Namespaces
Variants
Actions

Difference between revisions of "Picard variety"

From Encyclopedia of Mathematics
Jump to: navigation, search
m
m (MR/ZBL numbers added)
(One intermediate revision by the same user not shown)
Line 28: Line 28:
 
[[#References|[4]]]. Also, in that case the Picard variety $\fP(X)$ is
 
[[#References|[4]]]. Also, in that case the Picard variety $\fP(X)$ is
 
isomorphic to the quotient group of the space $\def\cO{{\mathcal{O}}} H^1(X,\cO_X)$ by the lattice
 
isomorphic to the quotient group of the space $\def\cO{{\mathcal{O}}} H^1(X,\cO_X)$ by the lattice
$H^1(X,\Z) \subset H^1(X\cO_X)$. In particular, the irregularity $q$ of $X$ coincides with $\dim H^1(X\cO_X) = \dim H^0(X,\Omega_X^1)$,
+
$H^1(X,\Z) \subset H^1(X,\cO_X)$. In particular, the irregularity $q$ of $X$ coincides with $\dim H^1(X,\cO_X) = \dim H^0(X,\Omega_X^1)$,
 
where $\Omega_X^1$ is the sheaf of regular $1$-forms. The latter result is true
 
where $\Omega_X^1$ is the sheaf of regular $1$-forms. The latter result is true
 
also in the case of non-singular projective curves over any
 
also in the case of non-singular projective curves over any
 
algebraically closed field as well as in the case of complete smooth
 
algebraically closed field as well as in the case of complete smooth
 
varieties over an algebraically closed field of characteristic $0$. In
 
varieties over an algebraically closed field of characteristic $0$. In
arbitrary characteristic one only has the Igusa inequality: $\dim H^1(X\cO_X) \ge q$ (an
+
arbitrary characteristic one only has the Igusa inequality: $\dim H^1(X,\cO_X) \ge q$ (an
 
example is known of a smooth algebraic surface $F$ of irregularity 1
 
example is known of a smooth algebraic surface $F$ of irregularity 1
having $\dim H^1(X\cO_X) = 2$
+
having $\dim H^1(X,\cO_X) = 2$
 
[[#References|[6]]]). This shows that a Picard variety is closely
 
[[#References|[6]]]). This shows that a Picard variety is closely
 
related to the theory of one-dimensional differential forms. E. Picard
 
related to the theory of one-dimensional differential forms. E. Picard
Line 56: Line 56:
 
valign="top"> E. Picard, "Sur les intégrales de différentielles
 
valign="top"> E. Picard, "Sur les intégrales de différentielles
 
totales algébriques" ''C.R. Acad. Sci. Paris'' , '''99''' (1884)
 
totales algébriques" ''C.R. Acad. Sci. Paris'' , '''99''' (1884)
pp. 961–963</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">
+
pp. 961–963 {{MR|}} {{ZBL|34.0459.03}} {{ZBL|34.0458.03}} {{ZBL|32.0419.01}} {{ZBL|32.0418.01}} {{ZBL|28.0560.01}} {{ZBL|16.0296.01}} {{ZBL|16.0293.01}} {{ZBL|17.0373.03}} {{ZBL|17.0332.03}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">
 
I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977)
 
I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977)
(Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD
+
(Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD
 
valign="top"> D. Mumford, "Abelian varieties" , Oxford Univ. Press
 
valign="top"> D. Mumford, "Abelian varieties" , Oxford Univ. Press
(1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">
+
(1974) {{MR|2514037}} {{MR|1083353}} {{MR|0352106}} {{MR|0441983}} {{MR|0282985}} {{MR|0248146}} {{MR|0219542}} {{MR|0219541}} {{MR|0206003}} {{MR|0204427}} {{ZBL|0326.14012}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">
 
P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" ,
 
P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" ,
Wiley (Interscience) (1978)</TD></TR><TR><TD valign="top">[5]</TD> <TD
+
Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD
 
valign="top"> C. Chevalley, "Sur la théorie de la variété de Picard"
 
valign="top"> C. Chevalley, "Sur la théorie de la variété de Picard"
''Amer. J. Math.'' , '''82''' (1960) pp. 435–490</TD></TR><TR><TD
+
''Amer. J. Math.'' , '''82''' (1960) pp. 435–490 {{MR|0118723}} {{ZBL|0127.37701}} </TD></TR><TR><TD
 
valign="top">[6]</TD> <TD valign="top"> J.-I. Igusa, "On some problems
 
valign="top">[6]</TD> <TD valign="top"> J.-I. Igusa, "On some problems
 
in abstract algebraic geometry" ''Proc. Nat. Acad. Sci. USA'' ,
 
in abstract algebraic geometry" ''Proc. Nat. Acad. Sci. USA'' ,
'''41''' : 11 (1955) pp. 964–967</TD></TR><TR><TD
+
'''41''' : 11 (1955) pp. 964–967 {{MR|0074085}} {{ZBL|0067.39102}} </TD></TR><TR><TD
 
valign="top">[7]</TD> <TD valign="top"> T. Matsusaka, "On the
 
valign="top">[7]</TD> <TD valign="top"> T. Matsusaka, "On the
 
algebraic construction of the Picard variety I" ''Jap. J. Math.'' ,
 
algebraic construction of the Picard variety I" ''Jap. J. Math.'' ,
'''21''' : 2 (1951) pp. 217–235</TD></TR><TR><TD valign="top">[8]</TD>
+
'''21''' : 2 (1951) pp. 217–235 {{MR|0062470}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD>
 
<TD valign="top"> C. Seshadri, "Variété de Picard d'une variété
 
<TD valign="top"> C. Seshadri, "Variété de Picard d'une variété
 
complète" ''Ann. Mat. Pura Appl.'' , '''57''' (1962)
 
complète" ''Ann. Mat. Pura Appl.'' , '''57''' (1962)
pp. 117–142</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">
+
pp. 117–142 {{MR|0138623}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">
 
C. Seshadri, "Universal property of the Picard variety of a complete
 
C. Seshadri, "Universal property of the Picard variety of a complete
 
variety" ''Math. Ann.'' , '''158''' : 3 (1965)
 
variety" ''Math. Ann.'' , '''158''' : 3 (1965)
pp. 293–296</TD></TR></table>
+
pp. 293–296 {{MR|0177988}} {{ZBL|0132.41501}} </TD></TR></table>
  
  
Line 99: Line 99:
 
<table><TR><TD valign="top">[a1]</TD> <TD
 
<table><TR><TD valign="top">[a1]</TD> <TD
 
valign="top"> S. Lang, "Abelian varieties" , Springer
 
valign="top"> S. Lang, "Abelian varieties" , Springer
(1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">
+
(1983) {{MR|0713430}} {{ZBL|0516.14031}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">
 
M. Rosenlicht, "Generalized Jacobian varieties" ''Ann. of Math.'' ,
 
M. Rosenlicht, "Generalized Jacobian varieties" ''Ann. of Math.'' ,
'''59''' (1954) pp. 505–530</TD></TR><TR><TD valign="top">[a3]</TD>
+
'''59''' (1954) pp. 505–530 {{MR|0061422}} {{ZBL|0058.37002}} </TD></TR><TR><TD valign="top">[a3]</TD>
 
<TD valign="top"> F. Oort, "A construction of generalized Jacobian
 
<TD valign="top"> F. Oort, "A construction of generalized Jacobian
 
varieties by group extensions" ''Math. Ann.'' , '''147''' (1962)
 
varieties by group extensions" ''Math. Ann.'' , '''147''' (1962)
pp. 277–286</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">
+
pp. 277–286 {{MR|0141667}} {{ZBL|0101.38502}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">
 
R. Hartshorne, "Algebraic geometry" , Springer (1977)
 
R. Hartshorne, "Algebraic geometry" , Springer (1977)
pp. 272</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">
+
pp. 272 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">
 
M. Flato, "Deformation view of physical theories" ''Czechoslovak
 
M. Flato, "Deformation view of physical theories" ''Czechoslovak
 
J. Phys.'' , '''B32''' (1982) pp. 472–475</TD></TR></table>
 
J. Phys.'' , '''B32''' (1982) pp. 472–475</TD></TR></table>

Revision as of 21:55, 30 March 2012

of a complete smooth algebraic variety $X$ over an algebraically closed field

The Abelian variety $\def\fP{{\frak{P}}} \fP(X)$ that parametrizes the quotient group ${\rm Div}^\alpha(X)/P(X)$ of the group ${\rm Div}^\alpha(X)$ of divisors that are algebraically equivalent to zero by the group of principal divisors $P(X)$, i.e. divisors of rational functions. From the point of view of the theory of sheaves, the Picard variety parametrizes the set of classes of isomorphic invertible sheaves with zero Chern class, i.e. $\fP(X)$ coincides with the connected component of the unit, ${\rm Pic}^0(X)$, of the Picard group ${\rm Pic}(X)$ of $X$. The structure of an Abelian variety on the group $\fP(X) = {\rm Dic}^\alpha(X)/P(X)$ is uniquely characterized by the following property: For any algebraic family of divisors $D$ on $X$ with base $S$ there exists a regular mapping $\phi:S\to\fP(X)$ for which $D(s)-D(s_0)\in\phi(s)$, where $s_0$ is a certain fixed point from $S_0$ [2]. The dimension $q=\dim\fP(X)$ is called the irregularity of $X$.

The classic example of a Picard variety is the Jacobi variety of a smooth projective curve. Another example is provided by a dual Abelian variety [3].

If $X$ is a smooth projective complex variety, $\fP(X)$ can be identified with the group of invertible analytic sheaves on $X$ with zero Chern class [4]. Also, in that case the Picard variety $\fP(X)$ is isomorphic to the quotient group of the space $\def\cO{{\mathcal{O}}} H^1(X,\cO_X)$ by the lattice $H^1(X,\Z) \subset H^1(X,\cO_X)$. In particular, the irregularity $q$ of $X$ coincides with $\dim H^1(X,\cO_X) = \dim H^0(X,\Omega_X^1)$, where $\Omega_X^1$ is the sheaf of regular $1$-forms. The latter result is true also in the case of non-singular projective curves over any algebraically closed field as well as in the case of complete smooth varieties over an algebraically closed field of characteristic $0$. In arbitrary characteristic one only has the Igusa inequality: $\dim H^1(X,\cO_X) \ge q$ (an example is known of a smooth algebraic surface $F$ of irregularity 1 having $\dim H^1(X,\cO_X) = 2$ [6]). This shows that a Picard variety is closely related to the theory of one-dimensional differential forms. E. Picard himself [1] started research on such forms on Riemann surfaces; he showed that the space $\dim H^0(X,\Omega_X^1)$ of everywhere-regular forms is finite dimensional.

The concept of a Picard variety can be extended to the case of a complete normal variety $X$. Studies have also been made on a Picard variety $\fP_c(X)$ corresponding to Cartier divisors and having good functorial properties, in contrast to $\fP(X)$ [9]. The variety $\fP_c(X)$ has been constructed for complete normal varieties $X$ [5], as well as for arbitrary projective varieties [8].

References

[1] E. Picard, "Sur les intégrales de différentielles

totales algébriques" C.R. Acad. Sci. Paris , 99 (1884)

pp. 961–963 Zbl 34.0459.03 Zbl 34.0458.03 Zbl 32.0419.01 Zbl 32.0418.01 Zbl 28.0560.01 Zbl 16.0296.01 Zbl 16.0293.01 Zbl 17.0373.03 Zbl 17.0332.03
[2]

I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977)

(Translated from Russian) MR0447223 Zbl 0362.14001
[3] D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974) MR2514037 MR1083353 MR0352106 MR0441983 MR0282985 MR0248146 MR0219542 MR0219541 MR0206003 MR0204427 Zbl 0326.14012
[4]

P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" ,

Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[5] C. Chevalley, "Sur la théorie de la variété de Picard" Amer. J. Math. , 82 (1960) pp. 435–490 MR0118723 Zbl 0127.37701
[6] J.-I. Igusa, "On some problems

in abstract algebraic geometry" Proc. Nat. Acad. Sci. USA ,

41 : 11 (1955) pp. 964–967 MR0074085 Zbl 0067.39102
[7] T. Matsusaka, "On the

algebraic construction of the Picard variety I" Jap. J. Math. ,

21 : 2 (1951) pp. 217–235 MR0062470
[8] C. Seshadri, "Variété de Picard d'une variété

complète" Ann. Mat. Pura Appl. , 57 (1962)

pp. 117–142 MR0138623
[9]

C. Seshadri, "Universal property of the Picard variety of a complete variety" Math. Ann. , 158 : 3 (1965)

pp. 293–296 MR0177988 Zbl 0132.41501


Comments

The Picard variety (over an algebraically closed field) has been constructed for Weil divisors by T. Matsusaka [7], by S. Chow (see [a1]) and by A. Weil (see [a1]), and for Cartier divisors by C. Chevalley ([5], [8] and [9]).

The Jacobian of a complete (possibly singular, possibly multiple) algebraic curve was constructed by M. Rosenlicht [a2] and F. Oort [a3], [a5].

References

[a1] S. Lang, "Abelian varieties" , Springer (1983) MR0713430 Zbl 0516.14031
[a2]

M. Rosenlicht, "Generalized Jacobian varieties" Ann. of Math. ,

59 (1954) pp. 505–530 MR0061422 Zbl 0058.37002
[a3] F. Oort, "A construction of generalized Jacobian

varieties by group extensions" Math. Ann. , 147 (1962)

pp. 277–286 MR0141667 Zbl 0101.38502
[a4]

R. Hartshorne, "Algebraic geometry" , Springer (1977)

pp. 272 MR0463157 Zbl 0367.14001
[a5]

M. Flato, "Deformation view of physical theories" Czechoslovak

J. Phys. , B32 (1982) pp. 472–475
How to Cite This Entry:
Picard variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Picard_variety&oldid=19590
This article was adapted from an original article by V.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article