Errata to "Principles of Algebraic Geometry" by Griffiths and Harris

Question

Griffiths' and Harris' book Principles of Algebraic Geometry is a great book with, IMHO, many typos and mistakes. Why don't we collaborate to write a full list of all of its typos, mistakes etc? My suggestions:

• Page 10 at the top, the definition of $\mathcal{O}_{n,z}$ is wrong (or at least written in a confusing way).

• Page 15, change of coordinates given for the projective spaces only work when $i < j$. It states that the given transitions also work in the case when $j< i$.

• Page 27, there needs to be a bar on the second entry of the $h_ij(z)$ operator defined. Also, shouldn't the title of this section be geometry of complex manifolds, instead of calculus on complex manifolds?

• Page 35, the definition of a sheaf is wrong. The gluing condition should be for any family of open sets, not just for pairs of open sets! I've seem PhD students presenting this definition of sheaf on pg seminars.

• Page 74, writes $D(\psi \wedge e)$, but $\psi$ and $e$ are in two different vector spaces, and one cannot wedge vectors in different vector spaces... I guess they mean tensor product.

• Page 130, the definition of divisor says it's a linear combination of codimension 1 irreducible subvarieties. By linear it means over $\mathbb{Z}$ not over the complex numbers (better should say, like Hartshorne, that $\operatorname{Div}$ is the free abelian group generated by the irreducible subvarieties).

• Page 180, equation $(\ast)$ has target a direct sum of line bundles, not tensor.

• Page 366, when it says "supported smooth functions over $\mathbb{R}^n$", are these complex valued or real valued functions?

• Page 440 top equation, Is it really correct?

• Page 445, the second phrase of the hypercohomology section, it says sheaves of abelian sheaves. Probably means set of abelian sheaves.

Answer 1

i think this is a good project, but the suggestions so far do not scratch the surface, they are mostly only the typos, not the mathematical errors. I would suggest that the book is a little like the fabled works of Lefschetz, i.e. the results are insightful and almost all correct, even if some proofs are lacunary. Thus reading the book as is, may be more valuable than the reading the result of filling the holes in the arguments. Nonetheless, trying to fill those holes may be very useful to the student.

Some arguments said to need elaboration or correction: poincare duality, kodaira vanishing, existence of rational curves on surfaces, Riemann singularities theorem, Clifford's theorem, Torelli's theorem.....

Nonetheless, the proof of Riemann - Roch is very clear, and follows exactly the historical account of Riemann and Roch, i.e. assuming the existence of differential forms of types 1 and 2. Moreover the discussion of Jacobian varieties is extremely valuable and helpful even if a few details are missing. This is a very useful book overall, especially if combined with reading the book on curves by Arbarello, Cornalba, Griffiths and Harris.

Answer 2

On page 38 at the bottom, the explicit formula for the coboundary operator is wrong. It should be:

\begin{equation*} (\delta\sigma)_{i_0, \dotsc, i_{p+1}} = \sum_{j=0}^{p+1} \left.(-1)^j \sigma_{i_0, \dotsc, \color{red}{\widehat{\imath_j}}, \dotsc, i_{p+1}} \right|_{_{U_{i_0} \cap \,\dotsb \,\cap\, U_{i_{p\color{red}{+1}}}}} \end{equation*}

Answer 3

This is relatively small, but the proof of the 'homotopy formula' on pages 384-385 has a error on page 385. The theorem/lemma is correct, but the offending lines are:

$ (\rho \phi)(z) = \overline{\partial}(K\rho\phi)(z) + K(\overline{\partial}(\rho\phi)(z)) $

Restricting to $V$,

$\phi(z) = \overline{\partial}(K\rho \phi)(z)$

The issue is that the homomorphism $K$ induced by the Bochner-Martinelli kernel is only a 'section-wise' homomorphism, and doesn't extend to a map of (pre-)sheaves. You have to do something else to get the homotopy formula.

Answer 4

Page 3, formula for $d\eta$: change minus sign to plus sign

... and so

$$ d\eta = \frac{1}{2\pi\sqrt{-1}} \frac{\partial f(w)}{\partial\overline{w}}\frac{dw\wedge d\overline{w}}{w-z}.$$

Page 144, the degree for T'(M) should have a $2\pi$ in the denominator and not $4\pi$:

... the classical Gauss-Bonnet theorem gives

$$ \deg T'(M) = \frac{1}{2\pi} \int_M K_M \cdot \Phi = \chi(M). $$

Answer 5

http://www.math.stonybrook.edu/~azinger/mat545-fall19/GHnotes.pdf Found an online document by Aleksey Zinger on Griffith and Harris errata which is by far the most comprehensive one I have seen.

Answer 6

I believe that the definition of a positive current at the top of page 386 should read $$ (-1)^{\frac{(n-p)(n-p-1)}{2}} i^{n-p} T(\eta \wedge \bar{\eta}) \geq 0. $$ I motivated this in the answer here.

Addendum: Here is why the definition as stated in the book cannot be correct as is and needs tweaking to begin with. For a real $(p,p)$-current $T$ (i.e. $\overline{T(\phi)} = T(\bar{\phi})$), complex-conjugating the expression in the book $$ i^{\frac{p(p-1)}{2}} T(\eta \wedge \bar{\eta}) $$ shows that this is not even a real number (depending on $p$ and $n$ and unless it is zero).