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
