Kähler–Einstein Fano 3-foldsManuscript · Web edition
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Kähler–Einstein Fano 3-folds

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Manuscript dated 8 October 2026Working partyOriginal PDF · 437 pagesReferences

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Abstract

We classify Kähler–Einstein smooth Fano threefolds of Picard rank at least two.

128 sections and references.

Introduction

  1. 1.Introduction PDF p. 3

Part I · Abban–Zhuang theory

  1. 2.Stability thresholds and refinement PDF p. 13
  2. 3.Elementary estimates for refined series PDF p. 21
  3. 4.Surface calculations and their normalization PDF p. 22
  4. 5.Identifying the section spaces and taking limits PDF p. 25
  5. 6.Keeping information from more than one divisor PDF p. 30

Part II · Family 2.5: cubic-surface fibrations

  1. 7.Geometry and the two remaining local cases PDF p. 33
  2. 8.The D5 point and the cubic section ring PDF p. 34
  3. 9.The E6 point PDF p. 44

Part III · Family 2.9: conic bundles

  1. 10.Geometry and the local estimates PDF p. 48
  2. 11.Expected vanishing orders on the threefold PDF p. 51
  3. 12.Refining to a cubic surface and then to a curve PDF p. 52
  4. 13.The limitations of the fibre curve flag PDF p. 57
  5. 14.Refinement by the exceptional plane of a point blow-up PDF p. 58
  6. 15.Blowing up the first base curve PDF p. 61
  7. 16.Complete decomposition in transverse position PDF p. 64
  8. 17.A uniform bound on the exceptional plane PDF p. 67
  9. 18.Weighted extraction of a double conic fibre PDF p. 73

Part IV · Family 2.10: quartic del Pezzo fibrations

  1. 19.Geometry and the local estimates PDF p. 81
  2. 20.The section ring and its filtrations PDF p. 82
  3. 21.Global reduction and nodal auxiliary sections PDF p. 85
  4. 22.The exceptional-surface estimates PDF p. 89

Part V · Family 2.11: blow-ups of cubic threefolds along lines

  1. 23.Geometry and the family theorem PDF p. 104
  2. 24.Three volume integrals and the reduced-fibre flags PDF p. 105
  3. 25.The special reduced fibre for second-kind lines PDF p. 107
  4. 26.Refinement by the exceptional divisor of the original blow-up PDF p. 109
  5. 27.The weighted extraction of a double fibre PDF p. 110
  6. 28.The refined series on the weighted exceptional surface PDF p. 112
  7. 29.A weighted extraction at a ramification point PDF p. 116
  8. 30.Exclusion of all centres and completion of the proof PDF p. 121
  9. 31.Comparison with a degree-two surface flag PDF p. 122

Part VI · Family 2.12: intersections of three bilinear divisors

  1. 32.Geometry and the reduction to point centres PDF p. 124
  2. 33.The ordinary point filtration PDF p. 125
  3. 34.Monomial degenerations and exact moment certificates PDF p. 129
  4. 35.Three cases on the ordinary exceptional plane PDF p. 131
  5. 36.The weighted extraction at a repeated residual direction PDF p. 136
  6. 37.Strict adjunction and completion of the proof PDF p. 141

Part VII · Family 2.13: genus-two sextics on quadrics

  1. 38.The scroll and the conic bundle PDF p. 143
  2. 39.Expected vanishing and quartic sections PDF p. 145
  3. 40.Reduced conic fibres PDF p. 147
  4. 41.Double conic fibres PDF p. 154
  5. 42.Completion of the proof for Family 2.13 PDF p. 163

Part VIII · Family 2.14: the elliptic quintic on the quintic del Pezzo threefold

  1. 43.The quintic del Pezzo pencil PDF p. 164
  2. 44.Fibre refinements and reduction of centres PDF p. 168
  3. 45.Ordinary lines in singular fibres PDF p. 170
  4. 46.Lines of the second kind: the ruled surface F2 PDF p. 173
  5. 47.The remaining lines of the second kind PDF p. 177
  6. 48.Ordinary double points of the fibres PDF p. 182
  7. 49.Points of type A2 PDF p. 186
  8. 50.Points of type A3 PDF p. 188
  9. 51.The point of type A4 PDF p. 192
  10. 52.Completion of the proof for Family 2.14 PDF p. 197

Part IX · Family 2.16: conics on intersections of two quadrics

  1. 53.The conic bundle and the remaining fibres PDF p. 198
  2. 54.Extracting a double conic fibre PDF p. 200
  3. 55.Expected orders and the surface threshold PDF p. 202
  4. 56.Completion for Family 2.16 PDF p. 205
  5. 57.A second proof for Family 2.16: the CM polarization PDF p. 207
  6. 58.Hilbert–Mumford inequalities for smooth pairs PDF p. 209
  7. 59.Comparing a GIT-stable filling with a K-polystable filling PDF p. 211
  8. 60.Completion of the GIT proof PDF p. 213

Part X · Family 2.17: line extractions and weighted points

  1. 61.K-stability of Family 2.17 PDF p. 214
  2. 62.Blowing up a contracted line PDF p. 215
  3. 63.Three refined surface calculations for Family 2.17 PDF p. 220
  4. 64.The other exceptional fibre PDF p. 224
  5. 65.Reduction to the triple-contact points PDF p. 230
  6. 66.The weighted point extraction and its actual section ring PDF p. 234
  7. 67.Binary spaces on the weighted exceptional surface PDF p. 237
  8. 68.The local threshold and the proof for Family 2.17 PDF p. 244

Part XI · Family 2.20: the K-polystable classification

  1. 69.The classification problem and the results proved here PDF p. 245
  2. 70.Pencils of binary quartics PDF p. 247
  3. 71.Points on smooth fibres of the conic bundle PDF p. 250
  4. 72.Reduced singular conic fibres: exact surface estimates PDF p. 254
  5. 73.The global conic-bundle model and double fibres PDF p. 256
  6. 74.The weighted moment at a double fibre in two explicit members PDF p. 261
  7. 75.The complete refinement at a double fibre PDF p. 265
  8. 76.A discriminant test for the parameter orbit PDF p. 270
  9. 77.Blowing up a component of a reduced fibre PDF p. 271
  10. 78.Special lines and points away from the fibre node PDF p. 276
  11. 79.Reduction to finitely many points and a triangle example PDF p. 281
  12. 80.The full refinement at a reduced-fibre node on E PDF p. 283
  13. 81.Surface flags at the remaining nodes PDF p. 288
  14. 82.The complete classification PDF p. 292

Part XII · Family 3.2: double conics in a projective bundle

  1. 83.The statement for Family 3.2 PDF p. 295
  2. 84.The distinguished surface and a double conic PDF p. 295
  3. 85.Reduction to the non-reduced conics PDF p. 297
  4. 86.The actual refined sections and their expected orders PDF p. 300
  5. 87.The refined threshold on the exceptional surface PDF p. 303
  6. 88.Completion of the proof for Family 3.2 PDF p. 305

Part XIII · Family 3.5: K-semistability and the complete smooth classification

  1. 89.Geometry and the scope of the result PDF p. 306
  2. 90.Two divisor refinements PDF p. 307
  3. 91.The totally ramified cases PDF p. 309
  4. 92.An audit of the singular-fibre refinement PDF p. 310
  5. 93.Weighted point extractions and their exact first moments PDF p. 312
  6. 94.The exceptional surface at an index-three point PDF p. 314
  7. 95.The exceptional surface at an index-four point PDF p. 321
  8. 96.Other centres on the singular fibres PDF p. 324
  9. 97.The mixed fibre PDF p. 329
  10. 98.The index-two endpoint of the mixed fibre PDF p. 330
  11. 99.A weighted extraction of the joining line PDF p. 334
  12. 100.The point extraction at an interior point PDF p. 337
  13. 101.Completion of the smooth classification PDF p. 342
  14. 102.Liu’s method: the CM line, GIT, and the reconstruction problem PDF p. 342

Part XIV · Family 3.6: double conics and weighted curve extractions

  1. 103.Geometry and the statement for Family 3.6 PDF p. 351
  2. 104.Reduction to the double conics PDF p. 352
  3. 105.The weighted extraction and its actual section ring PDF p. 356
  4. 106.The refined estimate on the exceptional surface PDF p. 359
  5. 107.Completion of the proof for Family 3.6 PDF p. 371

Part XV · Family 3.7: sextic fibres, quintic sections, and point extractions

  1. 108.Statement and method PDF p. 372
  2. 109.The pencil of sextic del Pezzo surfaces PDF p. 372
  3. 110.Fibre refinements and the remaining centres PDF p. 375
  4. 111.Quintic del Pezzo sections PDF p. 378
  5. 112.The singular point of an A1 fibre PDF p. 383
  6. 113.The singular point of an A2 fibre PDF p. 386
  7. 114.Points on the original exceptional divisor PDF p. 390
  8. 115.The smooth corner of an A1 fibre PDF p. 395
  9. 116.Proof for every smooth member PDF p. 397

Part XVI · Family 3.8

  1. 117.The two blow-ups and the quartic pencil PDF p. 398
  2. 118.Pencils, ramification, and geometric invariant theory PDF p. 399
  3. 119.The total ramification cases PDF p. 401
  4. 120.Anticanonical sections and the distinguished surface PDF p. 402
  5. 121.Reduction to exceptional rulings and joining lines PDF p. 405
  6. 122.The point of a fibre with ramification index two PDF p. 409
  7. 123.The point of a fibre with ramification index three PDF p. 412
  8. 124.The exceptional ruling over an index-two point PDF p. 416
  9. 125.The exceptional ruling over an index-three point PDF p. 422
  10. 126.The joining line of a fibre of type (2,2) PDF p. 423
  11. 127.The joining line in a fibre of type (3,1) PDF p. 427
  12. 128.The complete classification PDF p. 431

References

  1. References PDF p. 433