Corso di dottorato: Introduzione alle varietà abeliane complesse.
Docenti: Gian Pietro Pirola e Lidia Stoppino
La prima parte del corso comincerà giovedì 22 febbraio dalle 10 alle 13 
e continuerà tutti i giovedì, sempre 10-13.
Il corso sarà in italiano.
Sarà possibile per motivate ragioni di salute o di lontananza seguire il corso in streaming, mandate una mail a
per avere le credenziali.
Overview. A complex abelian variety is a complex torus with a natural structure of algebraic projective variety. Abelian varieties are at the same time a very much studied topic in algebraic geometry and also indispensable tools for much research on other topics in algebraic geometry and number theory. For example, they occur naturally when studying line bundles over an algebraic variety or the arithmetic of a number field. Special cases of abelian varieties are elliptic curves and the Jacobian of an algebraic curve.
In this course we will present an overview of the basics of the theory of abelian varieties, then we will focus on the geometry of curves inside abelian varieties.
The first part will last approximately 18 hours and will be taught by Lidia Stoppino.
More precisely, here is a tentative list of the topics that will be treated.
  • Complex tori. Isogenies and morphisms of complex tori. Cohomology of coplex tori.
  • Line bundles on complex tori. Appel-Humbert Theorem. The dual complex torus, The Poincarè bundle.
  • Rigidity theorem, Seesaw principle, theorem of the cube, Riemann-Roch theorem.
  • Polarizations and type of polarization. Lefschetz Theorem. Abelian varieties. Riemaniann conditions.
  • Maps from algebraic varieties to abelian varieties.
  • The Jacobian of a curve and the Abel-Jacobi map, the Albanese variety and the Albanese morphism.
  • A few words about moduli of polarized complex abelian varieties. Torelli map and Torelli theorem.
The references for this first part of the course are: [2,3,7,8,9,11,12,14]
In the second part of the course there will be a glimpse of some research topics. The study of the relationships between algebraic curves and abelian varieties has always been one of the main themes of algebraic geometry. As mentioned above, to any curve C of genus g one can associate the Jacobian J(C), which is an algebraic variety of dimension g. We will investigate the points of contact and those of distance between the two theories. The accepted philosophy is that as g grows, the geometry of a general abelian variety of dimension g becomes “rarefied”, while the geometry of Jacobians always reflects the presence of the curve.
This second part will last more or less 12 hours and will be held by Gian Pietro Pirola. The references for this part are the following: [1,4,5,6,9,10,13,15,16].
The program of the second part will tentatively be the following:
(1) The geometry of the ceresa Cycle (4 hours) (see [4],[5]).
(2) Exercise: infinitesimal method (2 hours).
(3) The genus of curves on a generic Abelian Variety (2 hours) (see [13],[10],[15]).
(4) The gonality problem and rational equivalence (2 hours) (see [6] and [16]).
(5) Hodge loci and the Jacobian locus (2 hours) (see [1]).
The final exam will consist in a follow-up seminar on a subject chosen with the teachers, followed by some questions on the arguments treated in the course.
Prerequisites: It would be extremely useful to have followed a basic course in algebraic geometry, and in particular to know something about Riemann surfaces. Also some basic knowledge about Hodge Theory (such as the one provided in the Ph.D. course by Alessandro Ghigi) would be useful.
[1] G. Balsi, B. Klinger, E. Ulmo, On the distribution of the Hodge Locus, arXiv 2107.08838
[2] C. Birkenhake, H. Lange, Complex abelian varieties. Second edition. G W 302. Springer-Verlag, Berlin, 2004.
[3] J.B. Bost, Introduction to compact Riemann surfaces, Jacobians, and abelian varieties. From number theory to physics (Les Houches, 1989), 64211, Springer, Berlin, 1992.
[4] G. Ceresa, C is not algebraically equivalent to C− in its Jacobian. Ann. of Math. (2)117(1983), no.2, 285 – 291.
[5] A. Collino, G. P. Pirola, The Griffiths infinitesimal invariant for a curve in its Jacobian, Duke Math. J. 78 (1995), no. 1, 59 – 88.
[6] E Colombo, O. Martin, J.C Naranjo, G. Pirola Degree of irrationality of a very general abelian variety. Int. Math. Res. Not. (2022), no. 11, 8295–8313.
[7] M.D.T Cornalba, Lezioni su superfici di Riemann e tori complessi. available here
[8] O. Debarre, Complex tori and abelian varieties, SMF/AMS Texts Monogr., 11, AMS Providence, Socie ́te ́ Mathe ́matique de France, Paris, 2005.
[9] G. Kempf, Complex abelian varieties and theta functions, Universitext Springer-Verlag, Berlin, 1991.
[10] V. Marcucci, On the genus of curves in a Jacobian variety, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), no. 3, 735–754.
[11] J. Milne, Abelian Varieties, available online.
[12] D. Mumford, Abelian varieties. With appendices by C. P. Ramanujan and Yuri Manin. Corrected reprint of the second (1974) edition. Tata Institute of Fundamental Research Studies in Mathematics, 5. Bombay; 2008.
[13] G. Pirola, Abel-Jacobi invariant and curves on generic abelian varieties. Abelian varieties (Egloff- stein, 1993), 237—249, de Gruyter, Berlin, 1995.
[14] R. Smith, The Jacobian variety of a Riemann surface and its theta geometry. Lectures on Riemann surfaces (Trieste, 1987), 350—427, World Sci. Publ., Teaneck, NJ, 1989.
[15] J. Tsimerman, Abelian Varieties are not quotients of low-dimension Jacobians, arxiv 2302.05860
[16] C. Voisin, Chow rings and gonality of general abelian varieties. Ann. H. Lebesgue 1, 313–332 (2018).