CONIFOLD LAB
Interactive geometry
A CALABI–YAU TRANSITION LABORATORY

Calabi–Yau Geometry at the Threshold.

A sphere disappears. Another takes its place. The global story lives in the relations.

3COMPLEX DIMENSIONS
6 real dimensions
SMOOTHINGS³ via a stereographic Hopf-fibre picture
THE SURGERY CORE

A three-sphere waiting to vanish.

Each linked circle is a Hopf fibre on S³, projected into R³.

Drag to orbit · use sliders below for keyboard control
smoothing / S³ resolution / S²
complex parameter μ decreases →← exceptional area A decreases
01

Keep the vanishing cycle.

The smoothing is locally diffeomorphic to . Its zero-section is the Lagrangian three-sphere.

02

Pass through the node.

The core contracts. The is where smooth topology can change.

03

Keep an exceptional curve.

A replaces the point by P¹ ≅ S². Its normal bundle is O(−1) ⊕ O(−1).

∂
THE IDEA BEHIND THE PICTURE

Local models tell you what can change.
Relations tell you whether the global geometry allows it.

FROM PICTURE TO PAPER

Keep the geometry.
Follow the argument.

Smoothability of Calabi–Yau threefolds

Full series & sources ↗
01 · LOCAL SINGULARITIES Non-smoothable Calabi–Yau threefolds from reflexive polytopes Bernd Johannes Wuebben · PDF ↗ 02 · DEFORMATIONS OF TORIC PAIRS Deformations of toric pairs and the smoothing of Batyrev Calabi–Yau threefolds Bernd Johannes Wuebben · PDF ↗ 03 · ISOLATED MIRROR PAIRS Doubly isolated Batyrev mirror pairs and non-smoothable Calabi–Yau threefolds Bernd Johannes Wuebben · PDF ↗ 04 · THE VANISHING-CYCLE OBSTRUCTION A vanishing-cycle obstruction to smoothing Calabi–Yau threefolds Bernd Johannes Wuebben · PDF ↗ 05 · THE SMOOTHING CRITERION Smoothing Calabi–Yau threefolds with nodes and del Pezzo cone points Bernd Johannes Wuebben · PDF ↗ 06 · MIRROR CONIFOLD RELATIONS Mirror Symmetry for Conifold Relations in Calabi–Yau Threefolds Bernd Johannes Wuebben · PDF ↗
A FEW GEOMETRIC WORDS

Objects before jargon.
MATHEMATICAL FRAMING

A precise model, with visible limits.

A Calabi–Yau threefold has complex dimension three and real dimension six. The canvas shows core cycles, projections, and an explicitly specified real slice. It does not render an entire compact threefold or compute a Ricci-flat metric.

The local model

Σ zⱼ² = μ,   z = x + iy

For real μ > 0, x·y = 0 and |x|² − |y|² = μ. Set u = x/|x|: y is tangent to the unit S³, giving the cotangent-bundle topology. The zero-section y = 0 is the vanishing sphere of radius √μ in the quadric coordinates.

The slice sets z₁ = u, z₂ = v, z₃ = iw, z₄ = 0, with u,v,w real; it satisfies u² + v² − w² = μ. At μ = 0 its drawing is a real two-dimensional double cone, not the six-dimensional conifold.

The small resolution is modelled by det A = xy − zw = 0, A = [[x,z],[w,y]], together with a kernel line or cokernel line in P¹. At A = 0 the fibre is all of P¹. The area A in the control is a separate Kähler parameter; it is not a continuation of the complex smoothing μ.

The animations

The topology movie depicts local exceptional curves collapsing to nodes, followed by vanishing spheres appearing on the smoothing. Endpoint Hodge numbers remain integers and are not interpolated. The tropical movie traces the belt, sketches the chain lifts and highlights coefficient support. Moving dots and changing diagram depth are visual guides, not a computed metric or a continuous deformation between mirror spaces.

The projections

The S³ view samples actual Hopf fibres on the unit sphere in C² and stereographically projects them into R³, scaling the picture by √μ. Only a finite collection of fibres is drawn. The S² view displays the exceptional core with latitude/longitude curves. Camera perspective is a visual aid.

Example data and attribution

Bernd Johannes Wuebben, Tropical 2-cycles and relations among vanishing spheres in Batyrev mirror families, manuscript dated September 25, 2026; §11.1, §11.2 and Table 1 of the manuscript. X₉’s circuit matrix and all three example bases are transcribed from those sections. Diagram layouts are schematic and do not reproduce the complete affine base or prove the lifting theorem. Relative tropical cycles supply the mirror 4-chain. The explicit tropical 3-chain lift on the resolution uses the stricter CBM cycle conditions; membership in K always gives an abstract homology relation there. The manuscript’s stated results are illustrated under its hypotheses, rather than independently verified here.

X₉ uses direct coefficients to test the circuit equation. X₁₉ and X₂₀ use combinations of the listed basis vectors, so those selected vectors always lie in K. A zero in one selected vector is different from a coordinate forced to be zero throughout K.

Primary references

  1. Smith–Thomas–Yau · Symplectic conifold transitions ↗
  2. Rossi · Geometric transitions ↗
  3. Castaño-Bernard–Matessi · Conifold transitions via affine geometry and mirror symmetry ↗
  4. Lee–Lin–Wang · Towards A + B theory in conifold transitions ↗