Applications of Grothendieck-Riemann-Roch?

Question

I am currently trying to learn a bit about Grothendieck-Riemann-Roch...

To try to get a better feeling for it, I am looking for examples of nice applications of GRR applied to a proper morphism $X \to Y$ where $Y$ is not a point. I already I know of a fair number of nice applications of HRR, i.e. GRR when $Y$ is a point. I've read through some of the relevant sections of Fulton's Intersection Theory book, but I've only found applications of HRR there, though it's very possible that I overlooked something.

I am also interested in seeing worked-out, explicit, concrete examples, with explicit Chow/cohomology classes.

Thanks much!

Answer 1

Check out Harris & Morrison's "Moduli of Curves", section 3E. There is a wealth of examples of applications of GRR coming from moduli theory, in which one applies it to projection from the universal family or some fibered power of the universal family. The basic idea in these cases is that both the base space and the total space are rather complicated beasts but the fibers of the morphisms are usually quite tractable, since they are just the gadgets you are trying to parametrize.

For more examples in the same vein, you could read the classic "Towards an enumerative geometry of the moduli space of curves" by David Mumford.

Answer 2

Here are four ``applications'' of the Grothendieck-Riemann-Roch theorem that I know of.

1. Moduli space of Enriques surfaces

The coarse moduli space of Enriques surfaces is known to be quasi-affine. A proof of this was given by Pappas using the Grothendieck-Riemann-Roch theorem in:

http://arxiv.org/abs/math/0701546

More precisely, it's the following result which is shown in the above article using GRR.

Theorem. The line bundle $R^0 f_\ast (\mathcal{L}\otimes \mathcal{L})$ is a torsion line bundle on $Y$.

2. Computing with the multiplication map on an abelian variety

Let $X$ be an abelian variety of dimension $g$. The following is based on the article

Heights for line bundles on arithmetic varieties

by J. Jahnel. (You can find it easily with Google.)

Let $p:X\times X \longrightarrow X$ be the projection onto the first coordinate. Similarly, let $q:X\times X\longrightarrow X$ be the projection onto the second coordinate. For any line bundle $\mathcal{F}$ on $X$, we define its Mumford line bundle on $X\times X$, denoted by $\Lambda$, as $$\Lambda := m^\ast \mathcal{F}\otimes (p^\ast \mathcal{F})^{-1} \otimes (q^\ast \mathcal{F})^{-1}.$$ The following theorem is a special case of Theorem 1.7 in Jahnel. Its proof uses GRR and is contained in the proof of Proposition 3.4.

Theorem. For any ample line bundle $\mathcal{L}$, we have that $$(\det q_!(\Lambda\otimes p^\ast\mathcal{L}))^{-1} = \left(\det q_!(m^\ast\mathcal{L} \otimes (q^\ast \mathcal{L})^{-1})\right)^{-1}$$ is an ample line bundle on $X$.

3. The weak Riemann-Hurwitz formula

Let $\pi:X\longrightarrow Y$ be a finite morphism of smooth quasi-projective varieties over an algebraically closed field.

Then, the Grothendieck-Riemann-Roch theorem applied to $\pi$ and $\mathcal{O}_X$ gives $$ch(\pi_\ast \mathcal{O}_X) = \pi_\ast( td(X/Y)).$$

In degree 0 this gives something we all know: $c_0(\pi_\ast \mathcal{O}_X)$ is the rank of $\pi_\ast \mathcal{O}_X$ whereas $$\pi_\ast (td(X/Y)_{0} = \pi_\ast (0) = \deg \pi.$$ That is, we get that the rank of the vector bundle $\pi_\ast \mathcal{O}_X$ equals $\deg \pi$.

In degree 1 it gives a weak version of the Riemann-Hurwitz theorem. Namely, it shows that $c_1(\pi_\ast \mathcal{O}_X) = \pi_\ast( td(X/Y)_{(1)})$ in the Chow ring of $Y$ (tensored with $\mathbf{Q}$). I call this version weak because you actually have an equality in the Chow ring of $X$ (tensored with $\mathbf{Q}$).

I should say that this isn't the complete picture yet. The ramification divisor appears when you do a local computation as in Chapter 3.6 Prop. 13 of Serre's book Local fields.

In higher degree, you can write out what GRR gives but I can't give a geometric interpretation of this. Maybe someone else can?

4. Heights for covers of algebraic surfaces in characteristic zero

Let $k$ be an algebraically closed field of characteristic zero.

Fix an smooth projective connected curve $C$ over $k$ and a flat projective morphism $h:X\longrightarrow C$ with $X$ connected and regular such that the generic fibre $X_\eta$ is nonsingular. Let $D\subset X$ be a simple normal crossings divisor. (This means that its components are nonsingular and meet transversally.)

We now define the set $Cov(C,X,h,D)$ as the set of finite morphisms $\pi:Y \longrightarrow X$ which arise as the normalization of $X$ in the function field of some finite etale morphism $V \longrightarrow X-D$ (with $V$ connected). For any element $\pi:Y \longrightarrow X$ of $Cov(C,X,h,D)$, we have that $\pi$ is finite flat and surjective and $Y$ is a normal integral complex algebraic surface with rational singularities.

You can define a height over $C$ on this set and give a nice formula for this height using the Grothendieck-Riemann-Roch theorem. This is all contained in the following

Theorem. Let $\pi:Y \longrightarrow X$ be an element of $Cov(C,X,h,D)$. Choose a resolution of singularities $\rho:Y^\prime\longrightarrow Y$ and write $f=h\circ \pi \circ \rho$. Then the first Chern class $c_1(f_! \mathcal{O}_{Y^\prime})$ equals $$f_\ast(td(Y^\prime)_{(2)}) - h_\ast(td(X)_{(1)})td(C)_{(1)} \deg \pi - h_\ast(c_1(\pi_\ast \mathcal{O}_Y))td(C)_{(1)}$$ in the class group of $C$ (tensored with $\mathbf{Q}$). Define the height over $C$ of $\pi$ to be $$ Height(\pi) = \deg c_1(f_! \mathcal{O}_{Y^\prime}).$$ This height is independent of the resolution $\rho$.

Proof. The formula for $c_1(f_! \mathcal{O}_{Y^\prime})$ is obtained by applying GRR to $(f,\mathcal{O}_{Y^\prime})$ and $(h,\pi_\ast \mathcal{O}_Y)$. The fact that the height is independent of $Y^\prime$ follows from the formula and the Hirzebruch-Riemann-Roch theorem. For details see the proof of Theorem 1.1 in http://arxiv.org/abs/0807.0184 .

Answer 3

Whenever you have a Fourier-Mukai transorm, if you want to compute the Chern character of an image of a sheaf, you need GRR.I think you can find examples in the Huybrecht's book "Fourier-Muaki transforms in algebraic geometry".

Answer 4

EDIT: Hm. I've just realized that this "answer" does not answer the question at all. This is an application of Grothendieck Duality, not of GRR. I suppose I was not reading the question carefully when I wrote this. Sigh. Sorry. Given the number of up-votes, it seems reasonable to leave it here, but I apologize for the possible confusion I may have caused with it.

---

Here is a simple proof of Kempf's criterion for rational singularities:

Theorem. (Kempf) Let $X$ be a normal variety over $\mathbb C$. Then $X$ has rational singularities (i.e., for a resolution $\phi:\widetilde X\to X$, $R^i\phi_*\mathcal O_{\widetilde X}=0$ for $i>0$) if and only if $X$ is Cohen-Macaulay and $\phi_*\omega_{\widetilde X}\simeq \omega_X$.

Proof. ($RHom$ stands for sheaf-RHom, $\omega^\cdot$ for the dualizing complex, $n=\dim X$).

First assume that $X$ has rational singularities. Then \begin{multline} \omega_X^\cdot\simeq RHom_X(\mathcal O_X,\omega_X^\cdot)\simeq RHom_X(R\phi_*\mathcal O_{\widetilde X},\omega_X^\cdot) \simeq_{\text{by Grothendieck duality}}\\\ \simeq R\phi_*RHom(\mathcal O_{\widetilde X}, \omega_{\widetilde X}^\cdot) \simeq R\phi_*\omega_{\widetilde X}[n]\simeq \phi_*\omega_{\widetilde X}[n] \end{multline} The last isomorphism follows by Grauert-Riemenschneider vanishing. The two ends of the displayed isomorphism shows that $\omega_X^\cdot$ has only one non-zero cohomology sheaf and hence $X$ is Cohen-Macaulay and that non-zero cohomology sheaf is $\phi_*\omega_{\widetilde X}\simeq \omega_X$.

The other direction is similar: \begin{multline} \mathcal O_X\simeq RHom_X(\omega_X^\cdot,\omega_X^\cdot)\simeq RHom_X(R\phi_*\omega_{\widetilde X}^\cdot,\omega_X^\cdot) \simeq_{\text{by Grothendieck duality}}\\\ \simeq R\phi_*RHom(\omega_{\widetilde X}^\cdot, \omega_{\widetilde X}^\cdot) \simeq R\phi_*\mathcal O_{\widetilde X} \end{multline}

Edit: A little bit more use of G̶R̶R̶ Grothendieck Duality(!) gives you interesting theorems about singularities. See for instance here.

Answer 5

Maybe the easiest non-trivial case to consider is when one has a double cover $f \colon X \to Y$ of smooth projective varieties. Such a double cover is defined by a smooth divisor $B$ in $Y$ which is $2$-divisible in the Picard group of $Y$, and by a choice of a square root $L$, namely $L^2 =B$. Then $$f_* \mathscr{O}_X=\mathscr{O}_Y \oplus L^{-1}.$$ Now take a line bundle $\mathscr{O}_X(D)$ in $X$. Then $f_{*} \mathscr O_X(D)$ is a rank $2$ vector bundle on $Y$, and its Chern classes can be recovered in terms of $D$ and $L$ by using Grothendieck-Riemann-Roch. I think these computations were first carried out by Schwarzenberger in the case of smooth surfaces, see [Vector bundles on algebraic surfaces and Vector bundles on the projective plane, Proc. London Math. Soc. 11 (1961)].

Proposition. Let $f \colon X \to Y$ and $L$ be as above and let $D$ be a divisor on $X$.

(1) The following equality holds in $\textrm{Pic} \, Y$:

$$c_1 (f_* \mathscr{O}_X(D))= [f_* D]-L.$$

(2) The following equality holds in $H^4(Y, \, \mathbb{Z}[1/2])$:

$$c_2 (f_* \mathscr{O}_X(D))= 1/2 ((f_* D)^2-f_*(D \cdot D)-(f_*D) \cdot L ).$$

Answer 6

I give Bismut, Gillet and Christophe Soulé observation about Grothendieck-Riemann-Roch and chern form of relative canonical bundle which is related to degeneration of Kahler-Einstein metric

One of the applications of Grothendieck-Riemann-Roch is study of degeneration of Kahler-Einstein metric.

Let $X$ be a smooth quasi-projective scheme over a field and $K_{0}(X)$ be the Grothendieck group of bounded complexes of coherent sheaves. Consider the Chern character as a functorial transformation $${\mbox{ch}}\colon K_{0}(X)\to A(X,{\mathbb {Q}})$$, where $A_{d}(X,{\mathbb {Q}})$ is the Chow group of cycles on $X$ of dimension $d$ modulo rational equivalence, tensored with the rational numbers. Now consider a proper morphism

$$f\colon X\to Y$$

between smooth quasi-projective schemes and a bounded complex of sheaves ${{\mathcal {F}}^{\bullet }}$ on $X$. Let $R$ be the right derived functor, then we have the following theorem of Grothendieck-Riemann-Roch.

The Grothendieck-Riemann-Roch theorem relates the pushforward map $$f_{\mbox{!}}=\sum (-1)^{i}R^{i}f_{*}\colon K_{0}(X)\to K_{0}(Y)$$

and the pushforward $f_{*}\colon A(X)\to A(Y)$ by the formula $${\mbox{ch}}(f_{\mbox{!}}{\mathcal {F}}^{\bullet }){\mbox{td}}(Y)=f_{*}({\mbox{ch}}({\mathcal {F}}^{\bullet }){\mbox{td}}(X))$$ Here $td(X)$ is the Todd genus of (the tangent bundle of) $X$. In fact, since the Todd genus is functorial and multiplicative in exact sequences, we can rewrite the Grothendieck-Riemann-Roch formula as $${\mbox{ch}}(f_{\mbox{!}}{\mathcal {F}}^{\bullet })=f_{*}({\mbox{ch}}({\mathcal {F}}^{\bullet }){\mbox{td}}(T_{f}))$$ where $T_f$ is the relative tangent sheaf of $f$, defined as the element $TX - f^*TY$ in $K^0(X)$. For example, when $f$ is a smooth morphism, $T_f$ is simply a vector bundle, known as the tangent bundle along the fibers of $f$.

Let $\pi : \mathcal X → \mathcal B$ be a proper, smooth holomorphic map of reduced complex spaces. Let $(\mathcal E, \mathcal h)$ be a hermitian vector bundle on $\mathcal X$ . Let $\lambda(\mathcal E)=\det f_!(\mathcal E)$ be determinant line bundle in the sense of in the derived category.

There exists a Quillen metric $h_Q$ on the determinant line bundle(due to Bismut, Gillet and Soule) such that the following equality holds for its first Chern form on the base $\mathcal B$ and the component in degree two of a fiber integral:

$$c_1(\lambda(\mathcal E), h^Q)=\left[\int_{\mathcal X/\mathcal B}td(\mathcal X/\mathcal B, \omega_{\mathcal X/\mathcal B})ch(\mathcal E,h)\right]_2$$

Here $ch$ and $td$ stand for the Chern and Todd character forms and $\omega_{\mathcal X/\mathcal B}$ is the fiberwise Kahler form

Now let $(L, h)$ be a hermitian line bundle and take $\mathcal E=(L-L^{-1})^{n+1}$

then the following is known result of Bismut, Gillet and Soule $$c_1(\lambda(\mathcal E), h^Q)=2^{n+1}\int_{\mathcal X/\mathcal B}c_1(L,h)^{n+1}$$

Now if we take $L=K_{\mathcal X/\mathcal B}=K_{\mathcal X}\otimes \pi^* K_{\mathcal B^{-1}}$ in previous formula,

If you take fibers are of general type i.e., $c_1({X_b})<0$ , then the left hand side of previous formula is canonical metric on moduli space of fibers which is Weil-Petersson metric . So if fibers are polarized CY or fibers be K-stable then we have the same formula(with adding - behind of integral) for canonical metric on moduli space of polarized CY or canonical metric on moduli space of K-stable fibers which we still call them Weil-Petersson metric.

See

1. Bismut, J.-M.: M´etriques de Quillen et d´egenerescence de vari´et´es k¨ahl´eriennes. C. R. Acad. Sci. Paris S´er. I. Math. 319, 1287–1291 (1994).

2. Bismut, J.-M.; Gillet, H.; Soule, Ch.: Analytic torsion and holomorphic determinant bundles I, II, III. Comm. Math. Phys. 115, 49–78, 79–126, 301–351 (1988)

3.Georg Schumacher, Positivity of relative canonical bundles and applications,October 2012, Volume 190, Issue 1, pp 1–56,Inventiones mathematicae

Answer 7

This is probably covered by Dan Petersen's answer, but the first application I ever saw was in the paper of Harris and Mumford, computing the canonical class of the moduli space of curves around 1981.

Answer 8

Following the answer of Sasha, I can give you here an explicit example of computations using GRR. Suppose you have a smooth projective surface $S$ and a $(-2)$-curve $i : C \hookrightarrow S$, that is $C^2=-2$, $C\simeq \mathbb{P}^1$. Then for any integer $a\in\mathbb{Z}$ the object $i_\ast O_C(a)$ is a spherical object in $D^b(S)$, and thus you can consider the spherical twist (in the sense of Seidel and Thomas) $T:= T_{\mathcal{O}_C(a)}$.

Now the action $T^H$ of $T$ on the cohomology $H^{\ast} (S,\mathbb{Q})$ (following Huybrechts, "Fourier-Mukai Transforms in Algebraic Geometry") is easily computable once you know the chern character $ch(i_*\mathcal{O}_C(a))$. I do these computations in cohomology:

By GRR for $i : C \hookrightarrow S$ you have

$$Todd(S) ch(i_\ast O_C(a)) = i_\ast (ch(O_C(a)) Todd(C)) = i_\ast([C]+(a+1)[x])$$ for $[x]$ the (cohomology) class of a point in $C\simeq \mathbb{P}^1$. You can get rid of $Todd(S)$ using that $K_S\cdot C=0$, and you obtain $$ch(i_\ast O_C(a)) = [C]+(a+1)[x].$$