Grothendieck–Riemann–Roch theorem

A vector bundle ${\displaystyle E\to C}$ of rank ${\displaystyle n}$ and degree ${\displaystyle d}$ (defined as the degree of its determinant; or equivalently the degree of its first Chern class) on a smooth projective curve over a field ${\displaystyle k}$ has a formula similar to Riemann-Roch for line bundles. If we take ${\displaystyle X=C}$ and ${\displaystyle Y=\{*\}}$ a point then the Grothendieck-Riemann-Roch formula can be read as

${\displaystyle {\begin{aligned}\mathrm {ch} (f_{!}{\mathcal {F}}^{\bullet })&=h^{0}(C,E)-h^{1}(C,E)\\f_{*}(\mathrm {ch} (E)\mathrm {td} (X))&=f_{*}((n+c_{1}(E))(1+(1/2)c_{1}(T_{C})))\\&=f_{*}(n+c_{1}(E)+(n/2)c_{1}(T_{C}))\\&=f_{*}(c_{1}(E)+(n/2)c_{1}(T_{C}))\\&=d+n(1-g)\end{aligned}}}$

hence

${\displaystyle \chi (C,E)=d+n(1-g).}$[6]

This formula also holds for coherent sheaves of rank ${\displaystyle n}$ and degree ${\displaystyle d}$.

One of the advantages of the Grothendieck–Riemann–Roch formula is it can be interpreted as a relative version of the Hirzebruch–Riemann–Roch formula. For example, a smooth morphism ${\displaystyle f\colon X\to Y}$ has fibers which are all equi-dimensional (and isomorphic as topological spaces when base changing to ${\displaystyle \mathbb {C} }$). This fact is useful in moduli-theory when considering a moduli space ${\displaystyle {\mathcal {M}}}$ parameterizing smooth proper spaces. For example, David Mumford used this formula to deduce relationships of the Chow ring on the moduli space of algebraic curves.[7]

For the moduli stack of genus ${\displaystyle g}$ curves (and no marked points) ${\displaystyle {\overline {\mathcal {M}}}_{g}}$ there is a universal curve ${\displaystyle \pi \colon {\overline {\mathcal {C}}}_{g}\to {\overline {\mathcal {M}}}_{g}}$ where ${\displaystyle {\overline {\mathcal {C}}}_{g}={\overline {\mathcal {M}}}_{g,1}}$ (is the moduli stack of curves of genus ${\displaystyle g}$ and one marked point. Then, he defines the tautological classes

${\displaystyle {\begin{aligned}K_{{\overline {\mathcal {C}}}_{g}/{\overline {\mathcal {M}}}_{g}}&=c_{1}(\omega _{{\overline {\mathcal {C}}}_{g}/{\overline {\mathcal {M}}}_{g}})\\\kappa _{l}&=\pi _{*}(K_{{\overline {\mathcal {C}}}_{g}/{\overline {\mathcal {M}}}_{g}}^{l+1})\\\mathbb {E} &=\pi _{*}(\omega _{{\overline {\mathcal {C}}}_{g}/{\overline {\mathcal {M}}}_{g}})\\\lambda _{l}&=c_{l}(\mathbb {E} )\end{aligned}}}$

where ${\displaystyle 1\leq l\leq g}$ and ${\displaystyle \omega _{{\overline {\mathcal {C}}}_{g}/{\overline {\mathcal {M}}}_{g}}}$ is the relative dualizing sheaf. Note the fiber of ${\displaystyle \omega _{{\overline {\mathcal {C}}}_{g}/{\overline {\mathcal {M}}}_{g}}}$over a point ${\displaystyle [C]\in {\overline {\mathcal {M}}}_{g}}$ this is the dualizing sheaf ${\displaystyle \omega _{C}}$. He was able to find relations between the ${\displaystyle \lambda _{i}}$ and ${\displaystyle \kappa _{i}}$ describing the ${\displaystyle \lambda _{i}}$ in terms of a sum of ${\displaystyle \kappa _{i}}$[7] (corollary 6.2) on the chow ring ${\displaystyle A^{*}({\mathcal {M}}_{g})}$ of the smooth locus using Grothendieck-Riemann-Roch. Because ${\displaystyle {\overline {\mathcal {M}}}_{g}}$ is a smooth Deligne–Mumford stack, he considered a covering by a scheme ${\displaystyle {\tilde {\mathcal {M}}}_{g}\to {\overline {\mathcal {M}}}_{g}}$ which presents ${\displaystyle {\overline {\mathcal {M}}}_{g}=[{\tilde {\mathcal {M}}}_{g}/G]}$ for some finite group ${\displaystyle G}$. He uses Grothendieck-Riemann-Roch on ${\displaystyle \omega _{{\tilde {\mathcal {C}}}_{g}/{\tilde {\mathcal {M}}}_{g}}}$ to get

${\displaystyle \mathrm {ch} (\pi _{!}(\omega _{{\tilde {\mathcal {C}}}/{\tilde {\mathcal {M}}}}))=\pi _{*}(\mathrm {ch} (\omega _{{\tilde {\mathcal {C}}}/{\tilde {\mathcal {M}}}})\mathrm {Td} ^{\vee }(\Omega _{{\tilde {\mathcal {C}}}/{\tilde {\mathcal {M}}}}^{1}))}$

Because

${\displaystyle \mathbf {R} ^{1}\pi _{!}({\omega _{{\tilde {\mathcal {C}}}_{g}/{\tilde {\mathcal {M}}}_{g}}})\cong {\mathcal {O}}_{\tilde {M}},}$

this gives the formula

${\displaystyle \mathrm {ch} (\mathbb {E} )=1+\pi _{*}({\text{ch}}(\omega _{{\tilde {\mathcal {C}}}/{\tilde {\mathcal {M}}}}){\text{Td}}^{\vee }(\Omega _{{\tilde {\mathcal {C}}}/{\tilde {\mathcal {M}}}}^{1})).}$

The computation of ${\displaystyle \mathrm {ch} (\mathbb {E} )}$ can then be reduced even further. In even dimensions ${\displaystyle 2k}$,

${\displaystyle {\text{ch}}(\mathbb {E} )_{2k}=0.}$

Also, on dimension 1,

${\displaystyle \lambda _{1}=c_{1}(\mathbb {E} )={\frac {1}{12}}(\kappa _{1}+\delta ),}$

where ${\displaystyle \delta }$ is a class on the boundary. In the case ${\displaystyle g=2}$ and on the smooth locus ${\displaystyle {\mathcal {M}}_{g}}$ there are the relations

${\displaystyle {\begin{aligned}\lambda _{1}&={\frac {1}{12}}\kappa _{1}\\\lambda _{2}&={\frac {\lambda _{1}^{2}}{2}}={\frac {\kappa _{1}^{2}}{288}}\end{aligned}}}$

which can be deduced by analyzing the Chern character of ${\displaystyle \mathbb {E} }$.

Closed embeddings ${\displaystyle f\colon Y\to X}$ have a description using the Grothendieck-Riemann-Roch formula as well, showing another non-trivial case where the formula holds.[8] For a smooth variety ${\displaystyle X}$ of dimension ${\displaystyle n}$ and a subvariety ${\displaystyle Y}$ of codimension ${\displaystyle k}$, there is the formula

${\displaystyle c_{k}({\mathcal {O}}_{Y})=(-1)^{k-1}(k-1)![Y]}$

Using the short exact sequence

${\displaystyle 0\to {\mathcal {I}}_{Y}\to {\mathcal {O}}_{X}\to {\mathcal {O}}_{Y}\to 0}$,

there is the formula

${\displaystyle c_{k}({\mathcal {I}}_{Y})=(-1)^{k}(k-1)![Y]}$

for the ideal sheaf since ${\displaystyle 1=c({\mathcal {O}}_{X})=c({\mathcal {O}}_{Y})c({\mathcal {I}}_{Y})}$.

Grothendieck-Riemann-Roch can be used in proving that a coarse moduli space ${\displaystyle M}$, such as the moduli space of pointed algebraic curves ${\displaystyle M_{g,n}}$, admits an embedding into a projective space, hence is a quasi-projective variety. This can be accomplished by looking at canonically associated sheaves on ${\displaystyle M}$ and studying the degree of associated line bundles. For instance, ${\displaystyle M_{g,n}}$[9] has the family of curves

${\displaystyle \pi \colon C_{g,n}\to M_{g,n}}$

with sections

${\displaystyle s_{i}\colon M_{g,n}\to C_{g,n}}$

corresponding to the marked points. Since each fiber has the canonical bundle ${\displaystyle \omega _{C}}$, there are the associated line bundles

${\displaystyle \Lambda _{g,n}(\pi )=\det(\mathbf {R} \pi _{*}(\omega _{C_{g,n}/M_{g,n}}))}$ and ${\displaystyle \chi _{g,n}^{(i)}=s_{i}^{*}(\omega _{C_{g,n}/M_{g,n}})}$.

It turns out that

${\displaystyle \Lambda _{g,n}(\pi )\otimes \left(\bigotimes _{i=1}^{n}\chi _{g,n}^{(i)}\right)}$

is an ample line bundle[9]^{pg 209}, hence the coarse moduli space ${\displaystyle M_{g,n}}$ is quasi-projective.