p83. For the Examples 4.28 (a) and (b), the inclusions $U_0\cap U_1 \hookrightarrow U_i$ are reversed. (Felipe Zaldivar)
p84. The "above diagram" refers to the top diagram. The \phi in the second diagram should be \varphi (Isac Hedén).
From Bhupendra Nath Tiwari: For AV, CFT, and CM
p2. The claim in the footnote that every abelian surface is a Jacobian variety is not quite true. See the preprints of E. Kani, "The moduli spaces of Jacobians isomorphic to a product of two elliptic curves" and "The existence of Jacobians isomorphic to a product of two elliptic curves". (Kuang-yen Shih)
p26. now shows that dim(H^1(A,IF_l)) >= 2g: ')' missing (Timo Keller).
From Tobias Barthel See pdf file (2 pages)
From Shaul Zemel In the proof of Theorem 10.15, p49, concerning the map from Hom(A,B) \tensor Z_l to the module Hom(T_l A,T_l B) (over Z_l), you start by proving that if e_1,...,e_n are linearly independent over Z in Hom(A,B) then their images are linearly independent over Z_l in Hom(T_l A,T_l B). But this immediately proves that n cannot exceed the rank of the latter over Z_l, i.e., 4dimAdimB (as can be even more clearly seen in Hom(V_l A,V_l B) over Q_l, after tensoring the latter with Q_l). Hence you immediately obtain the finiteness of the rank of Hom(A,B), and the desired bound, without the need to involve decomposition into simple Abelian varieties, different topologies, and polynomials. This is in fact similar (as you have indicated there for something else) to the fact that showing that if a (clearly torsion-free) subgroup of a real vector space of dimension n is discrete then it's free of rank not exceeding n.
Timo Keller points out that, in the proof of Theorem 10.15, p49, M should be defined to be a Z-submodule of End^0(A), not End(T_l(A)) (the degree map P is defined on End^0(A), not on End(T_l(A)). Also, when I write "Now choose the e_i to be a Q-basis for End^0(A)." I seem to be assuming that End^0(A) is finite dimensional over Q, which is what I'm trying to prove. The proof should be replaced by this.
From Everett Howe In Prop. 13.2(b), I found a small typo, probably carried over from copying the result from [1986b] and not changing all of the notation: the "f" in the exponent should be an alpha.
Tim Dokchitser points out that I prove Zarhin's trick (13.12) only
over an algebraically closed field , and then immediately apply it in (13.13) over a finite field.
This is doubly confusing because (13.10) is certainly false over nonalgebraically closed field (over such a field
an abelian variety need not be isogenous to a principally polarized abelian variety).
However, I believe everything is O.K. Specifically, the proof of Zarhin's trick requires only (13.8), and,
because this holds over an algebraically closed field, it holds over every perfect field (see my 1986
Storr's article Abelian Varieties 16.11 and 16.14).
From Sunil Chetty. Near the start of I 14 (Rosati involution): in (\alpha\beta)^\dagger = \beta\alpha there should be a dagger on each of \beta and \alpha.
From Roy Smith (on proofs of Torelli's theorem III 13)
You ask on your website for advice on conceptual proofs of Torelli. ... here goes.
There are many, and the one you give there is the least conceptual one, due I believe to Martens.
Of course you also wanted short, ....well maybe these are not all so short.
The one due to Weil is based on the fact that certain self
intersections of a jacobian theta divisor are reducible, and is
sketched in mumford's lectures on curves given at michigan. Indeed
about 4 proofs are sketched there.
The most geometric one, due to Andreotti - Mayer and Green is to
intersect at the origin of the jacobian, those quadric hypersurfaces
occurring as tangent cones to the theta divisor at double points, thus
recovering the canonical model of the curve as their base locus, with
some few exceptions.
To show this works, one can appeal to the deformation theoretic
results of Kempf. i.e. since the italians proved that a canonical
curve is cut out by quadrics most of the time, one needs to know that
the ideal of all quadrics containing the canonical curve is generated
by the ones coming as tangent cones to theta. the ones which do arise
that way cut out the directions in moduli of abelian varieties where
theta remains singular in codimension three.
But these equisingular deformations of theta embed into the
deformations of the resolution of theta by the symmetric product of
the curve, which kempf showed are equal to the deformations of the
curve itself. hence every equisingular deformation of theta(C) comes
from a deformation of C, and these are cut out by the equations in
moduli of abelian varieties defined by quadratic hypersurfaces
containing C. hence the tangent cones to theta determine C.
This version of Green's result is in a paper of smith and varley,
in compositio 1990.
Perhaps the shortest geometric proof is due to andreotti, who
computed the branch locus of the canonical map on the theta divisor,
and showed quite directly it equals the dual variety of the canonical
curve. this is explained in andreotti's paper from about 1958, and
quite nicely too, with some small errata, in arbarello, cornalba,
griffiths, and harris' book on geometry of curves.
There are other short proofs that torelli holds for general
curves, simply from the fact that the quadrics containing the
canonical curve occur as the kernel of the dual of the derivative of
the torelli map from moduli of curves to moduli of abelian varieties.
this is described in the article on prym torelli by smith and varley
in contemporary mat. vol. 312, in honor of c.h. clemens, 2002, AMS.
there is also a special argument there for genus 4, essentially using
zariski's main theorem on the map from moduli of curves to moduli of
jacobians.
There are also inductive arguments, based on the fact that the
boundary of moduli of curves of genus g contains singular curves of
genus g-1, and allowing one to use lower genus torelli results to
deduce degree torelli for later genera.
Then of course there is matsusaka's proof, derived from torelli's
original proof that given an isomorphism of polarized jacobians, the
theta divisor defines the graph of an isomorphism between their
curves.
For shortest most conceptual, I recommend the proof in Arbarello,
Cornalba, Griffiths Harris, i.e. Andreotti's, for conceptualness and
completeness in a reasonably short argument..
p.261. For an improved exposition of the proof of Theorem 5.1, see RG I, 1.29.
From Bhupendra Nath Tiwari: list of misprints for AGS, LAG, RG odt; doc
From Timo Keller
p. 21: ... with [A,B] = AB - BA,and <- space missing
p. 25, l.-3: ) missing at the end of the equation
I, Theorem 16.21. The centre of a reductive group is of multiplicative type, but need not be reduced (e.g., SL_p). The correct statement is that the radical of G (which is a torus) is the reduced connected group attached to the centre. (There may be other errors of this type, where I incorrectly translated from the characteristic zero case to the general case.) There will be a new version of the notes probably in March 2012.