       = {Pith},
  title        = {Pith review of: Convex Algebraic Geometry of Curvature Operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/INUGMJKP}},
  note         = {Machine review of arXiv:1908.03713}
}

abstract

We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry. More precisely, we determine in which dimensions $n$ this convex semialgebraic set is a spectrahedron or a spectrahedral shadow; in particular, for $n\geq5$, these give new counter-examples to the Helton--Nie Conjecture. Moreover, efficient algorithms are provided if $n=4$ to test membership in such a set. For $n\geq5$, algorithms using semidefinite programming are obtained from hierarchies of inner approximations by spectrahedral shadows and outer relaxations by spectrahedra.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

[AB08] S. B. Alexander and R. L. Bishop . Lorentz and semi-Riemannian spaces with Alexandrov curvature bounds . Comm. Anal. Geom., 16 (2008), 251–282. [Afs11] B. Afsari . Riemannian Lp center of mass: existence, uniqueness, and convexity . Proc. Amer. Math. Soc., 139 (2011), 655–673. [AH98] L. Andersson and R. How ard . Comparison and rigidity theorems in...

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[FCPS04] P

arXiv preprint arXiv:1905.02575, (2019). [FCPS04] P. T. Fletcher, Conglin Lu, S. M. Pizer, and Sarang Joshi . Principal geodesic analysis for the study of nonlinear statistics of shape . IEEE Transactions on Medical Imaging, 23 (2004), 995–1005. [Fin36] P. Finsler . ¨Uber das Vorkommen deﬁniter und semideﬁniter Formen in Scha ren quadratischer Formen. Com...

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[GS] D. R. Grayson and M. E. Stillman . Macaulay2, a software system for research in algebraic geometry. Available at https://faculty.math.illinois.edu/Macaulay2/. [Hil88] D. Hilbert. Ueber die Darstellung deﬁniter Formen als Summe von Formenq uadraten. Math. Ann., 32 (1888), 342–350. [HN09] J. W. Helton and J. Nie . Suﬃcient and necessary conditions for ...

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[Tho72] J. A. Thorpe . On the curvature tensor of a positively curved 4-manifold. In Proceed- ings of the 13th Biennial Seminar of the Canadian Mathematic al Congress (Dalhousie 28 R. G. BETTIOL, M. KUMMER, AND R. A. E. MENDES Univ., Halifax, N.S., 1971), Vol. 2 , 156–159. Canad. Math. Congr., Montreal, Que.,

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[Tho91] R. C. Thompson . Pencils of complex and real symmetric and skew matrices . Linear Algebra Appl., 147 (1991), 323–371. [W ei72] A. Weinstein. Remarks on curvature and the Euler integrand . J. Diﬀerential Geometry, 6 (1971/72), 259–262. [Zol79] S. M. Zoltek . Nonnegative curvature operators: some nontrivial example s. J. Diﬀer- ential Geom., 14 (197...

work page 1991

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[Cal64] E. Calabi . Linear systems of real quadratic forms . Proc. Amer. Math. Soc., 15 (1964), 844–846. [CKPP] D. Cifuentes, T. Kahle, P. A. Parrilo, and H. Peyrl . SOS: sums of squares. Version 2.0 . A Macaulay2 package available at https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages. [Del97] C. N. Delzell . Bad points for positive semideﬁ...

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Derinkuyu and M

CONVEX ALGEBRAIC GEOMETRY OF CUR V ATURE OPERATORS 27 [DP06] K. Derinkuyu and M. C. Pinar . On the S-procedure and some variants . Math. Meth- ods Oper. Res., 64 (2006), 55–77. [EH16] D. Eisenbud and J. Harris . 3264 and all that—a second course in algebraic geometry . Cambridge University Press, Cambridge,

work page 2006

[8] [1998]

[BJV17] M

Trans- lated from the 1987 French original, Revised by the authors. [BJV17] M. Banerjee, B. Jian, and B. C. Vemuri . Robust Fr´ echet Mean and PGA on Rie- mannian Manifolds with Applications to Neuroimaging . In M. Niethammer, M. Styner, S. Aylward, H. Zhu, I. Oguz, P.-T. Yap, and D. Shen, editors, Information Processing in Medical Imaging , 3–15, Cham,

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