Torus fibers and the weight filtration

Andrew Harder

REVIEW 3 major objections 5 minor 1 cited by

Torus fibers and the weight filtration

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For simple normal crossings log Calabi-Yau pairs and projective Calabi-Yau degenerations, a single real torus computes every odd-degree weight filtration of cohomology.

desk verdict A clean, genuinely new single-torus theorem for log Calabi-Yau pairs and Calabi-Yau degenerations; the main idea holds up and the flaws are presentation-level, not load-bearing. read the letter →

arxiv 1908.05110 v1 pith:XCOKQ5X3 submitted 2019-08-14 math.AG

Andrew Harder This is my paper · ORCID

classification math.AG MSC 14C3014D0714J2814J32

keywords mixedHodgestructureweightfiltrationlogCalabi-YaupairtorusfiberperverseLerayP=WequalitydegenerationK3surface

topics The Hodge Conjecture

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 for a simple normal crossings log Calabi-Yau pair, the odd-degree pieces of the weight filtration on the cohomology of the complement are exactly the kernel of restriction to one real torus, whose dimension is the codimension of the deepest stratum of the boundary. The same statement holds for the monodromy weight filtration of a projective Calabi-Yau degeneration, with a torus one dimension lower. The point is not just that one such torus exists but that it is canonical up to homotopy, so a single probe detects the weight filtration at every degree. This makes the weight filtration concretely computable and yields P=W-type equalities: on rational surfaces with nodal anticanonical boundary and on K3 surfaces, the perverse Leray filtration of a torus fibration coincides with the weight filtration.

What carries the argument

The mechanism is the profound torus: for each minimal stratum $Y_J$ of the boundary, take a point and loop around each of the $j$ components meeting there. The load-bearing step is Lemma 2.12: a rational curve contained in the $(j-1)$-stratum and meeting the $j$-strata transversally in exactly two points produces a homotopy between the two associated profound tori. A cited theorem on log Calabi-Yau pairs guarantees chains of such rational curves between every pair of minimal strata, so all profound tori are homotopic; together with the generalized Leray cycles this turns the direct sum of tori in Theorem 2.11 into a single torus. For degenerations, the same picture is built from the contraction map, whose preimage of a point in a deep stratum is a torus of dimension $\delta-1$, and the same curve-chain argument identifies all of those tori.

What would settle it

Exhibit an snc log Calabi-Yau pair whose boundary has two minimal strata of different cardinalities; then the attached profound tori have different dimensions, so the kernel to any single torus cannot equal $W_{2k-1}H^k(X\setminus Y;\mathbb{Q})$ for all $k$, and the theorem's conclusion fails.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: if $(X,Y)$ is an snc log Calabi-Yau pair and $\delta$ is the maximal number of components of $Y$ meeting nontrivially, then there is a real torus $T\subset X\setminus Y$ such that $W_{2k-1}H^k(X\setminus Y;\mathbb{Q})=\ker(H^k(X\setminus Y;\mathbb{Q})\to H^k(T;\mathbb{Q}))$ for all $k$; locally $T$ is $\{(\varepsilon e^{i\theta_1},\ldots,\varepsilon e^{i\theta_\delta},0,\ldots,0)\}$. For a projective Calabi-Yau degeneration, the analogous torus has dimension $\delta-1$ and computes the monodromy weight filtration $M_{2k-1}H^k(X_1;\mathbb{Q})$ in the same way. The proof shows that every torus attached to a stratum of the boundary can be deformed into a torus attached to a minimal stratum, and that in the Calabi-Yau case all these minimal-stratum tori are homotopic, because chains of rational curves connecting the minimal strata induce homotopies of the corresponding tori. Thus the a priori large collection of tori collapses to a single torus, and the weight filtration is visible as a single kernel.

Load-bearing premise

The construction rests on a theorem that every two minimal strata of an snc log Calabi-Yau boundary have the same number of components and are connected by chains of rational curves meeting the deepest strata transversally; if that is false, the profound tori are not necessarily homotopic and one torus would not compute the weight filtration.

Editorial extensions

If this is right

For any log Calabi-Yau pair, the dimension of each odd-weight graded piece is bounded by $\binom{\delta}{k}$, and the single torus controls the equality cases.
On a rational surface with a reduced nodal anticanonical divisor, the Lagrangian torus fibration satisfies $P_i = W_{2i} = W_{2i+1}$ for its perverse Leray and weight filtrations, giving a P=W-type equality.
On a K3 surface with an elliptic fibration, there is a semistable degeneration whose monodromy weight filtration matches the perverse Leray filtration of the fibration.
If a Betti moduli space admits a log Calabi-Yau compactification with $\delta = \dim$, Theorem 1.1(1) produces a real torus of the expected dimension computing the odd weight filtration, exactly the kind of object the P=W conjecture predicts at highest weight.

Reading between the lines

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

The collapse to one torus is essentially a property of the dual complex of the boundary being connected by rational curves; one could test the same rational-curve-chain condition on other families of log Calabi-Yau pairs to predict where a single torus should exist.
For compact holomorphic symplectic manifolds with Lagrangian torus fibrations, the K3 argument suggests that the monodromy operator built from an isotropic fiber class and a transverse class should reproduce the perverse Leray filtration; a numerical check on known examples would separate the cases where this holds from those where it fails.
The surface theorem suggests a two-dimensional converse: if a log Calabi-Yau surface admits a Lagrangian torus fibration whose general fiber is homotopic to the profound torus, then perverse Leray equals weight, so constructing such fibrations on other rational surfaces would extend the result.

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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

The thing to know: this is a genuinely useful paper. The main theorem says that for an snc log Calabi-Yau pair, a single real torus of dimension equal to the maximal depth of a stratum computes the odd weight filtration, and the analogous statement holds for projective Calabi-Yau degenerations. I don't see that statement in the cited literature; it's a real consolidation of the El Zein-Némethi and Clemens torus constructions, and the P=W-type applications for rational surfaces with nodal anticanonical boundary and for elliptic K3s go beyond what Gross and Zhang had.

The proof strategy is transparent: reduce to established results, get a direct sum over profound tori, then use Kollár's connectivity theorem to show all profound tori are homotopic. I checked the stress-test objection about (P^2, smooth cubic). I don't think it lands. That pair has delta = 1 and a single profound torus; homotopy independence within one stratum is already in Definition 2.7. If Kollár's theorem is being quoted for two points on the same stratum, the statement needs a 'distinct minimal strata' qualifier, but the needed reduction doesn't require that case. So the load-bearing step survives.

The real soft spots are presentation-level. Proposition 3.3's proof has an im/ker indexing error: formula (4) gives M_{2k-1}H^k = ker(N^k_k), not im, and the proof later writes M_{2d-2k-1} where the proposition needs M_{2d-2k}. The proposition itself is standard and the intended orthogonality argument is visible, but as written the proof is wrong. A referee should require fixing that. The surface section applies the snc theorem to nodal anticanonical divisors without saying why that's allowed; a reduced nodal curve is snc, so it's fine, but one line would remove the friction. The paper leans on Kollár, Clemens, Saito, and Soldatenkov more heavily than on its own machinery, and I did not verify those imports; nothing in the text indicates a circular dependence, and the sole self-reference [17] is confined to a remark.

Bottom line: this deserves a serious referee. I would send it out, ask for the Proposition 3.3 cleanup and a precise statement of the Kollár import, and expect it to come back as a solid contribution for Hodge theorists and P=W people.

Referee Report

3 major / 5 minor

Summary. The paper develops a torus-fiber description of the odd part of the weight filtration. Theorem 1.1 asserts that for a simple normal crossings log Calabi-Yau pair (X,Y), if δ is the maximal number of components of Y meeting at a point, there is a real torus T of dimension δ embedded in X\Y such that W_{2k-1}H^k(X\Y;Q) equals the kernel of the restriction map H^k(X\Y;Q) -> H^k(T;Q), and that an analogous statement holds for semistable Calabi-Yau degenerations with a torus of dimension δ-1 in the smooth fiber. The proof identifies the lowest weight homology classes with classes of tori obtained by generalized Leray cycles (El Zein-Némethi) or by Clemens contraction, then uses Kollár's connectedness theorem to show all tori attached to minimal strata are homotopic. The paper then proves P=W-type identifications for rational surfaces with nodal anticanonical divisor and for K3 surfaces, and formulates a conjecture for compact hyperkähler manifolds.

Significance. If the main theorem is correct, it provides a very concrete geometric description of the odd weight filtration in terms of a single embedded torus, with immediate dimension bounds and a bridge to the P=W conjecture via Simpson's and Auroux's conjectures. The surface and K3 applications are genuinely interesting and go beyond previously known cases (Zhang, Gross). The proof is mostly transparent and relies on classical external results; the paper also honestly identifies the special chain condition (*) that is needed. However, the correctness of the central reduction currently depends on a misstated version of Kollár's theorem and on an invalid proof of a duality statement, so the significance will be realized only after those points are repaired.

major comments (3)

[Theorem 2.13 and Theorem 2.14] Theorem 2.13 is false as stated. Taking X=P^2 and Y a smooth cubic gives an snc log CY pair with δ=1, and for J1=J2={1} and distinct points p1,p2 the asserted rational curve C⊂P^2 would have to meet the cubic transversely in exactly two points; Bezout's theorem forces deg(C)·3 = 2, impossible. Since the proof of Theorem 2.14 uses Theorem 2.13 to conclude that all profound tori are homotopic, the statement must be corrected (for instance, by requiring J1 and J2 to index distinct minimal strata, and by checking the curve-chain assertion against Kollár's original theorem), or Lemma 2.12 should be generalized so that subarcs of a rational curve with finitely many intersection points with Y^j give the required homotopies.
[Proposition 3.3] Proposition 3.3 has a proof that is inconsistent with formula (4). With ℓ=k, formula (4) gives M_{2k-1}H^k(X1;Q)=ker(N^k_k), not im(N^k_k), and for H^{2d-k} it gives M_{2d-1}H^{2d-k}=ker(N^k_{2d-k}), not M_{2d-2k-1}. The statement of Proposition 3.3 is in fact true by Poincaré duality, and Corollary 3.4 can be proved directly from formula (4) and duality, but the proof as written does not establish it; this needs to be rewritten before the Clemens-torus computation can be used.
[Theorem 4.5] Theorem 4.5 applies Theorem 1.1(1) to a pair (X,Y) in which Y is only assumed to be reduced and nodal, not necessarily snc, and its proof uses the weight spectral sequence with E1-term H^{2p+q}(Y^{-p}) as though Y were snc. Since Theorem 1.1 was proved only for snc divisors, this application needs an explicit reduction, e.g. by passing to an snc model of the same complement via the Gross-Hacking-Keel toric model invoked in Theorem 4.1, and by checking that the torus fiber of the Lagrangian fibration is homotopic to the profound torus in that model. As written, the P=W-type conclusion for nodal anticanonical surfaces is not rigorously derived.

minor comments (5)

[Section 2.2, proof of Theorem 2.11] The expression W_{-k}H_k should read W_{-2k}H_k.
[Definition 2.7] The phrase 'p1 and p1' should be 'p1 and p2'.
[Proposition 4.8] The vector η is used in the proof without being defined; one should state that η is a class with ⟨η,β⟩≠0 (or ⟨η,β⟩=1).
[Theorem 3.11] There is a duplicated word in 'contained in in A^{j-1}'.
[Remark 4.13] Remark 4.13 promises a proof in [17], which is listed as 'in preparation'; this should be marked as forthcoming work or removed if it is not needed for the current argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main torus theorem is derived from external results of El Zein–Némethi, Clemens, and Kollár, and the sole self-citation is not load-bearing.

full rationale

The derivation chain is self-contained in the relevant sense. Section 2 imports El Zein–Némethi's description of the lowest weight piece as generated by tori T_I (Propositions 2.5 and 2.8), consolidates those tori through an explicit local-coordinate homotopy argument (Proposition 2.9), and obtains the direct-sum kernel statement (Theorem 2.11) without any fitted parameter or self-referential definition. The passage to a single torus in Theorem 2.14 is explicitly conditional on the external Kollár result quoted as Theorem 2.13, citing [21, Theorem 10]; the paper does not derive that condition from its own conclusions. The degeneration case follows the same structure, using Clemens' contraction-map tori (Theorem 3.5) and Kawamata–Namikawa for the relevant homotopy lemma (Lemma 3.10), with Kollár's theorem imported externally once more. The P=W-type results in Section 4 rely on Symington, Gross–Hacking–Keel, Saito, and Soldatenkov; the equality of perverse Leray and weight filtrations is obtained by identifying a smooth fiber of the constructed fibration with a profound torus (Theorem 4.1) and then applying Theorem 1.1, rather than assuming the desired equality. The only self-citation, [17] (Harder, in preparation), appears in Remark 4.13 as a pointer to future work on a consequence and is not used in any proof, so it is not load-bearing. Whether Kollár's theorem is paraphrased accurately is a correctness question about an external result, not a circularity in the paper's reduction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The central results rest entirely on imported theorems from the algebraic geometry literature, listed above.

assumptions (6)

standard math Deligne's mixed Hodge theory and El Zein-Némethi's generalized Leray cycles (paper's Proposition 2.5).
Provides the fact that W_{-2k}H_k(X\Y;Q) is generated by the tori T_I, the starting point of Theorem 2.11.
domain assumption Kollár's Theorem 10 on P^1-connectedness of log canonical centers (paper's Theorem 2.13).
Load-bearing for showing all profound tori are homotopic; the main theorem fails if minimal strata are not connected by rational curves.
domain assumption Clemens' contraction theorem (paper's Theorem 3.5).
Identifies the image of the k-th power of monodromy logarithm with classes of tori T_I in a semistable degeneration.
domain assumption Saito's decomposition theorem and the perverse Leray filtration computation (paper's Proposition 4.4).
Used to identify P_{k-1}H^k with the kernel of restriction to a smooth fiber for the elliptic fibrations in Section 4.
domain assumption Livne-Moishezon classification of elliptic Lefschetz fibrations over P^1.
Used in Proposition 4.3 to embed the elliptic Lefschetz fibration over a disc into an algebraic elliptic surface.
domain assumption Torelli theorem for K3 surfaces and Soldatenkov's construction of a degeneration with monodromy N_{β,ρ} (Theorem 4.10).
Used to produce the semistable degeneration whose monodromy weight filtration is compared to the perverse Leray filtration for elliptic K3 surfaces.

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Pith. "Pith review of Torus fibers and the weight filtration." pith.science (2026). https://pith.science/paper/XCOKQ5X3

@misc{pith2026190805110,
  author       = {Pith},
  title        = {Pith review of: Torus fibers and the weight filtration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XCOKQ5X3}},
  note         = {Machine review of arXiv:1908.05110}
}

abstract

We show that if $(X,Y)$ is a simple normal crossings log Calabi--Yau pair, then there is a real torus of dimension equal to the codimension of the smallest stratum of $Y$ which can be used to construct $W_{2k-1}H^k(X \setminus Y;\mathbb{Q})$ for all $k$. We show that an analogous result holds for degenerations of Calabi--Yau varieties. We use this to show that P=W type results hold for pairs $(X,Y)$ consisting of a rational surface $X$ and a nodal anticanonical divisor $Y$, and for K3 surfaces.

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Forward citations

Cited by 1 Pith paper

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

P=W for Lagrangian fibrations and degenerations of hyper-K\"ahler manifolds
math.AG 2019-08 accept novelty 7.0 of 10

For every Lagrangian fibration of a projective hyper-Kähler manifold, the perverse filtration equals the monodromy weight filtration of an associated type III degeneration.

Reference graph

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Pith tools

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

[1] [17]

A. Harder. Cohomology of log symplectic pairs. in preparation, 2019

work page 2019

[2] [1]

D. Auroux. Mirror symmetry and T -duality in the complement of an anticanonical divisor. J. Gökova Geom. Topol. GGT, 1:51–91, 2007

work page 2007

[3] [2]

Beauville

A. Beauville. Variétés Kähleriennes dont la première cl asse de Chern est nulle. J. Diﬀerential Geom. , 18(4):755– 782 (1984), 1983

work page 1984

[4] [3]

P. Boalch. Wild character varieties, meromorphic Hitch in systems and Dynkin diagrams. In Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin , pages 433–454. Oxford University Press, 2018

work page 2018

[5] [4]

C.H. Clemens. Picard-Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singu - larities. Trans. Amer. Math. Soc. , 136:93–108, 1969

work page 1969

[6] [5]

Cox and S

D. Cox and S. Katz. Mirror symmetry and algebraic geometry , volume 68 of Mathematical Surveys and Mono- graphs. American Mathematical Society, Providence, RI, 1999

work page 1999

[7] [6]

de Cataldo, T

M. de Cataldo, T. Hausel, and L. Migliorini. Topology of H itchin systems and Hodge theory of character varieties: the case A1. Ann. of Math. (2) , 175( 3):1329–1407, 2012

work page 2012

[8] [7]

de Cataldo and L

M. de Cataldo and L. Migliorini. The decomposition theor em, perverse sheaves and the topology of algebraic maps. Bull. Amer. Math. Soc. (N.S.) , 46( 4):535–633, 2009

work page 2009