ProofAtlas

Symplectic geometry · algebraic geometry · mirror symmetry

Kontsevich’s Homological Mirror Symmetry Conjecture

Collaboration beta

For a compact mirror pair, homological mirror symmetry predicts that the symplectic Fukaya category and the algebraic category of twisted perfect complexes encode the same mathematics. Important families are known, but no uniform theorem covers the general compact smooth proper Calabi–Yau setting.

Perf(X,α)≃DπF(Y;Λ)
Known results and sources
A compact symplectic Calabi–Yau with Lagrangian cycles faces a smooth proper algebraic mirror across an unresolved categorical equivalence marked with a question mark.
Homological mirror symmetry predicts an equivalence between the symplectic and algebraic categories of compact mirror partners; the general statement remains open.

Research problem

Exact mathematical statement

For a compact mirror pair, let YY be a compact connected symplectic Calabi–Yau manifold of real dimension 2n2n, let Λ\Lambda be a Novikov field, and let DπF(Y;Λ)D^\pi\mathcal F(Y;\Lambda) be the split-closed derived Fukaya category with the required grading, convergence, bounding-cochain, smoothness, properness, and cyclic structures. Let XX be a smooth connected proper Calabi–Yau mirror of complex dimension nn, with the source-specified twist α\alpha. The compact HMS target is

Perf(X,α)≃DπF(Y;Λ).\operatorname{Perf}(X,\alpha) \simeq D^\pi\mathcal F(Y;\Lambda).

A complete result must construct the categories and mirror, produce an enhanced functor, prove full faithfulness and essential surjectivity, and justify analytic completion or algebraization. Wrapped, noncompact, Fano/Landau–Ginzburg, singular, relative, and partially wrapped variants are outside this exact statement.

Problem infographic

Problem at a glance

Problem-first diagram placing the compact symplectic and algebraic Calabi–Yau categories on opposite sides of a conjectural equivalence, with solved special families separated from the general construction barrier.
The exact compact HMS statement requires both categories, a mirror and enhanced functor, full faithfulness, essential surjectivity, and generation; special-family theorems do not provide a uniform general proof.

Current mathematical picture

Where work on Kontsevich’s Homological Mirror Symmetry Conjecture stands

Partially resolved

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureUnattached abstraction

The bottom-line work order explicitly prioritizes one named end-to-end certificate over additional abstract branches. A named surface or fourfold certificate remains viable if all source-listed construction, descent, compactness, and generation hypotheses are verified.

Route status · Narrowed route
Main reductionEven-dimensional exact frontier

The source reports that comparison data and a cyclic split reduce the correction to a pure product-graph object with a sharply constrained Boolean core.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeClose the surface one-generator compactness route on a named compact mirror pair.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Kontsevich’s Homological Mirror Symmetry Conjecture in numbers

4.7kretained lines of mathematical investigation4,735 in the current working snapshot
Argument development
3,789 · 80%
Explored or eliminated routes
106 · 2%
Computational analysis
175 · 4%
Open obligations
227 · 5%
Definitions and setup
438 · 9%
8selected mapped statements1routes investigated3open questions3contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byMaxim KontsevichProposed1994Status checkedAug 14, 2026
Source recordedAug 14, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

12 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

12 selected steps

Scroll horizontally to explore the route

Working route overview for Kontsevich’s Homological Mirror Symmetry ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Compact mirror partners should have equivalent algebraic and symplectic categories. — Depends on missing premiseCompact mirror partnersshould have equivalentalgebraic…Current reduction — Depends on missing premiseCurrent reductionEven-dimensional exact frontier — Depends on missing premiseEven-dimensional exactfrontierExact compact HMS target — Depends on missing premiseExact compact HMS targetSurface one-generator route — Depends on missing premiseSurface one-generator routeClosing target — Depends on missing premiseClosing targetExact scope boundary — Depends on missing premiseExact scope boundaryNo general proof claimed — Depends on missing premiseNo general proof claimedUnattached abstraction — stoppedUnattached abstractionClose the surface one-generator compactness route on a named compact mirror pair. — OpenClose the surfaceone-generator compactnessroute…Test the exact fourfold product-graph and reduced-block frontier on a named mirror class. — OpenTest the exact fourfoldproduct-graph andreduced-block…Supply the uniform construction and generation data required by the full compact conjecture. — OpenSupply the uniformconstruction and generationdata…
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routeUnattached abstraction

The bottom-line work order explicitly prioritizes one named end-to-end certificate over additional abstract branches. A named surface or fourfold certificate remains viable if all source-listed construction, descent, compactness, and generation hypotheses are verified.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Close the surface one-generator compactness route on a named compact mirror pair.Suggested move: Compute the single generator complex V_TG and prove it is perfect; only if that fails, build the full finite-algebra DVR heart.
Ready to work on
02
Test the exact fourfold product-graph and reduced-block frontier on a named mirror class.Suggested move: Compute the canonical detector, divisor components, Boolean core, normalized Fitting ideals, restricted Brauer classes, and any surviving H^1 and H^2 normal-channel data.
Ready to work on
03
Supply the uniform construction and generation data required by the full compact conjecture.Suggested move: Construct the graded Fukaya enhancement, represented compactified brane moduli, universal family, convergence and twisted descent, anchors, comparison windows, and Cardy generation for the intended mirror family.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusPartially resolved

Homological mirror symmetry has been proved for important compact mirror families, including the quartic surface and broader K3 and Greene–Plesser settings, often with carefully scoped foundational hypotheses. The general compact smooth proper Calabi–Yau equivalence across intended mirror pairs remains open.

[1][3][4]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. PreprintHacking and Keating reported HMS for projective K3 surfaces with integral Kähler class, extending earlier K3 cases.[4]
  2. PreprintSheridan and Smith reported HMS for generalized Greene–Plesser mirrors, with explicitly stated higher-dimensional foundational assumptions.[3]
  3. Peer reviewedSeidel proved a form of homological mirror symmetry for the quartic surface.[2]
  4. Historical sourceKontsevich’s ICM lecture formulated a refined mirror conjecture in homological-algebraic terms.[1]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusKontsevich’s homological mirror symmetry conjecture
Solved special caseQuartic-surface homological mirror symmetry

The quartic K3 surface is a compact special case, not a proof for all compact mirror pairs.

[2]
Solved special caseProjective-K3 homological mirror symmetry

The projective-K3 theorem addresses a broad surface family over a formal Laurent-series field while leaving general compact Calabi–Yau mirror pairs open.

[4]
Related problemGeneralized Greene–Plesser mirror pairs

Generalized Greene–Plesser results cover structured toric mirror families and document where higher-dimensional Fukaya-category foundations enter.

[3]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA formal construction of the relevant graded, split-closed, smooth, proper, cyclic Fukaya categories with convergence and bounding-cochain control.
  • Formalization targetFormal derived and twisted perfect categories at the required geometric level, including enhancements and Azumaya or gerbe conventions.
  • Formalization targetFormal mirror constructions and the full-faithfulness, essential-surjectivity, generation, and analytic-completion arguments for named compact mirror pairs.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements2 proposed statements3 open questions1 narrowed routes
Statements by mathematical role8 selected mapped statements
  • theorem candidate1 of 81
  • reduction3 of 83
  • lemma1 of 81
  • equivalence1 of 81
  • negative result2 of 82
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included
  • retained route statementCompact mirror partners should have equivalent algebraic and symplectic categories.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact compact HMS targetintermediate
  • retained route statementExact scope boundaryintermediate
  • retained route statementNo general proof claimedintermediate
  • retained route statementSurface one-generator routeintermediate
  • retained route statementEven-dimensional exact frontierintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureUnattached abstractionreported failure
  • Research targetClose the surface one-generator compactness route on a named compact mirror pair.open
  • Research targetTest the exact fourfold product-graph and reduced-block frontier on a named mirror class.open
  • Research targetSupply the uniform construction and generation data required by the full compact conjecture.open
  • Research targetGlobal construction and generationsuperseded
  • Narrowed routeUnattached abstractionThe bottom-line work order explicitly prioritizes one named end-to-end certificate over additional abstract branches. A named surface or fourfold certificate remains viable if all source-listed construction, descent, compactness, and generation hypotheses are verified.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeClose the surface one-generator compactness route on a named compact mirror pair.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointClose the surface one-generator compactness route on a named compact mirror pair.

Kontsevich’s Homological Mirror Symmetry Conjecture · ready to start

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Research contextPrepared context for any AI agent

For a compact mirror pair, homological mirror symmetry predicts that the symplectic Fukaya category and the algebraic category of twisted perfect complexes encode the same mathematics. Important families are known, but no uniform theorem covers the general compact smooth proper Calabi–Yau setting.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Homological Algebra of Mirror Symmetryoriginal source · Maxim Kontsevich · International Congress of Mathematicians proceedings / arXiv · 1994 preprint; 1995 proceedings · ARXIV alg-geom/9411018 · accessed Aug 14, 2026
  2. 2
    Homological Mirror Symmetry for the Quartic Surfacepeer reviewed result · Paul Seidel · Memoirs of the American Mathematical Society · 2015 · accessed Aug 14, 2026
  3. 3
    Homological mirror symmetry for generalized Greene–Plesser mirrorspreprint · Nick Sheridan, Ivan Smith · arXiv · 2017 · ARXIV 1709.08937 · accessed Aug 14, 2026
  4. 4
    Homological mirror symmetry for projective K3 surfacespreprint · Paul Hacking, Ailsa Keating · arXiv · 2025 · ARXIV 2503.05680 · accessed Aug 14, 2026

Important qualifications

  • This is a scoped status review of the compact Calabi–Yau categorical formulation and representative solved families, not an exhaustive bibliography across wrapped, Fano/Landau–Ginzburg, singular, or relative variants.
  • The 2025 projective-K3 result is recorded at its arXiv preprint posture; no peer-review status was inferred.
  • Scoped searches of current mathlib, Isabelle, and Coq public documentation found no end-to-end formalization of this HMS statement; that does not establish absence.

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