Canonical line bundles and Serre duality #
On an integral separated Noetherian curve with discrete valuation rings at its closed points, the divisor of a nonzero Weil differential determines a line bundle. When the curve has all the places of its function field and the ground field is algebraically closed in that function field, this line bundle is independent of the differential up to isomorphism.
This file constructs a representative InvertibleSheaf.canonicalBundle of that class and
transports the divisor-sheaf duality to arbitrary line bundles:
H¹(X, L)ᵛ ≃ H⁰(X, ω ⊗ L⁻¹) and H⁰(X, L)ᵛ ≃ H¹(X, ω ⊗ L⁻¹).
The equivalence is asserted as an existence result: no canonical trace or naturality is claimed.
The construction uses Weil differentials, not a relative dualizing complex, and does not yet
identify the canonical bundle with the sheaf of Kähler differentials or provide base change.
The degree of the canonical bundle is 2g - 2 on a proper curve. These results allow line-bundle
cohomology to be used without choosing divisor presentations in the statements.
References #
- J.-P. Serre, Algebraic Groups and Class Fields, Chapter II, Section 5.
- R. Hartshorne, Algebraic Geometry, Chapter III, Corollary 7.7, and Chapter IV, Section 1.
A canonical line bundle, obtained by choosing a divisor in the canonical class of Weil
differentials and forming its divisor sheaf. Its isomorphism class is characterized by
mk_canonicalBundle_eq_iff.
Equations
- One or more equations did not get rendered due to their size.
Instances For
A divisor represents the canonical line-bundle class precisely when its function-field divisor represents the canonical class of Weil differentials.
The canonical bundle is isomorphic to the sheaf of every canonical divisor.
Serre duality for line bundles. The dual of H¹(X, L) is linearly isomorphic to
H⁰(X, ω ⊗ L⁻¹) for any representative ω of the canonical line-bundle class.
The existence statement does not select a canonical pairing.
Serre duality in degree zero. The dual of H⁰(X, L) is linearly isomorphic to
H¹(X, ω ⊗ L⁻¹) for any representative ω of the canonical class.
The dimensions in Serre duality agree for every line bundle and every representative of the canonical class.
On a proper curve, the Euler-characteristic degree of the canonical line bundle is
2g - 2, where g = dim H¹(X, 𝒪_X).