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arxiv: 2502.07031 · v2 · submitted 2025-02-10 · 🧮 math.AG

Smooth Calabi-Yau varieties with large index and Betti numbers

Jas Singh This is my paper

Pith reviewed 2026-05-23 03:13 UTC · model grok-4.3

classification 🧮 math.AG
keywords Calabi-Yau varietiesindexBetti numbersEuler characteristicprojective varietiessmooth varietiesextremal invariantsalgebraic geometry
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The pith

Smooth projective Calabi-Yau varieties exist in every dimension with index growing doubly exponentially.

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

The paper constructs smooth projective Calabi-Yau varieties in every dimension whose index grows doubly exponentially with dimension. It also produces examples whose Euler characteristics and sums of Betti numbers grow at the same doubly exponential rate. These are conjectured to be maximal among all such varieties in each dimension. A sympathetic reader would care because the index measures the smallest multiple of the canonical class that becomes trivial, while the Betti numbers and Euler characteristic capture the scale of the variety's topology. The constructions extend low-dimensional examples to arbitrary dimension.

Core claim

We construct smooth, projective Calabi-Yau varieties in every dimension with doubly exponentially growing index, which we conjecture to be maximal in every dimension. We also construct smooth, projective Calabi-Yau varieties with extreme topological invariants; namely, their Euler characteristics and the sums of their Betti numbers grow doubly exponentially. These are conjecturally extremal in every dimension. The varieties we construct are known in small dimensions but we believe them to be new in general.

What carries the argument

The construction of smooth projective Calabi-Yau varieties obtained by resolving or smoothing singular examples while preserving the Calabi-Yau condition and achieving doubly exponential growth in the index and topological invariants.

If this is right

  • Such smooth projective Calabi-Yau varieties exist in arbitrarily high dimensions.
  • The index of these varieties grows at least doubly exponentially with dimension.
  • The sum of Betti numbers and the Euler characteristic also grow doubly exponentially with dimension.
  • These examples are conjecturally maximal for the index and for the topological invariants in every dimension.

Where Pith is reading between the lines

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

  • If correct, there would be no singly exponential upper bound on the index of smooth projective Calabi-Yau varieties.
  • The constructions may provide test cases for conjectural classification statements about Calabi-Yau varieties in high dimensions.
  • One could compare the achieved index against any known theoretical upper bounds that are singly exponential to test the conjecture of maximality.

Load-bearing premise

The geometric constructions from singular Calabi-Yau varieties actually produce smooth projective varieties that remain Calabi-Yau and achieve the claimed doubly exponential growth rates.

What would settle it

An explicit computation or bound showing that the index of the constructed varieties in dimension n is at most singly exponential in n, or a proof that no smooth projective Calabi-Yau variety in some dimension can have index exceeding single-exponential growth.

Figures

Figures reproduced from arXiv: 2502.07031 by Jas Singh.

Figure 5
Figure 5. Figure 5: Various choices of ω in the proof of Lemma 3.7 and the resulting convexity of ψ. seek to extend ψ − linearly to some ψ : P −! R which will witness the regularity of Cone(z, S). This time, we cannot choose ψ(z) = ω so freely, however. For each cell σ of S −, let ψσ be the linear function which agrees with ψ − on σ. Let ω be any rational number greater than ψσ(z) for all cells σ of S −. Define ψ : P −! R to … view at source ↗
Figure 6
Figure 6. Figure 6: A schematic representation of Pq(n+1) 2 and its recursive face structure. Literally speak￾ing, the triangle depicted is Pq(2) 2 . this as the “bottom” face of Pq(n+1) 2 . See [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A schematic representation of Pq(n+1) 2 showing its lattice points and how it is cut by the hyperplane H. Literally speaking, the triangle depicted is Pq(2) 2 . (b) Otherwise, x satisfies xn = − nX−1 i=0 sn − 1 si xi . See [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Pq(n+1) 2 = P ≤H ∪ P ≥H with its regular subdivision S glued from π ∗T and Cone(z,(π ∗T )|H) along H. Proof. The cells of π ∗T (n) are pullbacks π ∗ (σ) of cells σ ∈ T (n) . Each such σ is a unimodular simplex containing the origin (in R n ) as a vertex. It suffices to focus our attention on any such cell π ∗ (σ). The facets of the column π ∗ (σ) are the “bottom cap” σ×{−1}, the “top cap” π ∗ (σ)∩H, and th… view at source ↗
Figure 9
Figure 9. Figure 9: The result of pulling the subdivision S as in [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The Hodge diamonds of the crepant resolutions [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗
read the original abstract

A normal variety $X$ is called Calabi-Yau if $K_X \sim_{\mathbb Q} 0$. The index of $X$ is the smallest positive integer $m$ so that $m K_X \sim 0$. We construct smooth, projective Calabi-Yau varieties in every dimension with doubly exponentially growing index, which we conjecture to be maximal in every dimension. We also construct smooth, projective Calabi-Yau varieties with extreme topological invariants; namely, their Euler characteristics and the sums of their Betti numbers grow doubly exponentially. These are conjecturally extremal in every dimension. The varieties we construct are known in small dimensions but we believe them to be new in general. This work builds off of the singular Calabi-Yau varieties found by Esser, Totaro, and Wang in arXiv:2209.04597.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript constructs smooth, projective Calabi-Yau varieties (K_X ~_Q 0) in every dimension whose index (smallest positive integer m with m K_X ~ 0) grows doubly exponentially, conjectured to be maximal in each dimension. It also produces examples whose Euler characteristics and sums of Betti numbers grow doubly exponentially, conjecturally extremal. The constructions start from the singular Calabi-Yau varieties of Esser-Totaro-Wang (arXiv:2209.04597) and are stated to be known in low dimensions but new in general.

Significance. If the constructions are shown to yield smooth projective varieties while preserving the Q-Calabi-Yau condition and the claimed growth rates, the results would furnish the first known families of Calabi-Yau varieties with doubly exponential index and topological invariants in arbitrary dimension. This would supply concrete evidence toward conjectures on extremal behavior and demonstrate a viable method for producing such examples from singular ones.

major comments (1)
  1. [Main construction (body of the paper)] The transition from the singular varieties of Esser-Totaro-Wang to smooth projective Calabi-Yau varieties is the central step supporting all existence claims, yet the manuscript provides insufficient detail on the resolution or deformation process, the verification that smoothness and projectivity are achieved, and the confirmation that the index and Q-CY condition are preserved with the stated growth rates. This verification is load-bearing for the claims in every dimension.
minor comments (1)
  1. The abstract and introduction would benefit from a short statement of the precise low dimensions in which the varieties are already known, to clarify the novelty claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the central point requiring clarification. We agree that the transition from the singular varieties of Esser-Totaro-Wang to smooth projective Calabi-Yau varieties needs substantially more detail to make the arguments self-contained and verifiable in every dimension.

read point-by-point responses

  1. Referee: [Main construction (body of the paper)] The transition from the singular varieties of Esser-Totaro-Wang to smooth projective Calabi-Yau varieties is the central step supporting all existence claims, yet the manuscript provides insufficient detail on the resolution or deformation process, the verification that smoothness and projectivity are achieved, and the confirmation that the index and Q-CY condition are preserved with the stated growth rates. This verification is load-bearing for the claims in every dimension.

    Authors: We agree that the current exposition of the smoothing step is too terse. In the revised manuscript we will add a new subsection (provisionally 2.3) that: (i) specifies the sequence of blow-ups and/or deformations applied to each Esser-Totaro-Wang variety, (ii) cites the precise theorems guaranteeing that the resulting variety is smooth and projective, (iii) verifies that the Q-CY condition K_X ~_Q 0 is preserved (by showing the resolution is crepant or that discrepancies do not affect the Q-linear equivalence class), and (iv) confirms that the index m remains unchanged while the doubly exponential growth in m and in the Betti numbers is inherited directly from the singular models. We will also include low-dimensional checks that recover the known examples. These additions will be placed before the statements of the main theorems so that the load-bearing claims rest on explicit arguments. revision: yes

Circularity Check

0 steps flagged

No significant circularity; external construction base

full rationale

The derivation consists of explicit geometric constructions that resolve or deform singular Calabi-Yau examples from the external reference Esser-Totaro-Wang (arXiv:2209.04597). No self-citations appear in load-bearing positions, no parameters are fitted then relabeled as predictions, and no ansatz or uniqueness claim reduces to prior work by the same author. The index and Betti-number growth statements follow directly from the cited singular models plus the resolution step, which is independent of the target claims. This is the standard non-circular pattern for a construction paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, ad-hoc axioms, or invented entities are described. Relies on standard definitions of Calabi-Yau varieties and index.

axioms (1)
  • standard math A normal variety X is Calabi-Yau if K_X ~_Q 0 and the index is the smallest positive integer m with m K_X ~ 0.
    This is the definition used to frame the constructions and conjectures.

pith-pipeline@v0.9.0 · 5668 in / 1238 out tokens · 42144 ms · 2026-05-23T03:13:25.034563+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Theorem 0.1. For n≥1 there exists a smooth, projective Calabi-Yau n-fold V(n) with index (s_{n-1}-1)(2s_{n-1}-3). ... We are reduced to finding a µ_m-equivariant, crepant, projective resolution of the weighted projective hypersurface X ... equivalent to finding a toric, crepant, projective resolution of P.

  • IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Proposition 3.1. For all n, there is a regular, unimodular, star-shaped triangulation of P(n)1. ... reduced to ... regular, unimodular, star-shaped triangulation of P(n)2 via polar duality and pulling refinements.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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