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arxiv: 2304.01856 · v5 · submitted 2023-04-04 · 🧮 math.AG

Non-calibrated framed processes, derived equivalence and Homological Mirror Symmetry

Michele Rossi This is my paper

Pith reviewed 2026-05-24 09:01 UTC · model grok-4.3

classification 🧮 math.AG
keywords framed dualityuncalibrated f-processesderived equivalenceHomological Mirror Symmetrycomplete intersectionsBondal-Orlov-Kawamata conjectureK-equivalencemirror symmetry
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The pith

Uncalibrated framed processes generate D-equivalent multiple mirror models for projective complete intersections.

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

The paper establishes a mirror theorem for framed duality from the Homological Mirror Symmetry viewpoint by examining derived equivalence of models produced by uncalibrated f-processes. It offers a general method to construct numerous multiple mirror models for any projective complete intersection of non-negative Kodaira dimension, with these models linked by uncalibrated f-processes and hence D-equivalent or K-equivalent. Additionally, it supplies several pieces of evidence for the Bondal-Orlov-Kawamata conjecture equating D-equivalence with K-equivalence.

Core claim

The paper produces a mirror theorem for framed duality by D-equivalence of models from uncalibrated f-processes, a general construction yielding multiple mirrors for projective complete intersections of non-negative Kodaira dimension that are D-equivalent or K-equivalent, and evidences for the Bondal-Orlov-Kawamata conjecture.

What carries the argument

Uncalibrated f-processes, which generate multiple mirror models connected through derived equivalences viewed from Homological Mirror Symmetry.

Load-bearing premise

That the uncalibrated f-processes produce models whose derived equivalences can be established from the Homological Mirror Symmetry perspective.

What would settle it

A projective complete intersection of non-negative Kodaira dimension where two models connected by an uncalibrated f-process fail to be derived equivalent would disprove the claimed mirror theorem.

Figures

Figures reproduced from arXiv: 2304.01856 by Michele Rossi.

Figure 1
Figure 1. Figure 1: Interplay of calibrated and non-calibrated framed processes connecting the LT-mirror and the BB-mirror. The proof that Y ∨ LT is an elliptic curve, that is a smooth complete intersection, is deferred to Remark 2.5. Remark 2.3. Recalling Def. 1.7 in [21] of a calibrated partitioned f-process, the one given by (P 3 , a = a1 + a2) ! (X, b = b1 + b2) as described in Proposition 2.1 and sketched in the upper pa… view at source ↗
read the original abstract

The present paper is aimed to discussing three kinds of problems: (1) producing some ``mirror theorem'' for the recent mirror symmetric construction, called \emph{framed} duality ($f$-duality), described in \cite{R-fTV} and \cite{R-fpCI}: this is performed from the point of view proposed by Homological Mirror Symmetry (HMS), by studying \emph{derived equivalence} ($D$-equivalence) of multiple mirror models produced by means of a, so-called, \emph{uncalibrated $f$-process}; (2) proposing a general construction giving a big number of multiple mirror models to, in principle, any projective complete intersection of non-negative Kodaira dimension: these multiple mirrors turn out to be each other connected by means of uncalibrated $f$-processes and then, after (1), $D$-equivalent or $K$-equivalent, in the sense of Kawamata \cite{Kawamata}; (3) presenting a number of evidences for the Bondal-Orlov-Kawamata conjecture that $D$-equivalence is $K$-equivalence, and viceversa.

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

0 major / 2 minor

Summary. The paper discusses three problems: (1) producing a mirror theorem for framed duality (f-duality) from the Homological Mirror Symmetry viewpoint by studying derived equivalence of multiple mirror models from uncalibrated f-processes; (2) proposing a general construction for multiple mirror models to projective complete intersections of non-negative Kodaira dimension, connected by uncalibrated f-processes leading to D-equivalence or K-equivalence; (3) presenting evidences for the Bondal-Orlov-Kawamata conjecture that D-equivalence is equivalent to K-equivalence.

Significance. If the results hold, this work advances the understanding of mirror symmetry by providing a general construction for multiple mirrors and linking it to derived equivalences via HMS. The explicit construction with concrete examples and evidences for the conjecture represent a positive contribution to the field.

minor comments (2)
  1. The abstract references the author's prior works on f-duality without a brief self-contained recap of the key definitions; adding a short summary in the introduction would improve accessibility.
  2. Notation for uncalibrated f-processes and the distinction between D-equivalence and K-equivalence could be made more uniform across the text for clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of its three main contributions, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proposes an explicit general construction for multiple mirror models of projective complete intersections via uncalibrated f-processes, then invokes HMS to obtain D-equivalence (hence K-equivalence per Kawamata). Prior self-citations define the base f-duality, but the current manuscript supplies the new construction, concrete examples, and direct evidences for the Bondal-Orlov-Kawamata conjecture. No quoted step reduces a prediction or central claim to a fitted input, self-definition, or unverified self-citation chain; the argument remains self-contained against external HMS benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted from the provided text.

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Reference graph

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