Documentation

DerivedAlgGeo.AlgebraicGeometry.Numerical.Models.Fourfold.CalabiYau

The sextic Calabi–Yau fourfold #

A smooth sextic hypersurface X ⊂ ℙ⁵, and the model that makes AlgebraicGeometry.Numerical.CalabiYauFourfold inhabited.

The data: ∫_X H⁴ = 6, K_X = 0, and the Chern classes of a degree-six hypersurface in ℙ⁵, which the normal-bundle sequence 0 → T_X → T_{ℙ⁵}|_X → O_X(6) → 0 and the Whitney sum formula give as c(X) = (1+H)⁶/(1+6H), hence c₂ = 15H² and c₄ = 435H⁴. Adjunction is the c₁ half of that same computation: K_X = (K_{ℙ⁵} + 6H)|_X = 0. Hence

td(X) = 1 + 0 + (5/4)H² + 0 + (1/3)H⁴,

with td₂ = c₂/12 = 15/12 and td₄ = (3c₂² − c₄)/720 = (675 − 435)/720 = 1/3. The top coefficient is the one worth checking: ∫_X td₄ = 6 · (1/3) = 2, which is χ(O_X) for a Calabi–Yau fourfold, where h^{0,i} = 1, 0, 0, 0, 1. A threefold's χ(O_X) is 0; a fourfold's is 2, and that difference is the whole reason CalabiYauFourfold.IsCalabiYau carries a value rather than a vanishing.

The lattice, and why its Euler characteristics are the check #

As on the quintic, the linear-section generators are classical varieties whose Euler characteristics are known independently:

generatorwhat it isχ
[O_X]the Calabi–Yau fourfold2
[O_{X∩H}]a smooth sextic threefold in ℙ⁴, h^{3,0} = C(5,4) = 5−4
[O_{X∩H²}]a smooth sextic surface in ℙ³, χ(O) = C(5,3) + 111
[O_{X∩H³}]a plane sextic curve, genus 10−9
[O_pt]a point1

All five agree with what sexticChi returns, which is what makes td₂ and td₄ above checked numbers rather than transcribed ones.

Main definitions #

Main results #

References #

source
def AlgebraicGeometry.Numerical.Examples.sexticTodd :
ℕ → ℚ

The Todd coefficients of a sextic Calabi–Yau fourfold against the powers of H.

Equations
Instances For
    source
    def AlgebraicGeometry.Numerical.Examples.sexticChi :
    FourfoldNum →+ ℤ

    The Euler characteristic on the sextic in linear-section coordinates. See the module docstring for the five generator values this encodes.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      source
      @[reducible]
      noncomputable def AlgebraicGeometry.Numerical.Examples.sexticNumericalVariety :
      NumericalVarietyData 4 (RankOneRing 4) FourfoldNum

      The model. A smooth sextic fourfold in ℙ⁵: ∫H⁴ = 6, td = 1 + (5/4)H² + (1/3)H⁴.

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For
        source
        theorem AlgebraicGeometry.Numerical.Examples.sexticNumericalVariety_satisfiesHRR :
        sexticNumericalVariety.SatisfiesHRR

        Substituting the sextic Todd coefficients into the rank-one HRR polynomial recovers the integer-valued Euler formula in linear-section coordinates.

        source
        theorem AlgebraicGeometry.Numerical.Examples.sextic_isCalabiYau :
        CalabiYauFourfold.IsCalabiYau sexticNumericalVariety

        The model really is a Calabi–Yau fourfold: td₁ = td₃ = 0 and ∫_X td₄ = χ(O_X) = 2.

        The third condition is the one with content — it is the only place the degree ∫H⁴ = 6 and the Todd coefficient 1/3 have to agree — and it is why this file computes c₄ = 435H⁴ rather than normalising td₄ to whatever makes the answer come out.

        source
        theorem AlgebraicGeometry.Numerical.Examples.sexticChi_structureSheaf :
        sexticChi ![1, 0, 0, 0, 0] = 2

        χ(O_X) = 2 — the Calabi–Yau fourfold signature, read off the coordinates.

        source
        theorem AlgebraicGeometry.Numerical.Examples.sexticChi_hyperplaneSection :
        sexticChi ![0, 1, 0, 0, 0] = -4

        χ(O_{X∩H}) = −4: a hyperplane section of the sextic fourfold is a smooth sextic threefold in ℙ⁴, whose only nonzero Hodge number off the structure sheaf is h^{3,0} = C(d−1,4) = 5, so χ(O) = 1 − 5.

        source
        theorem AlgebraicGeometry.Numerical.Examples.sexticChi_surfaceSection :
        sexticChi ![0, 0, 1, 0, 0] = 11

        χ(O_{X∩H²}) = 11: a codimension-two linear section is a smooth sextic surface in ℙ³, with χ(O) = C(d−1,3) + 1 = 11.

        source
        theorem AlgebraicGeometry.Numerical.Examples.sexticChi_curveSection :
        sexticChi ![0, 0, 0, 1, 0] = -9

        χ(O_{X∩H³}) = −9: a codimension-three linear section is a plane sextic curve, of genus (6−1)(6−2)/2 = 10, so χ(O) = 1 − 10.

        source
        theorem AlgebraicGeometry.Numerical.Examples.sexticChi_point :
        sexticChi ![0, 0, 0, 0, 1] = 1

        χ(O_pt) = 1.

        source
        theorem AlgebraicGeometry.Numerical.Examples.sexticChi_lineBundle (m c e f : ℤ) (hc : 2 * c = m * (m + 1)) (he : 6 * e = m * (m + 1) * (m + 2)) (hf : 4 * f = m * (m + 1) * (m + 2) * (m + 3)) :
        4 * sexticChi ![1, m, c, e, f] = m ^ 4 + 15 * m ^ 2 + 8

        Riemann–Roch on the sextic reproduces χ(O_X(m)) = (m⁴ + 15m² + 8)/4.

        O(mH) is the class (1, m, c, e, f) with 2c = m(m+1), 6e = m(m+1)(m+2) and 4f = m(m+1)(m+2)(m+3). The denominator there is 4 rather than the 24 of ℙ⁴ because the point class is [pt] = H⁴/6, so f = 6·C(m+3,4) and 24·C(m+3,4) clears to 4f; it is an integer because a product of four consecutive integers is divisible by 24.

        At m = 1 this gives 6, which is h⁰(O_X(1)) = 6 for the sextic embedded in ℙ⁵; at m = 0 it gives 2 = χ(O_X).