Canonical bundle | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

In mathematics, the canonical bundle of a non-singular algebraic variety of dimension over a field is the line bundle , which is the nth exterior power of the cotangent bundle on . Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle . Equivalently, it is the line bundle of holomorphic n-forms on . This is the dualising object for Serre duality on . It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor on giving rise to the canonical bundle — it is an equivalence class for linear equivalence on , and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor − with canonical. The anticanonical bundle is the corresponding inverse bundle . When the anticanonical bundle of is ample, is called a Fano variety. Adjunction formula Suppose that X is a smooth variety and that D is a smooth divisor on X. The adjunction formula relates the canonical bundles of X and D. It is a natural isomorphism In terms of canonical classes, it is This formula is one of the most powerful formulas in algebraic geometry. An important tool of modern birational geometry is inversion of adjunction, which allows one to deduce results about the singularities of X from the singularities of D. On a singular variety , there are several ways to define the canonical divisor. If the variety is normal, it is smooth in codimension one. In particular, we can define canonical divisor on the smooth locus. This gives us a unique Weil divisor class on . It is this class, denoted by that is referred to as the canonical divisor on Alternately, again on a normal variety , one can consider , the 'th cohomology of the normalized dualizing complex of . This sheaf corresponds to a Weil divisor class, which is equal to the divisor class defined above. In the absence of the normality hypothesis, the same result holds if is S2 and Gorenstein in dimension one. If the canonical class is effective, then it determines a rational map from V into projective space.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (37)

Related people (4)

Zsolt Patakfalvi, Maciej Emilian Zdanowicz, Claudio Scrucca, Francesca Carocci

Related units (2)

Related concepts (20)

Ample line bundle

In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety X amounts to understanding the different ways of mapping X into projective space.

Divisor (algebraic geometry)

In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-r subvariety need not be definable by only r equations when r is greater than 1.

Del Pezzo surface

In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general type, whose canonical class is big. They are named for Pasquale del Pezzo who studied the surfaces with the more restrictive condition that they have a very ample anticanonical divisor class, or in his language the surfaces with a degree n embedding in n-dimensional projective space , which are the del Pezzo surfaces of degree at least 3.

Related courses (5)

MATH-473: Complex manifolds

The goal of this course is to help students learn the basic theory of complex manifolds and Hodge theory.

MATH-535: Algebraic geometry III - selected topics

This course is aimed to give students an introduction to the theory of algebraic curves, with an emphasis on the interplay between the arithmetic and the geometry of global fields. One of the principl

MATH-410: Riemann surfaces

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex

Related lectures (33)

Ample line bundle

Divisor (algebraic geometry)

Del Pezzo surface

Albanese and Fourier Transform: Positive Characteristic

Explores Albanese varieties, Fourier-Mukai transforms, and positive characteristic applications.

Modular Forms: Dimension Formula MATH-511: Number theory II.a - Modular forms

Explores modular forms, discussing pullback maps, meromorphic differentials, and the Riemann-Roch theorem.

Varieties with nef anti-canonical: Surjective Albanese

Presents a proof that smooth projective varieties with nef anti-canonical divisor have surjective Albanese morphism.

The Nonvanishing Problem for varieties with nef anticanonical bundle

Nikolaos Tsakanikas

We prove that if (X, A) is a threefold pair with mild singularities such that -(KX + A) is nef, then the numerical class of -(KX + A) is effective. ...

Berlin2023

Arithmetic and geometric deformations of threefolds

Stefano Filipazzi, Fabio Bernasconi

We show that mixed-characteristic and equicharacteristic small deformations of 3-dimensional canonical (resp., terminal) singularities with perfect residue field of characteristic p>5 are canonical (resp., terminal). We discuss applications to arithmetic a ...

Hoboken2023

A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves

Francesca Carocci

We construct a modular desingularisation of (M) over bar (2,n)(P-r, d)(main). The geometry of Gorenstein singularities of genus two leads us to consider maps from prestable admissible covers; with this enhanced logarithmic structure, it is possible to desi ...

GEOMETRY & TOPOLOGY PUBLICATIONS2023