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Realizations of homology classes and projection areas

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Pith's one-line read Rational surface classes in products of projective spaces exist exactly when a simple eigenvalue condition holds.

desk verdict Strong, original paper whose central theorem is stated in the wrong cohomological degree; fix the grading and the zero-row case and it deserves publication. read the letter →

arxiv 2505.08881 v2 pith:WVUXX4GV submitted 2025-05-13 math.AG math.COmath.MG

Daoji Huang , June Huh , Mateusz Micha{l}ek , Botong Wang , Shouda Wang This is my paper · ORCID
classification math.AGmath.COmath.MG MSC 14C2514C1552A3914J2814M25
keywords realizablehomologyclassesalgebraicSteenrodproblemLorentzianmatricestriangularhyperfieldPlückerrelationsmixedvolumesprojectionareasconvexbodies
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 proves that two classical realization questions share one numerical answer. On the algebraic side, a rational $2$-cycle in a product of projective spaces is the fundamental class of an irreducible algebraic surface exactly when the intersection matrix of the hyperplane classes on it is Lorentzian: nonnegative entries and at most one positive eigenvalue. On the convex side, six nonnegative numbers arise as the areas of the coordinate projections of a convex body in $\mathbb{R}^4$ exactly when the same Lorentzian condition holds for the six entries placed on the off-diagonal of a zero-diagonal $4\times 4$ matrix. The criterion also characterizes the six projection degrees of irreducible surfaces in $(\mathbb{P}^1)^4$, and the authors extend it to all products of projective spaces, resolving the algebraic Steenrod problem in that setting. If correct, the result replaces the search for geometric representatives by a single spectral count.

What carries the argument

The central object is the matrix $L(\eta)_{ij}=\int_\eta H_iH_j$, and the criterion is that this matrix be Lorentzian, meaning all entries are nonnegative and at most one eigenvalue is positive. For $n=4$ this is equivalent to membership in $\Delta(\mathbb{T}_2)$, the cone defined by the triangular-hyperfield Plücker relations $\sqrt{p_{ij}p_{kl}}\leq \sqrt{p_{ik}p_{jl}}+\sqrt{p_{il}p_{jk}}$. The sufficiency proof has two engines. The toric-convex engine uses mixed volumes: wherever $p$ is a rational point of $\Delta(\mathbb{T}_2)$, it is of the form $A\wedge B$ for rational polytopes $A,B$ (Theorem 3.7), and toric geometry converts such polytope data into an irreducible surface in $P^m$ (Theorem 2.4). The second engine is the prolific-surface construction (Theorem 6.3): for every $k$ there is a smooth projective surface $Y$ of Picard rank $k$ with no negative curves and all nef divisors semiample; Lemma 6.9, via Meyer's theorem on rational quadratic forms, embeds any rational Lorentzian matrix into the Néron–Severi space of $Y$, and semiample divisors define maps to projective spaces that realize the class.

What would settle it

Exhibit any rational sextuple $p=(p_{12},\ldots,p_{34})$ satisfying the triangle inequalities $\sqrt{p_{ij}p_{kl}}\le \sqrt{p_{ik}p_{jl}}+\sqrt{p_{il}p_{jk}}$ for which no irreducible complex surface in $(\mathbb{P}^1)^4$ has the six projection degrees $p_{ij}$; for instance, computing the Hilbert scheme of surfaces with that class and showing it is empty would settle it. Such a sextuple would contradict Theorem 1.6 and, with it, the Lorentzian criterion of Theorem 1.8.

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Extended reading notes

Core claim

The central claim is Theorem 1.8: for $\eta \in H^4(P^m,\mathbb{Q})$, where $P^m=\prod_i \mathbb{P}^{m_i}$ and $H_i$ is the hyperplane class pulled back from the $i$-th factor, the class $\eta$ is a nonnegative rational multiple of the fundamental class of an irreducible algebraic surface if and only if the matrix $L(\eta)_{ij}=\int_\eta H_iH_j$ is Lorentzian. The proof establishes two four-dimensional incarnations of the same condition: the six coordinate projections of a convex body in $\mathbb{R}^4$ can have areas $p_{12},\ldots,p_{34}$ exactly when the square roots of the opposite products satisfy the triangle inequalities defining $\Delta(\mathbb{T}_2)$ (Theorem 1.4), and the same inequalities characterize the integer sextuples that occur as projection degrees of an irreducible surface in $(\mathbb{P}^1)^4$ (Theorem 1.6). A key intermediate result, Theorem 3.7, identifies the rational points of $\Delta(\mathbb{T}_2)$ with vectors $A\wedge B$ of mixed projection areas of pairs of rational polytopes. The sufficiency direction for arbitrary products of projective spaces is carried by prolific surfaces: smooth surfaces of arbitrarily large Picard rank with no negative curves and every nef divisor semiample, whose Néron–Severi spaces absorb every rational Lorentzian matrix and then map to $P^m$.

Load-bearing premise

The load-bearing premise is that for every $k$ there exists a prolific surface: a smooth projective surface of Picard rank $k$, with no negative curves and every nef divisor semiample, and the proof of its existence requires the base field to be uncountable or of characteristic zero; if such surfaces do not exist in the needed generality, the step that embeds a Lorentzian matrix into a Néron–Severi space has no target space, and the rational-realization proof collapses.

Editorial extensions

If this is right

  • In $(\mathbb{P}^1)^4$, rational realizability is completely decided by six triangle-type inequalities; over $\mathbb{Z}$ these are necessary but not sufficient, with the extra obstruction $p_{kl}\leq p_{ik}p_{jl}+p_{il}p_{jk}$ whenever $p_{ij}>0$ (Proposition 5.2).
  • The algebraic Steenrod problem over $\mathbb{Q}$ is solved for every product of projective spaces: an eigenvalue check on an $n\times n$ matrix decides whether a rational 2-cycle is the class of an irreducible surface.
  • The same matrix criterion characterizes the six coordinate-projection areas of convex bodies in $\mathbb{R}^4$, and on rational points it matches the mixed projection-area vectors of pairs of rational polytopes (Theorem 3.7).
  • For $(\mathbb{P}^1)^n$ with $2\leq n\leq 11$, every primitive integral Lorentzian zero-diagonal matrix is realized up to a prime factor: for every prime $p$, either $\eta$ or $p\eta$ is an integral surface class (Theorem 6.13).
  • A quadratic form is a volume polynomial over an algebraically closed field if and only if it is Lorentzian (Corollary 6.11).

Reading between the lines

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

  • If Theorem 1.8 is correct, rational realizability in products of projective spaces is algorithmically decidable by linear algebra on $L(\eta)$, in sharp contrast to the integral version, which remains open in $(\mathbb{P}^1)^4$ and involves genuine arithmetic obstructions.
  • A natural next experiment is dimension five for the convex conjecture: search for $5\times 5$ zero-diagonal Lorentzian matrices that are not the coordinate-projection volumes of any convex body in $\mathbb{R}^5$; even a numerical near-counterexample would clarify how far the criterion extends beyond $d=4$.
  • The prolific-surface construction requires an uncountable or characteristic-zero base field (Remark 6.8). Testing whether prolific surfaces of every Picard rank exist over finite fields would delimit whether the Lorentzian criterion can hold in positive characteristic.
  • Conjecture 7.3 proposes that Lorentzian signature is sufficient for realizable 2-cycles on any smooth projective variety, provided the class is universally pseudoeffective; the blow-up and matroid examples in Section 7 show where that hypothesis is needed, and testing it on other homogeneous spaces would be a concrete next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies the parallel realizability problems of projection-area vectors of convex bodies in R^4 and homology classes of irreducible surfaces in products of projective spaces. The convex-geometric answer is that the six projection areas are exactly the points of the triangular-hyperfield Grassmannian ∆(T2), with a rational version for pairs of rational polytopes (Theorems 1.4, 3.6, 3.7). The algebraic answer characterizes realizable surface classes in (P^1)^4 over Q by the same set (Theorem 1.6), and Theorem 1.8 extends this to products ∏P^{m_i} by the condition that the matrix L(η)_{ij}=∫_η H_iH_j be Lorentzian. The proof combines the Hodge index theorem, mixed volumes and toric geometry, Meyer's theorem on rational quadratic forms, explicit prolific surfaces, and K3 period maps, and ends with conjectures on universally pseudoeffective classes.

Significance. If the intended statements are established, the paper makes a strong contribution: Theorem 1.6 gives a complete and surprisingly clean numerical criterion for a class in H_4((P^1)^4,Q) to be algebraic, and Theorem 1.8 generalizes this to all products of projective spaces through Lorentzian matrices. The necessity arguments via the Hodge index theorem and the Lorentzian mixed-volume reformulation are elegant, and the sufficiency constructions for (P^1)^4 are supported by explicit rational polytopes. The paper also contains credible and falsifiable predictions (Conjectures 1.9 and 7.3) and a nontrivial result for Grassmannians (Theorem 7.7). The reliance on external results (Hodge index, Meyer, Morrison, Shenfeld–van Handel, Ciliberto–van der Geer) is appropriate and acknowledged. However, as printed, Theorem 1.8 is stated in the wrong cohomological degree and is false as written, and the proof has an unhandled zero-row case; these issues are repairable but must be fixed before the central claims can be accepted.

major comments (4)
  1. [§1, Theorem 1.8 and Definition 1.5] The theorem is stated for η∈H^4(P^m,Q), but the objects under study are surfaces. In a d-dimensional product X=P^m, the Poincaré dual of an irreducible surface class lies in H^{2d-4}(X,Q), not in H^4(X,Q) unless d=4. As written the statement is false: take X=P^2×P^1 and η=H_1H_2∈H^4(X,Q). Then L(η)=[[0,1],[1,0]] is Lorentzian, but η is Poincaré dual to a curve (a line in P^2 times a point in P^1), not to any surface. The proof itself works with [φ(Y)]∈H_2(P^m), which corresponds to the H^{2d-4} convention. Please restate Theorem 1.8 with η∈H^{2d-4}(P^m,Q), or equivalently with η∈H_2(P^m,Q) and the Poincaré-duality convention made explicit, and adjust the abstract and Section 6 accordingly.
  2. [§6.2, proof of Theorem 1.8, after Lemma 6.9] The sufficiency proof contains an unhandled case when L(η) has a zero row. The step 'Choose an ample divisor H on Y such that ∫_Y A_i H is nonzero for all i' is impossible when some A_i equals zero, which occurs exactly when the i-th row of L(η) vanishes. For example, in (P^1)^4 the class η=H_3H_4 is realizable over Q and has L(η) with L_34=1 and all other entries zero, so the construction would force A_1=A_2=0. This is repairable by an induction on n that discards factors whose rows are zero and realizes the pulled-back class in the remaining product, but as written the proof is incomplete.
  3. [§3, Lemma 3.4] The displayed polytopes in the second statement do not have the claimed self-mixed volumes. For instance, A_2=conv(e_1+e_2,e_3,e_4) gives (p12,p13,p14,p23,p24,p34)=(0,1,1,1,1,1), not (1,1,1,1,1,0); A_7=conv(0,e_3,e_4) gives (0,0,0,0,0,1), not (1,0,0,0,0,0). The stated vectors are realizable as A^A by other lattice triangles (for example, conv(e_1,e_2,e_3+e_4) realizes (1,1,1,1,1,0), and conv(0,e_1,e_2) realizes (1,0,0,0,0,0)), so the lemma's conclusion is likely true, but the proof's computations need correction. Since Lemma 3.4 is used for the boundary cases of Theorems 3.6, 3.7, and indirectly 1.4, this is a load-bearing proof issue.
  4. [§1, Theorem 1.4 and §3, Lemma 3.4] The paper does not specify whether 'convex body in R^4' is meant in the standard full-dimensional sense. The zero-entry boundary points are realized in Lemma 3.4 by polytopes of dimension at most 2, e.g. (1,0,0,0,0,0) by conv(0,e_1,e_2). If 'convex body' means compact convex set with nonempty interior, then Theorem 1.4(2) and Theorem 3.6 are false: no full-dimensional A can realize (1,0,0,0,0,0), since the vanishing of all projections except π12 forces A into a lower-dimensional affine subspace. If the intended meaning is 'compact convex set', the definition and Question 1.1 should say so explicitly.
minor comments (4)
  1. [§1, Definition 1.5] The notation [V] is used both for the homology fundamental class and, via the stated abuse, for its Poincaré dual. Since Theorem 1.8 currently switches between H_4 and H^4 conventions, the Poincaré-duality convention should be stated explicitly in Definition 1.5.
  2. [§5, proof of Theorem 5.5] There is a typo 'alwyas' in the sentence preceding the proof; it should read 'always'.
  3. [§6.1, Remark 6.8] The phrase 'the ground field to be uncountable or of characteristic 0' should be 'the ground field is uncountable or of characteristic 0'.
  4. [§4, Proposition 4.1] The 4×16 matrices L and M are difficult to parse visually; adding a brief explanation of how the columns are ordered, or a link to a machine-checkable verification of the projection areas, would help readers verify the computation.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; minor self-citations for context only.

full rationale

The central derivation chain is self-contained against external mathematics. Theorem 1.4 is proved by explicit construction of convex polytopes (Section 4) using the Shenfeld–van Handel equality conditions, an external result. Theorem 1.6 follows from Theorem 2.4 (toric Bernstein–Khovanskii–Kushnirenko construction) plus Theorem 3.7 (explicit mixed-volume realization), and its necessity from Hodge Index (Corollary 2.2) and the direct determinant computation in Proposition 3.1. Theorem 1.8 is derived from the prolific-surface construction (Theorem 6.3: E×C with C a complete intersection, using Severi/Ciliberto–van der Geer), Meyer's theorem on rational quadratic forms (Lemma 6.9), and the semiampleness argument; it does not assume the conclusion. The Lorentzian/Plücker equivalence (Prop 3.1) is proved in the paper, not merely cited from [BHKL]. Self-citations [BH20] and [BHKL] are used for context (Lorentzian polynomial terminology, triangular-hyperfield interpretation, homeomorphism of ∆(T2) to a ball) and are not load-bearing premises. Two non-circular correctness gaps should be noted separately: Theorem 1.8 as printed states H^4(P^m), but for total dimension d≠4 surface classes lie in H_{2d-4}(P^m) by Poincaré duality, so the statement is ill-defined/false as written (e.g., P^2×P^1); and the proof step 'choose an ample divisor H with ∫_Y A_iH≠0 for all i' fails when a row of L(η) is zero (e.g., η=H_3H_4 in (P^1)^4). These are repair issues, not circularity, and do not raise the circularity score beyond the minor-self-citation level.

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

No empirically fitted constants or hand-tuned numbers appear. All inputs are the candidate tuple or matrix; the parameter s in Proposition 4.1 is the input coordinate, and c = sqrt(1-1/s) is determined by s. The ample divisor H is a generic choice, not a fitted value. 'Prolific surface' and 'triangular hyperfield point set' are definitions of classes within existing objects, not new postulated entities.

assumptions (7)
  • standard math Bernstein–Khovanskii–Kushnirenko theorem (Theorem 2.3, cited to CLO05 Section 7.5)
    Converts mixed volumes of Newton polytopes into numbers of solutions of Laurent systems; used in Theorem 2.4 to identify intersection numbers with mixed volumes.
  • standard math Hodge Index Theorem (Section 2, Proposition 2.1, cited to Hartshorne)
    Yields the at-most-one-positive-eigenvalue condition for the intersection matrix of an irreducible surface, giving necessity in Theorems 1.4, 1.6, 1.8.
  • standard math Meyer's theorem: indefinite rational quadratic forms in at least five variables have nontrivial rational zeros (cited to Cassels)
    Load-bearing in Lemma 6.9, which embeds a candidate rational intersection matrix into the Neron-Severi space of a prolific surface.
  • standard math Morrison's result: every even lattice of signature (1,n-1) with n≤11 occurs as the Neron-Severi group of a complex algebraic K3 surface (Mor84)
    Used in Proposition 6.14 and Theorem 6.13 to realize integral Lorentzian matrices as intersection matrices of divisors on K3 surfaces.
  • standard math Shenfeld–van Handel equality case of Minkowski's quadratic inequality (SvH22, Theorem 1.3; SvH23)
    Used in Section 4 to constrain boundary convex bodies and to derive the vertex matrices L and M in Proposition 4.1; the final existence check is independent.
  • standard math Severi–Ciliberto–van der Geer theorem on Jacobians of curves on surfaces, and the characteristic-zero version due to Koch (Koc18)
    Used in Lemma 6.7 to show the Neron-Severi pullback map is surjective for Y=E×C.
  • standard math Alexandrov–Fenchel inequality for mixed volumes (Sch14, Section 7.3)
    Used in Corollary 2.5 to show that mixed volumes of projections form a Lorentzian matrix, proving necessity in Theorem 1.4.

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Pith. "Pith review of Realizations of homology classes and projection areas." pith.science (2026). https://pith.science/paper/WVUXX4GV

@misc{pith2026250508881,
  author       = {Pith},
  title        = {Pith review of: Realizations of homology classes and projection areas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WVUXX4GV}},
  note         = {Machine review of arXiv:2505.08881}
}
abstract

The relationship between convex geometry and algebraic geometry has deep historical roots, tracing back to classical works in enumerative geometry. In this paper, we continue this theme by studying two interconnected problems regarding projections of geometric objects in four-dimensional spaces: (1) Let $A$ be a convex body in $\mathbb{R}^4$, and let $(p_{12}, p_{13}, p_{14}, p_{23}, p_{24}, p_{34})$ be the areas of the six coordinate projections of $A$ in $\mathbb{R}^2$. Which tuples of six nonnegative real numbers can arise in this way? (2) Let $S$ be an irreducible surface in $(\mathbb{P}^1)^4$, and let $(p_{12}, p_{13}, p_{14}, p_{23}, p_{24}, p_{34})$ be the degrees of the six coordinate projections from $S$ to $(\mathbb{P}^1)^2$. Which tuples of six nonnegative integers can arise in this way? We show that these questions are governed by the Pl\"ucker relations for the Grassmannian $\text{Gr}(2,4)$ over the triangular hyperfield $\mathbb{T}_2$. We extend our analysis by determining the homology classes in $(\mathbb{P}^m)^n$ proportional to the fundamental classes of irreducible algebraic surfaces, resolving the algebraic Steenrod problem in this setting. Our results lead to several conjectures on realizable homology classes in smooth projective varieties and on the projection volumes of convex bodies.

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Pith tools

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