The canonical bundle is a holomorphic line bundle on a complex manifold which is determined
by its complex structure . On a coordinate
chart nonvanishing bundle
section transition function between coordinate
charts is given by the determinant of the Jacobian of the coordinate
transformation .
The canonical bundle is defined in a similar way to the holomorphic tangent bundle . In fact, it is the exterior power of the
dual bundle to the holomorphic
tangent bundle .
The canonical-ampleness conjecture asks whether this bundle is ample on every compact connected Brody hyperbolic Kähler manifold
of positive complex dimension . The hyperbolicity hypothesis
means that every entire curve is constant. It does
not require a metric with negative holomorphic
sectional curvature .
See also Brody Hyperbolic Manifold ,
Canonical Divisor ,
Fujita
Freeness Conjecture ,
Kobayashi
Canonical-Ampleness Conjecture ,
Kodaira Dimension
Portions of this entry contributed by Todd
Rowland
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References Demailly, J.-P. Complex Analytic and Differential Geometry. June 21, 2012. https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf . Diverio,
S. "Kobayashi Hyperbolicity, Negativity of the Curvature and Positivity of the
Canonical Bundle." In Hyperbolicity Properties of Algebraic Varieties
(Ed. S. Diverio). Paris: Soc. Math. France, pp. 209-252, 2021. https://arxiv.org/abs/2011.11379 . Referenced
on Wolfram|Alpha Canonical Bundle
Cite this as:
Weisstein, Eric W. , with contributions by Todd Rowland . "Canonical Bundle." From MathWorld https://mathworld.wolfram.com/CanonicalBundle.html
Subject classifications
,
it is spanned by the nonvanishing bundle
section 
.
The transition function between coordinate
charts is given by the determinant of the Jacobian of the coordinate
transformation.
th exterior power of the
dual bundle to the holomorphic
tangent bundle.
