Applications of Grothendieck-Riemann-Roch? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T10:26:09Z http://mathoverflow.net/feeds/question/43768 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43768/applications-of-grothendieck-riemann-roch Applications of Grothendieck-Riemann-Roch? Kevin Lin 2010-10-27T07:25:14Z 2011-04-29T17:57:18Z <p>I am currently trying to learn a bit about Grothendieck-Riemann-Roch...</p> <p>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 <em>not</em> 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.</p> <p>I am also interested in seeing worked-out, explicit, concrete examples, with explicit Chow/cohomology classes.</p> <p>Thanks much!</p> http://mathoverflow.net/questions/43768/applications-of-grothendieck-riemann-roch/43770#43770 Answer by Dan Petersen for Applications of Grothendieck-Riemann-Roch? Dan Petersen 2010-10-27T07:45:20Z 2010-10-27T07:45:20Z <p>Check out Harris &amp; 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. </p> <p>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.</p> http://mathoverflow.net/questions/43768/applications-of-grothendieck-riemann-roch/43777#43777 Answer by Sasha for Applications of Grothendieck-Riemann-Roch? Sasha 2010-10-27T09:08:21Z 2010-10-27T09:08:21Z <p>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".</p> http://mathoverflow.net/questions/43768/applications-of-grothendieck-riemann-roch/43809#43809 Answer by Francesco Polizzi for Applications of Grothendieck-Riemann-Roch? Francesco Polizzi 2010-10-27T14:58:49Z 2010-10-27T15:11:17Z <p>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</p> <p>$f_* \mathcal{O}_X=\mathcal{O}_Y \oplus L^{-1}$.</p> <p>Now take a line bundle $\mathcal{O}_X(D)$ in $X$.</p> <p>Then $f_{*} O_X(D)$ 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, Proc. London Math. Soc. 11 (1961)] and [Vector bundles on the projective plane, same journal]. </p> <p><strong>Proposition</strong></p> <p>Let $f \colon X \to Y$ and $L$ be as above, $D$ a divisor on $X$.</p> <p><strong>(1)</strong> We have the following equality in $\textrm{Pic}(Y)$:</p> <p>$c_1$ $(f_* O(D))$= $[f_* D]-L$.</p> <p><strong>(2)</strong> We have the following equality in $H^4(Y, \mathbb{Z}[1/2])$:</p> <p>$c_2$ $(f_* O(D))$= $1/2((f_* D)^2-f_*(D \cdot D)-(f_*D) \cdot L)$.</p> http://mathoverflow.net/questions/43768/applications-of-grothendieck-riemann-roch/43816#43816 Answer by Sándor Kovács for Applications of Grothendieck-Riemann-Roch? Sándor Kovács 2010-10-27T15:43:47Z 2010-10-27T18:38:50Z <p>Here is a simple proof of Kempf's criterion for rational singularities:</p> <p><strong>Theorem.</strong> (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 $\phi_*\omega_{\widetilde X}\simeq \omega_X$.</p> <p><strong>Proof.</strong> ($RHom$ stands for sheaf-RHom, $\omega^\cdot$ for the dualizing complex, $n=\dim X$).</p> <p>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 $\phi_*\omega_{\widetilde X}\simeq \omega_X$.</p> <p>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} </p> <p><strong>Edit:</strong> A little bit more use of GRR gives you interesting theorems about singularities. See for instance <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.dmj/1092749293" rel="nofollow">here</a>. </p> http://mathoverflow.net/questions/43768/applications-of-grothendieck-riemann-roch/43857#43857 Answer by Ariyan Javanpeykar for Applications of Grothendieck-Riemann-Roch? Ariyan Javanpeykar 2010-10-27T19:21:53Z 2011-04-29T17:57:18Z <p>Here's four applications'' of the Grothendieck-Riemann-Roch theorem that I know of.</p> <p><strong>1. Moduli space of Enriques surfaces</strong></p> <p>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: </p> <p><a href="http://arxiv.org/abs/math/0701546" rel="nofollow">http://arxiv.org/abs/math/0701546</a></p> <p>More precisely, it's the following result which is shown in the above article using GRR.</p> <p><strong>Theorem.</strong> The line bundle $R^0 f_\ast (\mathcal{L}\otimes \mathcal{L})$ is a torsion line bundle on $Y$.</p> <p><strong>2. Computing with the multiplication map on an abelian variety</strong></p> <p>Let $X$ be an abelian variety of dimension $g$. The following is based on the article </p> <p><em>Heights for line bundles on arithmetic varieties</em></p> <p>by J. Jahnel. (You can find it easily with Google.)</p> <p>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 <em>Mumford line bundle</em> 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.</p> <p><strong>Theorem.</strong> 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 $X$.</p> <p><strong>3. The weak Riemann-Hurwitz formula</strong></p> <p>Let $\pi:X\longrightarrow Y$ be a finite morphism of smooth quasi-projective varieties over an algebraically closed field. </p> <p>Then, the Grothendieck-Riemann-Roch theorem applied to $\pi$ and $\mathcal{O}_X$ gives <code>$$ch(\pi_\ast \mathcal{O}_X) = \pi_\ast( td(X/Y)).$$</code></p> <p>In degree 0 this gives something we all know: <code>$c_0(\pi_\ast \mathcal{O}_X)$</code> is the rank of <code>$\pi_\ast \mathcal{O}_X$</code> whereas <code>$$\pi_\ast (td(X/Y)_{0} = \pi_\ast (0) = \deg \pi.$$</code> That is, we get that the rank of the vector bundle <code>$\pi_\ast \mathcal{O}_X$</code> equals $\deg \pi$.</p> <p>In degree 1 it gives a weak version of the Riemann-Hurwitz theorem. Namely, it shows that <code>$c_1(\pi_\ast \mathcal{O}_X) = \pi_\ast( td(X/Y)_{(1)})$</code> 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 $\mathbf{Q}$). </p> <p>Ow and maybe 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 <em>Local fields</em>.</p> <p>In higher degree, you can write out what GRR gives but I can't give a geometric interpretation of this. Maybe someone else can?</p> <p><strong>4. Heights for covers of algebraic surfaces in characteristic zero</strong> </p> <p>Let $k$ be an algebraically closed field of characteristic zero.</p> <p>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.)</p> <p>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.</p> <p>You can define a <em>height over $C$</em> on this set and give a nice formula for this height using the Grothendieck-Riemann-Roch theorem. This is all contained in the following</p> <p><strong>Theorem.</strong> 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 <code>$$f_\ast(td(Y^\prime)_{(2)}) - h_\ast(td(X)_{(1)})td(C)_{(1)} \deg \pi - h_\ast(c_1(\pi_\ast \mathcal{O}_Y))td(C)_{(1)}$$</code> in the class group of $C$ (tensored with $\mathbf{Q}$). Define the <em>height over $C$</em> of $\pi$ to be $$Height(\pi) = \deg c_1(f_! \mathcal{O}_{Y^\prime}).$$ This height is independent of the resolution $\rho$.</p> <p><em>Proof.</em> The formula for <code>$c_1(f_! \mathcal{O}_{Y^\prime})$</code> is obtained by applying GRR to <code>$(f,\mathcal{O}_{Y^\prime})$</code> and <code>$(h,\pi_\ast \mathcal{O}_Y)$</code>. The fact that the height is independent of <code>$Y^\prime$</code> follows from the formula and the Hirzebruch-Riemann-Roch theorem. For details see the proof of Theorem 1.1 in <a href="http://arxiv.org/abs/0807.0184" rel="nofollow">http://arxiv.org/abs/0807.0184</a> .</p> http://mathoverflow.net/questions/43768/applications-of-grothendieck-riemann-roch/44302#44302 Answer by roy smith for Applications of Grothendieck-Riemann-Roch? roy smith 2010-10-31T03:16:26Z 2011-04-27T00:32:16Z <p>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.</p>