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The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c0201201.png" /> of divisors, with respect to linear equivalence on an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c0201202.png" />, which are divisors of differential forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c0201203.png" /> of maximal degree. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c0201204.png" /> is a non-singular algebraic variety and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c0201205.png" />, then in local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c0201206.png" /> a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c0201207.png" /> can be written as
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The class $K_X$ of [[Divisor (algebraic geometry)|divisors]], with respect to linear equivalence on an [[algebraic variety]] $X$, which are divisors of [[differential form]]s $\omega$ of maximal degree. If $X$ is a non-singular algebraic variety of dimension $n$, then in local coordinates $x_1,\ldots,x_n$ a form $\omega$ can be written as
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$$
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\omega = f(x_1,\ldots,x_n) \, dx_1 \wedge \cdots \wedge dx_n \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c0201208.png" /></td> </tr></table>
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The divisor $(\omega)$ of $\omega$ is locally equal to the divisor $(f)$ of this rational function $f$. This construction does not depend on the choice of local coordinates and gives the divisor $(\omega)$ of $\omega$ on all of $X$. Since for any other form $\omega'$ of the same degree as $\omega$, $\omega' = g\omega$, it follows that $(\omega') = (g) + (\omega)$, and corresponding divisors are equivalent. The canonical class $K_X$ thus constructed is the first [[Chern class]] of the sheaf $\Omega_X^n$ of regular differential forms of degree $n$. Its numerical characteristics (degree, index, self-intersections, etc.) are effectively calculable invariants of the algebraic variety.
  
The divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c0201209.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012010.png" /> is locally equal to the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012011.png" /> of this rational function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012012.png" />. This construction does not depend on the choice of local coordinates and gives the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012014.png" /> on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012015.png" />. Since for any other form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012016.png" /> of the same degree as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012018.png" />, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012019.png" />, and corresponding divisors are equivalent. The canonical class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012020.png" /> thus constructed is the first [[Chern class|Chern class]] of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012021.png" /> of regular differential forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012022.png" />. Its numerical characteristics (degree, index, self-intersections, etc.) are effectively calculable invariants of the algebraic variety.
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If $X$ is a non-singular projective curve of genus $g$, then $\deg K_X = 2g-2$. For elliptic curves and, more generally, for Abelian varieties, $K_X = 0$. If $X$ is a non-singular hypersurface of degree $d$ in projective space $\mathbf{P}^n$, then $K_X = (d-n-1)H$, where $H$ is a hyperplane section of it.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012023.png" /> is a non-singular projective curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012024.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012025.png" />. For elliptic curves and, more generally, for Abelian varieties, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012027.png" /> is a non-singular hypersurface of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012028.png" /> in projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020120/c02012031.png" /> is a hyperplane section of it.
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See also [[Canonical imbedding]].
 
 
See also [[Canonical imbedding|Canonical imbedding]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Iitaka, "Algebraic geometry, an introduction to birational geometry of algebraic varieties" , Springer (1982) {{MR|}} {{ZBL|0491.14006}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Iitaka, "Algebraic geometry, an introduction to birational geometry of algebraic varieties" , Springer (1982) {{MR|}} {{ZBL|0491.14006}} </TD></TR>
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</table>
  
 
[[Category:Algebraic geometry]]
 
[[Category:Algebraic geometry]]
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{{TEX|done}}

Revision as of 20:35, 19 October 2017

The class $K_X$ of divisors, with respect to linear equivalence on an algebraic variety $X$, which are divisors of differential forms $\omega$ of maximal degree. If $X$ is a non-singular algebraic variety of dimension $n$, then in local coordinates $x_1,\ldots,x_n$ a form $\omega$ can be written as $$ \omega = f(x_1,\ldots,x_n) \, dx_1 \wedge \cdots \wedge dx_n \ . $$

The divisor $(\omega)$ of $\omega$ is locally equal to the divisor $(f)$ of this rational function $f$. This construction does not depend on the choice of local coordinates and gives the divisor $(\omega)$ of $\omega$ on all of $X$. Since for any other form $\omega'$ of the same degree as $\omega$, $\omega' = g\omega$, it follows that $(\omega') = (g) + (\omega)$, and corresponding divisors are equivalent. The canonical class $K_X$ thus constructed is the first Chern class of the sheaf $\Omega_X^n$ of regular differential forms of degree $n$. Its numerical characteristics (degree, index, self-intersections, etc.) are effectively calculable invariants of the algebraic variety.

If $X$ is a non-singular projective curve of genus $g$, then $\deg K_X = 2g-2$. For elliptic curves and, more generally, for Abelian varieties, $K_X = 0$. If $X$ is a non-singular hypersurface of degree $d$ in projective space $\mathbf{P}^n$, then $K_X = (d-n-1)H$, where $H$ is a hyperplane section of it.

See also Canonical imbedding.

References

[1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001


Comments

References

[a1] S. Iitaka, "Algebraic geometry, an introduction to birational geometry of algebraic varieties" , Springer (1982) Zbl 0491.14006
How to Cite This Entry:
Canonical class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Canonical_class&oldid=42136
This article was adapted from an original article by A.N. Parshin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article