The tilt slope and the Bayer–Macrì–Toda quantity on a threefold #
At n = 3 the twisted Chern character gives the three twisted degrees
∫H²·ch₁^β, ∫H·ch₂^β and ∫ch₃^β, and with them the tilt slope ν_{α,β}
and the quantity
Q_{α,β}(E) = α²·Δ_H(E) + 4(H·ch₂^β(E))² − 6(H²·ch₁^β(E))·(ch₃^β(E))
whose nonnegativity on tilt-semistable objects is the Bayer–Macrì–Toda conjecture.
This is a conjecture, and it is false in general #
Read this before consuming BMTData for anything.
The inequality is Conjecture 1.3.1 of Bayer–Macrì–Toda, Bridgeland stability conditions on threefolds I: Bogomolov–Gieseker type inequalities, J. Algebraic Geom. 23 (2014) 117–163. It appears as Conjecture 4.1 of Bayer–Macrì–Stellari, The space of stability conditions on abelian threefolds, and on some Calabi–Yau threefolds, Invent. Math. 206 (2016) 869–933 — the numbering used by the papers that assume it per-threefold.
It is known for:
| case | reference |
|---|---|
ℙ³ | Macrì, Algebra Number Theory 8 (2014) 173–190 |
the smooth quadric hypersurface in ℙ⁴ | Schmidt |
| all Fano threefolds of Picard rank one | Li |
| abelian threefolds | Maciocia–Piyaratne; Bayer–Macrì–Stellari |
And it is false on the blow-up of ℙ³ at a point — Schmidt,
Counterexample to the Generalized Bogomolov–Gieseker Inequality for
Threefolds, Int. Math. Res. Not. IMRN 2017, no. 8, 2562–2566.
So BMTData is not merely unproved here. It is unprovable as stated for all
threefolds, and nothing in this repository will ever inhabit it in general. A
consumer must take it as a hypothesis about one specific threefold and must
never treat it as a fact.
Contrast this with BogomolovGiesekerData. That structure is also supplied
rather than proved, but what it supplies is true — the obstruction there is
that the proof runs through restriction theorems this repository does not have.
Here the obstruction is that the statement is false. The two look alike in Lean
and are entirely different in status, and that difference is the reason this
section exists.
TiltSemistable is opaque #
For the same reason Semistable is in BogomolovGieseker.lean, and one step
further out: tilt-semistability is a property of an object of a tilted heart,
and the numerical layer has neither hearts nor objects. It is a field.
Main results #
The three twisted degrees #
∫_X H²·ch₁^β(E).
Equations
- AlgebraicGeometry.Numerical.Threefold.degH1Beta V P β E = V.ring.degree (AlgebraicGeometry.Numerical.chBetaComp V P β E 1 * P.cls ^ 2)
Instances For
∫_X H·ch₂^β(E).
Equations
- AlgebraicGeometry.Numerical.Threefold.degH2Beta V P β E = V.ring.degree (AlgebraicGeometry.Numerical.chBetaComp V P β E 2 * P.cls)
Instances For
∫_X ch₃^β(E).
Equations
- AlgebraicGeometry.Numerical.Threefold.deg3Beta V P β E = V.ring.degree (AlgebraicGeometry.Numerical.chBetaComp V P β E 3)
Instances For
The H-twisted discriminant on a threefold,
Δ_H = (H²·ch₁^β)² − 2(H³·ch₀)(H·ch₂^β).
ch₀ carries no β: the twist sums over j ≤ 0 in degree zero, so
ch₀^β = ch₀ = rank.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The threefold tilt discriminant is the Mukai self-pairing, at
coefficient ring ℚ with the multiplication of ℚ as the bilinear form.
The rank slot carries ∫H³, the codimension-three analogue of the ∫H² in
Surface.discrH_eq_selfPairing. The triple is the three twisted degrees at
β, and discrHBeta_beta_inert below says the value does not depend on which
β that is.
The tilt slope ν_{α,β}(E) = (H·ch₂^β − (α²/2)·H³·ch₀) / (H²·ch₁^β).
Junk where H²·ch₁^β vanishes, the same convention slopeH uses at rank
zero.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The Bayer–Macrì–Toda quantity
Q_{α,β} = α²·Δ_H + 4(H·ch₂^β)² − 6(H²·ch₁^β)(ch₃^β).
Defining it is all this does. Whether it is nonnegative is BMTData, and see
the module docstring before assuming so.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The β = 0 and α = 0 specialisations #
β is inert in the discriminant #
The two twisted degrees each move with β, and the discriminant is the one
combination of them that does not. degH1Beta_eq_sub and degH2Beta_eq_sub
are the two expansions; discrHBeta_beta_inert is the cancellation.
∫H²·ch₁^β = ∫H²·ch₁ − β·∫H³·ch₀.
∫H·ch₂^β = ∫H·ch₂ − β·∫H²·ch₁ + (β²/2)·∫H³·ch₀.
β is inert in the tilt discriminant.
Each twisted degree moves with β — see the two expansions above — and the
combination (∫H²ch₁^β)² − 2(∫H³ch₀)(∫H·ch₂^β) is the one in which every β
term cancels: the linear terms cancel against each other and the quadratic ones
against the β²/2 in ch₂^β.
This is the threefold instance of the B-twist invariance
ChernCharacter.discriminant_twist already records on the divisorial side.
Q written out in untwisted components.
At α = 0 the tilt slope is the plain ratio of the two twisted degrees.
At α = 0, Q loses its discriminant term.
The conjecture, supplied and uninhabited #
The Bayer–Macrì–Toda inequality, as supplied data.
Carries a choice of tilt-semistability predicate as well as a proposition, so
…Data rather than …Statement, matching BogomolovGiesekerData.
It is false in general. See the module docstring: the conjecture fails on
the blow-up of ℙ³ at a point (Schmidt, IMRN 2017). This structure is therefore
not merely unproved here but unprovable as stated for all threefolds, and
nothing in this repository inhabits it. Any consumer must hypothesise it for one
specific threefold.
That is the difference from BogomolovGiesekerData, which is unproved here but
true.
Tilt-semistability at parameters
(α, β), supplied.Opaque for the same reason
Semistableis, and one step further out: this is a property of an object of a tilted heart, and the numerical layer has neither hearts nor objects.The conjectural inequality. Supplied, not proved, and false for some threefolds — see the structure's docstring.