Quadratic formula for Euler characteristics of powers

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OAI.MaximalSeshadri.Geometry.Surface.arbitrary_power_euler_quadratic

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

maximal-seshadrioai-math

For $L$ ample on $S$ and any line bundle $M$ : $2\chi(M^n)=n(n-1)M^2+2n(\chi(M)-\chi(\mathcal O_S))+2\chi(\mathcal O_S)$ .

Notation. $L^2$ is selfIntersection, namely $\chi(L^2)-2\chi(L)+\chi(\mathcal O_S)$ . $\chi$ 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 TopologicalSpace
open MaximalSeshadri.Frames MaximalSeshadri.Projective

Formal statement

theorem Surface.arbitrary_power_euler_quadratic (S : Surface)
    (L : LineBundle S.scheme) (hL : L.IsAmple) (M : LineBundle S.scheme) (n : ℕ) :
    2*eulerCharacteristic S.structureMap 2 (M.pow n).sheaf =
      (n : ℤ)*((n : ℤ)-1)*selfIntersection S M +
      2*(n : ℤ)*(eulerCharacteristic S.structureMap 2 M.sheaf -
        eulerCharacteristic S.structureMap 2 (O S.scheme)) +
      2*eulerCharacteristic S.structureMap 2 (O S.scheme) := 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.Surface.arbitrary_power_euler_quadratic`, https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Intersection/GeneralSurfaceEuler.lean#L20

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