I am looking at the section of chern class computation from Robert Friedman- Algebraic surfaces and holomorphic vector bundles.

If $X$ is smooth quasi-projective variety, $Z$ is a reduced, irreducible subvariety of codimension $r$ in $X$. $j:Z\hookrightarrow X$ denote inclusion map. Then

$$c_i(j_* O_Z)=0 \textrm{ for } i<r \textrm{ and } c_r(j_* O_Z)=(-1)^{r-1}(r-1)![Z]$$ where $[Z]$ is the cycle associated to $Z$. Similary, if $V$ is a vector bundle on rank $n$ on $Z$, then $$c_i(j_* V)=0 \textrm{ for } i<r \textrm{ and } c_r(j_* V)=(-1)^{r-1}(r-1)!n[Z].$$

This is statement given in the book. What about higher chern classes? Can we say if $c_{i}(j_*O_X)=c_{i}(j_*V)=0$ for $i>r$. Is this true?

  • $\begingroup$ All Chern classes can be computed from Grothendieck--Riemann--Roch Theorem. $\endgroup$ – Sasha Aug 19 '16 at 9:22
  • $\begingroup$ @Sasha, I am just curious to know why no mention is made about $c_i(j_*V)$ for $i>r$. Will they be zero? Or is it possible that they are non-zero $\endgroup$ – user349424 Aug 19 '16 at 9:24
  • 1
    $\begingroup$ They are nonzero in general. $\endgroup$ – Sasha Aug 19 '16 at 9:35

The classes will in general be nonzero. You can check this in the simple case where $Z$ is a global complete intersection (say of codimension two), since then you can write down an explicit resolution for $O_Z$. A general formula (in the codimension two case) can be found in Fulton, "Intersection Theory", Example 15.3.5 — see here.

Fulton, William. Intersection theory. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 2. Springer-Verlag, Berlin, 1998. xiv+470 pp.


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