Euler characteristic across a Cartier divisor

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OAI.MaximalSeshadri.Geometry.cartier_euler_difference

by wurtle · Nov 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

maximal-seshadrioai-math

Let $S$ be a surface with ample $A$ , $M,N$ line bundles, $\varphi:M\to N$ a monomorphism, and $I$ an ideal sheaf whose subscheme is integral of dimension one and which is locally generated by the coefficient of $\varphi$ . Then $\chi(N)-\chi(M)=\chi(I\text{-subscheme},N|_{I})$ , with the two-term Euler characteristic on $S$ and the one-term one on the curve.

Notation. $\chi$ is eulerCharacteristic: the alternating sum of $\mathbb C$ -dimensions of $\operatorname{Ext}^n(\mathcal O,-)$ , for $n\le2$ on a surface and $n\le1$ on a curve. IsAmple means that for each point $x$ and open $V\ni x$ , some positive power has a global section with nonvanishing open containing $x$ , inside $V$ , and affine. A Surface is an integral scheme smooth of relative dimension 2 over $\operatorname{Spec}\mathbb C$ with a closed immersion into some complex projective space over $\mathbb C$ .

Formalization Note The statement is verbatim from OpenAI's Lean development, with Definitions files of this development imported in place of the upstream modules; these are supporting code for the statement.

Preamble

import Mathlib
import Definitions.Def_MaximalSeshadriConstants
import Definitions.Def_SeshadriPrelim6

namespace OAI

section

namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.Projective

Formal statement

theorem cartier_euler_difference (S : Surface) (A : LineBundle S.scheme) (hA : A.IsAmple)
    (M N : LineBundle S.scheme) (φ : M.sheaf ⟶ N.sheaf) [Mono φ]
    (I : S.scheme.IdealSheafData) [IsIntegral I.subscheme]
    (hd : topologicalKrullDim I.subscheme = 1)
    (heq : ∀ x : S.scheme, ∃ U : S.scheme.affineOpens, x ∈ U.1 ∧
      ∃ e : M.sheaf.restrict U.1.ι ≅ O U.1.toScheme,
      ∃ d : N.sheaf.restrict U.1.ι ≅ O U.1.toScheme,
        I.ideal U = Ideal.span {U.1.topIso.hom
          (endValue (e.inv ≫ (Scheme.Modules.restrictFunctor U.1.ι).map φ ≫ d.hom))}) :
    eulerCharacteristic S.structureMap 2 N.sheaf - eulerCharacteristic S.structureMap 2 M.sheaf =
      eulerCharacteristic (I.subschemeι ≫ S.structureMap) 1
        ((Scheme.Modules.pullback I.subschemeι).obj N.sheaf) := by
  sorry

end
end MaximalSeshadri.Geometry
end
end OAI

Source

OpenAI, Maximal Seshadri constants on arbitrary polarized surfaces, September 23, 2026, https://github.com/openai/math (paper: preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026/main.pdf; Lean: lean/OAI/AlgebraicGeometry/Seshadri, Apache License 2.0); Lean declaration `OAI.MaximalSeshadri.Geometry.cartier_euler_difference`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Intersection/CartierEuler.lean#L18

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