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Does there exist a definition of Chern character (or Chern classes) for a coherent sheaf $\mathscr{F}$ on a singular variety $X$? In this case I might not be able to find a projective resolution for $\mathscr{F}$.

It seems that Vakil define it in his notes https://math.stanford.edu/~vakil/245/245class19.pdf on page 1, but I think he is implicitly assuming $X$ to be non-singular.

Thank you in advance.

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  • $\begingroup$ One possibility is to consider Chern classes of the sheaf restricted to the nonsingular locus of $X$, and then take their pushforward to $X$. $\endgroup$ – Sasha Oct 21 '19 at 16:04
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    $\begingroup$ You can not push forward classes from an open subset. The Chern character does not exist, in general; the closest that comes to my mind is the Riemann-Roch map. $\endgroup$ – Angelo Oct 21 '19 at 16:58
  • $\begingroup$ In some cases you can define some Chern classes of $X$ by pushing forward from a resolution of singularities. This is the case for instance if your $X$ is regular in codimension $k\in\mathbb N$: then it makes sense to define $c_\ell(X)$ to be $\pi_*c_\ell(\tilde X)$, where $\pi\colon\tilde X\to X$ is a resolution of singularities. $\endgroup$ – diverietti Oct 21 '19 at 20:39
  • $\begingroup$ There exists a Chern character of coherent sheaves with values in $CH_*(X) \otimes \mathbb{Q}$, due to Fulton and MacPherson. I believe that existence of integral Chern classes of coherent sheaves in the singular setting with values in $CH_*(X)$ is not known. In fact, even $c_2$ for singular surfaces is not clear to me. $\endgroup$ – Evgeny Shinder Oct 21 '19 at 21:16
  • $\begingroup$ The Fulton-Macpherson paper I had in mind is in fact by Baum-Fulton-Macpherson: numdam.org/article/PMIHES_1975__45__101_0.pdf. $\endgroup$ – Evgeny Shinder Oct 22 '19 at 21:46
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Let me summarize what is known about Chern classes and the Chern character on singular varieties, expanding on the comments to the question.

  1. On a normal variety the first Chern class is easily defined by removing the singular locus (which is of codimension at least two) and closing up the first Chern class on the nonsingular part. Then $c_1(\mathcal F)$ is an element of the Chow group $CH_{\dim(X)-1}(X)$, isomorphic to the class group of Weil divisors.

(ADDED: If a variety is non-normal then there is no homomorphism $c_1$ from the Grothendieck group of coherent sheaves to $CH_{\dim(X)-1}(X)$. For instance, if $X$ is a nodal union of two $\mathbb{P}^1$ intersecting in a point, $CH_0(X) = \mathbb{Z}$, and the structure sheaves of the two components are forced to have $c_1$ equal to $1/2$)

  1. In general there is no way to define $c_2$ for coherent sheaves in Chow groups with integer coefficients which would satisfy the usual axioms of Chern classes. Given that Chow groups of singular varieties in general do not form a ring, it is not exactly clear what axioms one would impose on $c_2$. However, for quotient singularites (such as cone over a conic), when Chow groups form a ring rationally, $c_2$ is forced to be non-integral for structure sheaves of Weil divisors. See Langer's paper ``Chern classes of reflexive sheaves on normal surfaces" for the construction of $c_2$ (rationally!) satisfying some of the expected axioms. Langer also presents an example due to Kawamata, showing that multiplicativity of the total Chern class under exact sequences can not hold true.

  2. Baum-Fulton-Macpherson in http://www.numdam.org/article/PMIHES_1975__45__101_0.pdf construct Chern character for singular varieties as a homomorphism from K-theory of coherent sheaves to Chow groups with rational coefficients. The construction proceeds by embedding $X$ into a smooth variety $M$ and constructing the Chern character as an element in Chow groups of $M$ with supports $X$, which is isomorphic to Chow groups of $X$ and then showing independence of $M$. This Chern character provides an isomorphism between rational K-group of coherent sheaves and rational Chow groups. Given all the problems with $c_2$ explained in (2), it's amazing that this works!

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