View source for Picard variety

''of a complete smooth algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p0726901.png" /> over an algebraically closed field''

The Abelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p0726902.png" /> that parametrizes the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p0726903.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p0726904.png" /> of divisors that are algebraically equivalent to zero by the group of principal divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p0726905.png" />, 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. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p0726906.png" /> coincides with the connected component of the unit, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p0726907.png" />, of the [[Picard group|Picard group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p0726908.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p0726909.png" />. The structure of an Abelian variety on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269010.png" /> is uniquely characterized by the following property: For any algebraic family of divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269012.png" /> with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269013.png" /> there exists a regular mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269014.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269016.png" /> is a certain fixed point from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269017.png" /> [[#References|[2]]]. The dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269018.png" /> is called the irregularity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269019.png" />.

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

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269020.png" /> is a smooth projective complex variety, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269021.png" /> can be identified with the group of invertible analytic sheaves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269022.png" /> with zero Chern class [[#References|[4]]]. Also, in that case the Picard variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269023.png" /> is isomorphic to the quotient group of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269024.png" /> by the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269025.png" />. In particular, the irregularity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269027.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269029.png" /> is the sheaf of regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269030.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269031.png" />. In arbitrary characteristic one only has the Igusa inequality: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269032.png" /> (an example is known of a smooth algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269033.png" /> of irregularity 1 having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269034.png" /> [[#References|[6]]]). This shows that a Picard variety is closely related to the theory of one-dimensional differential forms. E. Picard himself [[#References|[1]]] started research on such forms on Riemann surfaces; he showed that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269035.png" /> of everywhere-regular forms is finite dimensional.

The concept of a Picard variety can be extended to the case of a complete normal variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269036.png" />. Studies have also been made on a Picard variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269037.png" /> corresponding to Cartier divisors and having good functorial properties, in contrast to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269038.png" /> [[#References|[9]]]. The variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269039.png" /> has been constructed for complete normal varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072690/p07269040.png" /> [[#References|[5]]], as well as for arbitrary projective varieties [[#References|[8]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Picard,   "Sur les intégrales de différentielles totales algébriques"  ''C.R. Acad. Sci. Paris'' , '''99'''  (1884)  pp. 961–963</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.R. Shafarevich,   "Basic algebraic geometry" , Springer  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Mumford,   "Abelian varieties" , Oxford Univ. Press  (1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , Wiley (Interscience)  (1978)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C. Chevalley,   "Sur la théorie de la variété de Picard"  ''Amer. J. Math.'' , '''82'''  (1960)  pp. 435–490</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J.-I. Igusa,   "On some problems in abstract algebraic geometry"  ''Proc. Nat. Acad. Sci. USA'' , '''41''' :  11  (1955)  pp. 964–967</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  T. Matsusaka,   "On the algebraic construction of the Picard variety I"  ''Jap. J. Math.'' , '''21''' :  2  (1951)  pp. 217–235</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  C. Seshadri,   "Variété de Picard d'une variété complète"  ''Ann. Mat. Pura Appl.'' , '''57'''  (1962)  pp. 117–142</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  C. Seshadri,   "Universal property of the Picard variety of a complete variety"  ''Math. Ann.'' , '''158''' :  3  (1965)  pp. 293–296</TD></TR></table>



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

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Lang,   "Abelian varieties" , Springer  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Rosenlicht,   "Generalized Jacobian varieties"  ''Ann. of Math.'' , '''59'''  (1954)  pp. 505–530</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Oort,   "A construction of generalized Jacobian varieties by group extensions"  ''Math. Ann.'' , '''147'''  (1962)  pp. 277–286</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Hartshorne,   "Algebraic geometry" , Springer  (1977)  pp. 272</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Flato,   "Deformation view of physical theories"  ''Czechoslovak J. Phys.'' , '''B32'''  (1982)  pp. 472–475</TD></TR></table>