# Work on Abelian Varieties

Abelian varieties are from a complex analytic point of view the simplest possible spaces — just tori and thus groups. But the curious thing is that tori don't fit easily into projective space. The sidebar shows the Kummer quartic with its sixteen double points: a 2-dimensional principally polarized Abelian variety mod inversion mapped to 3-space via |2Θ|. The interaction of an ample line bundle with the group structure on an Abelian variety is the subject of the first paper below as well as volume three of my "Tata Lectures on Theta". I think this is a surprising theory because it leads to very explicit equations both for individual Abelian varieties and for their moduli space.

I have sometimes thought that such an explicit description might have applications, e.g. to number theory. Some of the other papers concern the lifting of Abelian varieties from characteristic *p* to characteristic 0 and some concern the theory of locally symmetric spaces which are moduli spaces for various families of Abelian varieties.

- On the Equations Defining Abelian Varieties I, II, III,
*Inventiones Mathematicae*, 1966,**1**, pp. 287-384; 1967,**3**, pp. 75-135 and pp. 215-244.Scanned reprint I and DASH reprint I; Scanned reprint II and DASH reprint II; Scanned reprint III and DASH reprint III - Families of Abelian Varieties, in
*Proc. of Symposium in Pure Math.*,**9**, Amer. Math. Soc., 1966. Scanned reprint - Abstract Theta Functions, mimeographed notes of lectures delivered at the Bowdoin Summer School, 1967. Scanned manuscript
- Degeneration of Algebraic Theta Functions, mimeographed notes of lectures delivered at the Bowdoin Summer School, 1967. Scanned manuscript
- Deformations and Liftings of Finite Commutative Group Schemes (with F. Oort),
*Inventiones Mathematicae*, 1968,**5**, pp. 317-334.Scanned reprint and DASH reprint - Bi-extensions of Formal Groups, in
*Algebraic Geometry*, Oxford University Press, 1969, pp. 307-322.Scanned reprint - A Note on Shimura's paper "Discontinuous Groups and Abelian Varieties",
*Math. Annalen*, 1969,**181**, pp. 345-351.Scanned reprint and DASH reprint *Abelian Varieties*, (with C. P. Ramanujam), Oxford University Press India, based on lectures at the Tata Institute in 1967-68 1st edition 1970; 2nd edition 1974, enlarged, with contributions from Yuri Manin; 3rd edition 2010 republished by the Tata Institute of Fundamental Research and distributed by the American Math Society.- An Analytic Construction of Degenerating Abelian Varieties over Complete Rings,
*Composito Math.*, 1972,**24**, pp. 239-272.Scanned reprint - A Rank 2 Vector Bundle on
**P**^{4}with 15,000 Symmetries (with G. Horrocks),*Topology*, 1973,**12**, pp. 63-81. Sections of this bundle define an Abelian surface in**P**^{4}. Scanned reprint - A New Approach to Compactifying Locally Symmetric Varieties, in
*Discrete Subgroups of Lie Groups*, (Proc. of International Colloquium Bombay, 1973), Oxford University Press, 1975, pp. 211-224. Scanned reprint *Toroidal Embeddings I*(with George Kempf, Finn Knudsen and Bernard Saint-Donat), Lecture Notes in Mathematics 339, Springer-Verlag 1973.*Smooth Compactifications of Locally Symmetric Varieties*(with Avner Ash, Michael Rapoport, Yung-Shen Tai), Lie Groups: History Frontiers and Applications, Vol. 4, Math. Sci. Press (a do-it-yourself press run by Bob Hermann) 1975, reprinted by Cambridge University Press, 2010. This is a continuation of the previous paper and book, applying that theory in particular to moduli of abelian varieties.- Hirzebruch's Proportionality Theorem in the non-compact case,
*Invent. Math.*, 1977,**42**., pp. 239-272.Scanned reprint and DASH reprint *Tata Lectures on Theta*(with C. Musili, Madhav Nori, Peter Norman, Emma Previato and Michael Stillman), Birkhauser-Boston, Part I, 1982, Part II, 1983, Part III, 1991, based on lectures given at the Tata Institute in 1978-79, I, II, III- On the Kodaira Dimension of the Siegel Modular Variety, in
*Algebraic Geometry - Open Problems*, (Ravello, 1982),*Lecture Notes in Mathematics 997*, Springer-Verlag 1983, pp.348-375. Scanned manuscript