Variations of the Albanese morphisms

Authors:
Gian Pietro Pirola and Francesco Zucconi

Journal:
J. Algebraic Geom. **12** (2003), 535-572

DOI:
https://doi.org/10.1090/S1056-3911-03-00359-X

Published electronically:
January 21, 2003

MathSciNet review:
1966026

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Abstract |
References |
Additional Information

Abstract: We estimate the number of moduli of an $n$-dimensional variety $X$ through the variation of its Albanese morphism. Refining upon our methods, we work out the classical Castelnuovo bound concerning the number $m$ of moduli of irregular surfaces with birational Albanese map. We interpret our variation by means of higher Abel-Jacobi mappings theory and under the only hypothesis that $X$ has a generically finite morphism to an Abelian variety $A$, we can bound from below the geometrical genus $p_{g}(X)$ in terms of the dimensions of $A$ and $X$. Using the same framework, we characterize the hyperelliptic locus in ${\mathcal {M}}_g$ as the only close subvariety ${\mathcal {H}}$ inside the moduli space of curves with $\dim {\mathcal {H}} \geq 2g-1$ and torsion Abel-Jacobi image of the Ceresa cycle at its generic point.

BCV F. Bardelli, C. Ciliberto, A. Verra, *Curves of minimal genus on a general Abelian variety,* Comp. Math. 96 (1995), 115-147.
CGGH F. Carlson, M. Green, P. Griffiths, J. Harris, *Infinitesimal variations of Hodge structure (I),* Comp. Math. 50 (1983), 109-205.
C G. Castelnuovo, *Sul numero dei moduli di una superficie irregolare I, II,* Rend. Accad. Lincei 7 (1949), 3-7, 8-11.
Ca1 F. Catanese, *On the moduli spaces of surfaces of general type,* J. Diff. Geometry 19 (1984), 483-515.
Ca2 F. Catanese, *Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations,* Inv. Math. 104 (1991), 263-289.
Ce G. Ceresa, *$C$ is not algebraic equivalent to $C^{-}$ in its Jacobian,* Ann. of Math. (2) 117 (1983), 285-291.
Cl H. Clemens, *Some results about Abel-Jacobi zmappings,* in Topics in Transcendental Algebraic Geometry, Ann. of Math. Stud. 106, Princeton Univ. Press, Princeton, (1984), 289-304.
CP A. Collino, G. P. Pirola, *The Griffiths infinitesimal invariant for a curve in its Jacobian,* Duke Math. Jour. 78, No 1 (1995), 59-88.
Fa N. Fakhruddin, *Algebraic cycles on generic Abelian varieties,* Comp. Math. 100 (1996), 101-119.
Gr M. L. Green, *Infinitesimal methods in Hodge theory. Algebraic cycles and Hodge theory (Torino, 1993),* L.N.M., 1594, Springer, Berlin (1994), 1-92.
G1 P. Griffiths, *Periods of integrals on algebraic manifolds I, II,* Amer. J. Math 90 (1968), 568-626, 805-865.
G2 P. Griffiths, *On the periods of certain rational integrals I, II,* Ann. of Math 90 (1969), 460-541.
G3 P. Griffiths, *Periods of integrals on algebraic manifold III,* Publ. Math IHES 38 (1970), 125-180.
G4 P. Griffiths, *Infinitesimal variations of Hodge structures III: determinantal varieties and the infinitesimal invariant of normal functions,* Comp. Math. 50 (1983), 267-324.
Ho E. Horikawa, *On deformations of holomorphic maps II,* J. Math. Soc. Japan Vol. 26, No. 4 (1974), 647-667.
Ke G. Kempf, *Complex Abelian varieties and theta functions,* Universitext, Springer-Verlag, Berlin (1991).
Ko J. Kollár, *Subadditivity of the Kodaira dimension: Fibers of general type.* Adv. Stu. Pure Math 10 (1987), 361-398.
Ii S. Itaka, *Algebraic Geometry,* Springer Verlag, New York, Heidelberg (1982).
Ml S. Mac Lane, *Homology,* Springer Verlag, Berlin-Göttingen-Heidelberg (1963).
No M.V. Nori, *Algebraic Cycles and Hodge Theoretic Connectivity,* Inv. Math. 111 (1993), 349-373.
P G. P. Pirola, *Abel-Jacobi invariant and curves on generic Abelian varieties,* Abelian varieties (Egloffstein, 1993), de Gruyter, Berlin (1995), 223-232.
Ra Z. Ran, *Deformation of maps,* in *Algebraic curves and projective geometry.* Proc. Trento 1988, L.N.M. 1389, Springer, Berlin (1989), 221-232.
Re I. Reider, *Bounds on the number of moduli for irregular surfaces of general type,* Manus. Math. 60, No. 2 (1988), 647-667.
Ue K. Ueno, *Classification theory of algebraic varieties and compact complex spaces,* L.N.M., 439, Springer-Verlag, Berlin-New York (1975).
Vi E. Viehweg, *Weak positivity and the additivity of Kodaira dimension for certain algebraic fiber spaces,* Adv. Stu. Pure Math 1, Algebraic Varieties and Analytic Varieties (1983), 329-353.
Vo C. Voisin, *Une Remarque Sur l’Invariant Infinitésimal Des Fonctions Normales*, C. R. Acad. Sci. Paris, t. 307, Série I (1988), 157-160.

BCV F. Bardelli, C. Ciliberto, A. Verra, *Curves of minimal genus on a general Abelian variety,* Comp. Math. 96 (1995), 115-147.
CGGH F. Carlson, M. Green, P. Griffiths, J. Harris, *Infinitesimal variations of Hodge structure (I),* Comp. Math. 50 (1983), 109-205.
C G. Castelnuovo, *Sul numero dei moduli di una superficie irregolare I, II,* Rend. Accad. Lincei 7 (1949), 3-7, 8-11.
Ca1 F. Catanese, *On the moduli spaces of surfaces of general type,* J. Diff. Geometry 19 (1984), 483-515.
Ca2 F. Catanese, *Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations,* Inv. Math. 104 (1991), 263-289.
Ce G. Ceresa, *$C$ is not algebraic equivalent to $C^{-}$ in its Jacobian,* Ann. of Math. (2) 117 (1983), 285-291.
Cl H. Clemens, *Some results about Abel-Jacobi zmappings,* in Topics in Transcendental Algebraic Geometry, Ann. of Math. Stud. 106, Princeton Univ. Press, Princeton, (1984), 289-304.
CP A. Collino, G. P. Pirola, *The Griffiths infinitesimal invariant for a curve in its Jacobian,* Duke Math. Jour. 78, No 1 (1995), 59-88.
Fa N. Fakhruddin, *Algebraic cycles on generic Abelian varieties,* Comp. Math. 100 (1996), 101-119.
Gr M. L. Green, *Infinitesimal methods in Hodge theory. Algebraic cycles and Hodge theory (Torino, 1993),* L.N.M., 1594, Springer, Berlin (1994), 1-92.
G1 P. Griffiths, *Periods of integrals on algebraic manifolds I, II,* Amer. J. Math 90 (1968), 568-626, 805-865.
G2 P. Griffiths, *On the periods of certain rational integrals I, II,* Ann. of Math 90 (1969), 460-541.
G3 P. Griffiths, *Periods of integrals on algebraic manifold III,* Publ. Math IHES 38 (1970), 125-180.
G4 P. Griffiths, *Infinitesimal variations of Hodge structures III: determinantal varieties and the infinitesimal invariant of normal functions,* Comp. Math. 50 (1983), 267-324.
Ho E. Horikawa, *On deformations of holomorphic maps II,* J. Math. Soc. Japan Vol. 26, No. 4 (1974), 647-667.
Ke G. Kempf, *Complex Abelian varieties and theta functions,* Universitext, Springer-Verlag, Berlin (1991).
Ko J. Kollár, *Subadditivity of the Kodaira dimension: Fibers of general type.* Adv. Stu. Pure Math 10 (1987), 361-398.
Ii S. Itaka, *Algebraic Geometry,* Springer Verlag, New York, Heidelberg (1982).
Ml S. Mac Lane, *Homology,* Springer Verlag, Berlin-Göttingen-Heidelberg (1963).
No M.V. Nori, *Algebraic Cycles and Hodge Theoretic Connectivity,* Inv. Math. 111 (1993), 349-373.
P G. P. Pirola, *Abel-Jacobi invariant and curves on generic Abelian varieties,* Abelian varieties (Egloffstein, 1993), de Gruyter, Berlin (1995), 223-232.
Ra Z. Ran, *Deformation of maps,* in *Algebraic curves and projective geometry.* Proc. Trento 1988, L.N.M. 1389, Springer, Berlin (1989), 221-232.
Re I. Reider, *Bounds on the number of moduli for irregular surfaces of general type,* Manus. Math. 60, No. 2 (1988), 647-667.
Ue K. Ueno, *Classification theory of algebraic varieties and compact complex spaces,* L.N.M., 439, Springer-Verlag, Berlin-New York (1975).
Vi E. Viehweg, *Weak positivity and the additivity of Kodaira dimension for certain algebraic fiber spaces,* Adv. Stu. Pure Math 1, Algebraic Varieties and Analytic Varieties (1983), 329-353.
Vo C. Voisin, *Une Remarque Sur l’Invariant Infinitésimal Des Fonctions Normales*, C. R. Acad. Sci. Paris, t. 307, Série I (1988), 157-160.

Additional Information

**Gian Pietro Pirola**

Affiliation:
Dipartimento di Matematica, Università degli studi di Pavia, Strada Ferrata 1, 27100 Pavia, Italia

MR Author ID:
139965

Email:
pirola@dimat.unipv.it

**Francesco Zucconi**

Affiliation:
Dipartimento di Matematica e Informatica, Università degli studi di Udine, Via delle Scienze 206, 33100 Udine, Italia

Email:
zucconi@dimi.uniud.it

Received by editor(s):
December 3, 2000

Published electronically:
January 21, 2003

Additional Notes:
The first author was partially supported by: (1) Cofin 99: Spazi di moduli e teoria delle rappresentazioni (Murst); (2) GNSAGA; (3) Far 2000 (Pavia): Varietà algebriche, calcolo algebrico, grafi orientati e topologici. The second author was partially supported by: (1) SC.D.I.M.I. cecu 04118 ’99 Ricerca Dipartimentale, Università di Udine; (2) Cofin 99: Spazi di moduli e teoria delle rappresentazioni(Murst); (3) Royal Society-Accademia Nazionale dei Lincei 2000 grant to do research in Great Britain