Numerical Chern-character data on surfaces #
This file constructs the surface components which can be recovered from determinants, intersection numbers, and Euler characteristics without introducing a Chow group.
For a two-term perfect determinant package, the virtual rank is the difference of the ranks in
the resolution and c₁ is its determinant class in Pic X. An IntersectionContext in
dimension two supplies the structure-sheaf Euler function and the symmetric Picard intersection
pairing. These determine the degree-level Todd terms by
td₁ · L = χ(L) - χ(O_X) - L² / 2, deg td₂ = χ(O_X),
and hence
deg ch₂(F) = χ(F) - rank(F) deg td₂ - td₁ · c₁(F).
The output is deliberately degree-level in codimension two. Recovering an element of an
arbitrary Layer A ring A² from its degree requires an explicit representability or
nondegeneracy hypothesis; no such assumption is hidden here.
As in Divisors.Determinant, arbitrary coherent sheaves are not silently declared perfect.
The API applies to coherent sheaves carrying explicit two-term perfect determinant data, and
rank additivity for three independently chosen resolutions remains an explicit hypothesis in
the short-exact theorem.
Rank and first Chern class #
The virtual rank of a coherent sheaf presented by a two-term finite locally free resolution.
Equations
Instances For
The determinant first Chern class, in the additive form of the Picard group.
Equations
Instances For
The numerical first Chern class as the functional obtained by intersecting the determinant class with every Picard class.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Todd pairings extracted from the structure-sheaf polynomial #
The degree-one Todd contribution paired with a Picard class:
td₁ · L = χ(L) - χ(O_X) - L²/2.
The polarization identity for the degree-two Picard Euler function proves that this expression
is additive in L.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The top Todd degree on a surface is the Euler characteristic of its structure sheaf.
Equations
Instances For
Surface Chern character #
The numerical degree of ch₂(F), recovered from Euler characteristic, rank, determinant,
and the structure-sheaf Todd data.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The three surface Chern-character coordinates recovered without a Chow ring.
- rank : ℤ
- first : Additive (Scheme.Modules.Pic X)
- secondDegree : ℚ
Instances For
Package the virtual rank, determinant first Chern class, and degree of ch₂.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The defining surface Riemann--Roch identity for the recovered degree-level components.
Isomorphism invariance and exact-sequence additivity #
Numerical first Chern classes are additive for a determinant-compatible short exact sequence.
Picard-valued first Chern classes are additive for a determinant-compatible short exact sequence.
ch₂ is additive on a determinant-compatible coherent short exact sequence once the
independently chosen two-term resolutions are also known to have additive virtual ranks.
Structure sheaves and line bundles #
The structure sheaf has ch₂ = 0 for any explicit determinant presentation with rank one
and trivial determinant class.
The Picard Euler function takes the trivial class to the actual Euler characteristic of the structure sheaf.
The two equivalent descriptions of the top Todd degree: χ(O_X) and the Picard Euler
function at the trivial class.
A rank-one coherent sheaf whose determinant class is L and whose Euler characteristic is
the Picard Euler value of L has the usual line-bundle formula deg ch₂ = L²/2.
The Euler-value hypothesis is supplied, for actual representatives in the intersection context,
by its TwistContext.realization field and is explicit for any independently chosen coherent
representative.
Discriminant and Layer A compatibility #
The degree of the surface discriminant Δ(F) = c₁(F)² - 2 rank(F) ch₂(F).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Compatibility with the degree of the Layer A numerical discriminant. The hypotheses state
exactly that a proposed Layer A class realizes the rank, c₁², and degree-level ch₂
constructed here.