Picard variety

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

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)
[a2]	M. Rosenlicht, "Generalized Jacobian varieties" Ann. of Math. , 59 (1954) pp. 505–530
[a3]	F. Oort, "A construction of generalized Jacobian varieties by group extensions" Math. Ann. , 147 (1962) pp. 277–286
[a4]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 272
[a5]	M. Flato, "Deformation view of physical theories" Czechoslovak J. Phys. , B32 (1982) pp. 472–475