The quintic Calabi–Yau threefold #
A smooth quintic hypersurface X ⊂ ℙ⁴, and the model that makes
AlgebraicGeometry.Numerical.CalabiYauThreefold inhabited. Until this file,
CalabiYauThreefold.chi_eq and CalabiYauThreefold.chi_eq_of_chComp_eq were conditional on
a NumericalVarietyData 3 A N satisfying IsCalabiYau existing at all, which the module
docstring of Numerical/Specializations/Threefold.lean recorded as the outstanding gap.
The data: ∫_X H³ = 5, K_X = 0, and ∫_X c₂(X)·H = 50, so
td(X) = 1 + 0 + (5/6)H² + 0.
The (5/6) is forced: td₂ = c₂/12 must satisfy ∫_X H·td₂ = 50/12, and ∫_X H³ = 5
turns that into the coefficient 50/(12·5) = 5/6. The top coefficient is zero because
td₃ = c₁c₂/24 = 0, i.e. χ(O_X) = 0 — the numerical signature of a Calabi–Yau threefold.
The lattice, and why its Euler characteristics are the check #
Linear-section coordinates (see Models/Threefold/LinearSection.lean) make χ an integer
by construction, and here they also make it verifiable, because each generator is a
classical variety with a classical Euler characteristic:
| generator | what it is | χ |
|---|---|---|
[O_X] | the Calabi–Yau threefold | 0 |
[O_{X∩H}] | a smooth quintic surface in ℙ³ | 5 |
[O_{X∩H²}] | a plane quintic curve, genus 6 | −5 |
[O_pt] | a point | 1 |
Those four numbers are computed independently of this file — C(d−1,3) + 1 = 5 for a degree
d = 5 surface in ℙ³, and 1 − g = 1 − 6 for a plane quintic — and they are exactly what
quinticChi returns. That agreement is what makes the Todd coefficient above a checked
number rather than a chosen one.
Main definitions #
quinticNumericalVariety— the model.
Main results #
quintic_isCalabiYau— the model satisfiesCalabiYauThreefold.IsCalabiYau.quinticChi_lineBundle—χ(O_X(m)) = 5m(m² + 5)/6, which is5atm = 1and0atm = 0, as the quintic's embedding inℙ⁴by the complete system|H|, withh⁰(O_X(1)) = 5, requires.
References #
- Hartshorne, Algebraic Geometry, I.7 and V.1
- Griffiths–Harris, Principles of Algebraic Geometry, ch. 4
The Todd coefficients of a quintic Calabi–Yau threefold against the powers of H.
Equations
Instances For
The Euler characteristic on the quintic in linear-section coordinates. See the module docstring for the four generator values this encodes.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The model. A smooth quintic threefold in ℙ⁴: ∫H³ = 5, td = 1 + (5/6)H².
Equations
- One or more equations did not get rendered due to their size.
Instances For
Substituting the quintic Todd coefficients into the rank-one HRR polynomial recovers the integer-valued Euler formula in linear-section coordinates.
The model really is a Calabi–Yau threefold: td₁ = 0 and ∫_X td₃ = χ(O_X) = 0.
With this presentation and its property witnesses, every theorem in the
CalabiYauThreefold namespace — in particular
chi_eq_of_chComp_eq, that χ cannot see the rank — is a statement about an object that
exists.
χ(O_X) = 0 — the Calabi–Yau signature, read off the coordinates.
χ(O_{X∩H}) = 5: a hyperplane section of the quintic is a smooth quintic surface in
ℙ³, and a degree-d surface in ℙ³ has χ(O) = C(d−1,3) + 1, which is 5 at d = 5.
χ(O_{X∩H²}) = −5: a codimension-two linear section of the quintic is a plane quintic
curve, of genus (5−1)(5−2)/2 = 6, so χ(O) = 1 − 6.
χ(O_pt) = 1.
Riemann–Roch on the quintic reproduces χ(O_X(m)) = 5m(m² + 5)/6.
O(mH) is the class (1, m, c, e) with 2c = m(m+1) and 6e = 5m(m+1)(m+2); the extra
factor of 5 relative to ℙ³ is the point class [pt] = H³/5. Stated cleared of
denominators to stay in ℤ.
At m = 1 this gives 5, which is h⁰(O_X(1)) = 5 for the quintic embedded in ℙ⁴; at
m = 0 it gives 0 = χ(O_X). Both are independent checks on the Todd coefficient.