Euler-Maclaurin formula and Riemann-Roch - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T05:53:33Z http://mathoverflow.net/feeds/question/10667 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch Euler-Maclaurin formula and Riemann-Roch VA 2010-01-04T05:10:05Z 2012-12-14T02:25:23Z <p>Let $Df$ denote the derivative of a function $f(x)$ and $\bigtriangledown f=f(x)-f(x-1)$ be the discrete derivative. Using the Taylor series expansion for $f(x-1)$, we easily get $\bigtriangledown = 1- e^{-D}$ or, by taking the inverses, $$\frac{1}{\bigtriangledown} = \frac{1}{1-e^{-D}} = \frac{1}{D}\cdot \frac{D}{1-e^{-D}}= \frac{1}{D} + \frac12+ \sum_{k=1}^{\infty} B_{2k}\frac{D^{2k-1}}{(2k)!} ,$$ where $B_{2k}$ are Bernoulli numbers.</p> <p>(<b>Edit:</b> I corrected the signs to adhere to the most common conventions.)</p> <p>Here, $(1/D)g$ is the opposite to the derivative, i.e. the integral; adding the limits this becomes a definite integral $\int_0^n g(x)dx$. And $(1/\bigtriangledown)g$ is the opposite to the discrete derivative, i.e. the sum $\sum_{x=1}^n g(x)$. So the above formula, known as Euler-Maclaurin formula, allows one, sometimes, to compute the discrete sum by using the definite integral and some error terms. </p> <p>Usually, there is a nontrivial remainder in this formula. For example, for $g(x)=1/x$, the remainder is Euler's constant $\gamma\simeq 0.57$. Estimating the remainder and analyzing the convergence of the power series is a long story, which is explained for example in the nice book "Concrete Mathematics" by Graham-Knuth-Patashnik. But the power series becomes finite with zero remainder if $g(x)$ is a polynomial. OK, so far I am just reminding elementary combinatorics.</p> <p>Now, for my question. In the (Hirzebruch/Grothendieck)-Riemann-Roch formula one of the main ingredients is the Todd class which is defined as the product, going over Chern roots $\alpha$, of the expression $\frac{\alpha}{1-e^{-\alpha}}$. This looks so similar to the above, and so suggestive (especially because in the Hirzebruch's version $$\chi(X,F) = h^0(F)-h^1(F)+\dots = \int_X ch(F) Td(T_X)$$ there is also an "integral", at least in the notation) that it makes me wonder: is there a connection?</p> <p>The obvious case to try (which I did) is the case when $X=\mathbb P^n$ and $F=\mathcal O(d)$. But the usual proof in that case is a residue computation which, to my eye, does not look anything like Euler-Maclaurin formula. </p> <p>But is there really a connection?</p> <p><hr></p> <p><b>An edit after many answers:</b> Although the connection with Khovanskii-Pukhlikov's paper and the consequent work, pointed out by Dmitri and others, is undeniable, it is still not obvious to me how the usual Riemann-Roch for $X=\mathbb P^n$ and $F=\mathcal O(d)$ follows from them. It appears that one has to prove the following nontrivial </p> <p><b>Identity:</b> The coefficient of $x^n$ in $Td(x)^{n+1}e^{dx}$ equals $$\frac{1}{n!} Td(\partial /\partial h_0) \dots Td(\partial /\partial h_n) (d+h_0+\dots + h_n)^n |_{h_0=\dots h_n=0}$$</p> <p>A complete answer to my question would include a proof of this identity or a reference to where this is shown. (I did not find it in the cited papers.) I removed the acceptance to encourage a more complete explanation.</p> http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch/10691#10691 Answer by Dmitri for Euler-Maclaurin formula and Riemann-Roch Dmitri 2010-01-04T10:42:25Z 2010-01-04T12:04:52Z <p>As far as I understand this connection was observed (and generalised) by Khovanskii and Puhlikov in the article </p> <p>A. G. Khovanskii and A. V. Pukhlikov, A Riemann-Roch theorem for integrals and sums of quasipolynomials over virtual polytopes, Algebra and Analysis 4 (1992), 188–216, translation in St. Petersburg Math. J. (1993), no. 4, 789–812.</p> <p>This is related to toric geometry, for which some really well written introduction articles are contained on the page of David Cox <a href="http://www3.amherst.edu/~dacox/" rel="nofollow">http://www3.amherst.edu/~dacox/</a></p> <p>Since 1992 many people wrote on this subject, for example </p> <p>EXACT EULER MACLAURIN FORMULAS FOR SIMPLE LATTICE POLYTOPES</p> <p><a href="http://arxiv.org/PS_cache/math/pdf/0507/0507572v2.pdf" rel="nofollow">http://arxiv.org/PS_cache/math/pdf/0507/0507572v2.pdf</a></p> <p>Or Riemann sums over polytopes <a href="http://arxiv.org/PS_cache/math/pdf/0608/0608171v1.pdf" rel="nofollow">http://arxiv.org/PS_cache/math/pdf/0608/0608171v1.pdf</a></p> http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch/10696#10696 Answer by Ilya Nikokoshev for Euler-Maclaurin formula and Riemann-Roch Ilya Nikokoshev 2010-01-04T11:42:20Z 2010-01-04T15:11:02Z <p>Yes, this is a big area of research. I'll add some references to the ones Dmitri provides.</p> <p>Here are references from a question about <em><a href="http://mathoverflow.net/questions/4541/moment-map-for-toric-actions-online-references" rel="nofollow">Moment map for toric actions</a></em>:</p> <ul> <li><a href="http://projecteuclid.org/euclid.jdg/1214459754" rel="nofollow"><em>Riemann-Roch for toric orbifolds</em></a> by Victor Guillemin</li> <li>to learn about toric geometry, draft of a book <a href="http://www.cs.amherst.edu/~dac/toric.html" rel="nofollow"><em>Toric Varieties</em></a> by Cox et al</li> </ul> <p>More on the topic itself:</p> <ul> <li><a href="http://arxiv.org/abs/math/0608171" rel="nofollow"><em>Riemann sums over polytopes</em></a> by Victor Guillemin and Shlomo Sternberg</li> <li><a href="http://arxiv.org/abs/math/0507572" rel="nofollow"><em>Exact Euler Maclaurin formulas for simple lattice polytopes</em></a> by Shlomo Sternberg et al.</li> <li>the original <a href="http://mi.mathnet.ru/eng/aa339" rel="nofollow">paper by Pukhlikov and Khovanskii</a> (page in English, full text in Russian)</li> </ul> <p>A series of papers <a href="http://arxiv.org/find/math/1/au:+Vergne%5FM/0/1/0/all/0/1" rel="nofollow">on arXiv by Michèle Vergne</a>, especially:</p> <ul> <li><a href="http://arxiv.org/abs/math/0103097" rel="nofollow"><em>Residues formulae for volumes and Ehrhart polynomials of convex polytopes</em></a>, arXiv:math/0103097</li> <li><a href="http://arxiv.org/abs/math/0507256" rel="nofollow"><em>Local Euler-Maclaurin formula for polytopes</em></a>, arXiv:math/0507256</li> <li><a href="http://arxiv.org/abs/0803.2810" rel="nofollow"><em>Paradan's wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator</em></a></li> </ul> <p>Also papers by Brion and Vergne, which seem to be missing from arXiv (<a href="http://scholar.google.com/scholar?hl=en&amp;q=brion+and+vergne+&amp;btnG=Search&amp;as%5Fsdt=2000&amp;as%5Fylo=&amp;as%5Fvis=0" rel="nofollow">Google Scholar</a>, thanks to Steve). </p> http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch/10699#10699 Answer by John Mangual for Euler-Maclaurin formula and Riemann-Roch John Mangual 2010-01-04T12:10:46Z 2010-01-04T12:10:46Z <p>Last year, Leonhard Euler posted a note, <a href="http://front.math.ucdavis.edu/0806.4096" rel="nofollow">Finding the sum of any series from a given general term</a> on the arXiv. In recent years, this idea has been extended to sums over lattice approximations of convex polytopes $\Delta \cap \mathbb{Z}^n$ as shown in the other responses.</p> http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch/10703#10703 Answer by Steve Huntsman for Euler-Maclaurin formula and Riemann-Roch Steve Huntsman 2010-01-04T14:25:58Z 2010-01-12T02:20:57Z <p>I thought I'd give a more explicit answer showing how the Todd class appears. Let $Td(x) := \frac{x}{1-e^{-x}} = -\sum_{j=0}^\infty B_j \frac{x^j}{j!}$. Now for $a,b \in \mathbb{Z}$, $z \in \mathbb{R}$, $|z| &lt;&lt; 1$, we have that $Td(\partial_h)e^{hz} = -\sum_{j=0}^\infty B_j \frac{\partial_h^{(j)}}{j!}e^{hz} = -\sum_{j=0}^\infty B_j \frac{z^j}{j!}e^{hz} = Td(z)e^{hz}$. So </p> <p><code>$Td(\partial_g)|_{g=0} Td(\partial_h)|_{h=0} \int_{a-g}^{b+h} e^{xz} dx$</code></p> <p><code>$= Td(\partial_g)|_{g=0} Td(\partial_h)|_{h=0} \frac{e^{(b+h)z} - e^{(a-g)z}}{z}$</code> </p> <p><code>$= \frac{Td(z)e^{bz} - Td(-z)e^{az}}{z} = \frac{e^{bz}}{1-e^{-z}} + \frac{e^{az}}{1-e^z}$</code> </p> <p><code>$= \sum_{k=a}^b e^{kz}$</code>. </p> <p>It follows for suitable functions $f$ (as VA pointed out below) that <code>$\sum_{k=a}^b f(k) = Td(\partial_g)|_{g=0} Td(\partial_h)|_{h=0} \int_{a-g}^{b+h} f(x) dx$</code>.</p> <p><hr /></p> <p>As far as references:</p> <p>Brion and Vergne give a good treatment of the problem. Their key paper is available at <a href="http://www.jstor.org/pss/2152855" rel="nofollow">http://www.jstor.org/pss/2152855</a></p> <p>Ewald's introduction to toric varieties takes place in the context of convex polytopes and is more concrete than others (e.g., Fulton): see <a href="http://books.google.com/books?id=bz8SfJId3BgC" rel="nofollow">http://books.google.com/books?id=bz8SfJId3BgC</a></p> <p>[PPS--I used this work to complete a structure theory for the equilibrium hybridization thermodynamics of DNA about 7 or 8 years ago: see <a href="http://mathoverflow.net/questions/10493/the-matrix-tree-theorem-for-weighted-graphs/10500#10500" rel="nofollow">http://mathoverflow.net/questions/10493/the-matrix-tree-theorem-for-weighted-graphs/10500#10500</a>]</p> http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch/12138#12138 Answer by VA for Euler-Maclaurin formula and Riemann-Roch VA 2010-01-17T20:30:24Z 2010-01-17T23:06:13Z <p>Euler-Maclaurin's formula transforms the integral $I=\int_a^b f(x)dx$ into the finite sum $S=\sum_a^b f(x)$, for two integers $a,b$. As Dmitri pointed out, in 1993 Khovanskii and Pukhlikov gave a multi-dimensional generalization of Euler-Maclaurin which, in particular says the following: </p> <p>Let $P$ be an $n$-dimensional polytope in $\mathbb R^n\supset\mathbb Z^n$ with integral vertices, and further assume that $P$ defines a nonsingular toric variety (i.e. $P$ is simplicial and at every vertex the integral generators of the edges give a basis in $\mathbb Z^n$). Let us say the facets of $P$ are defined by the inequalities $l_j(x)\le a_j$ for some primitive integral linear functions $l_j(x_1,\dots,x_n)$. Denote by $P(h)$ the polytope defined by the inequalities $l_j(x)\le a_j+h_j$. Finally, let $$I(f,h)= \int_{P(h)} f(x)dx, \quad S(f)= \int_{P\cap \mathbb Z^n} f(x).$$ Then for any quasipolynomial $f(x)$ (a sum of products of polynomial and exponential functions) one has $$S(f) = \prod_j Td(\partial / \partial h_j)\ I(f,h)\ |_{h_j=0}.$$</p> <p>Here is how the Hirzebruch-Riemann-Roch for the sheaf $\mathcal F=\mathcal O(d)$ on $X=\mathbb P^n$ follows from the Khovanskii-Pukhlikov's version of Euler-Maclaurin's formula:</p> <p>Taking $P$ to be a simplex of side $d$ and $f(x)=1$, the Khovanskii-Pukhlikov's formula gives $$h^0(\mathbb P^n, \mathcal O(d)) = \prod_{j=0}^n Td(\partial/\partial h_j) \frac{(d+h_0+\dots+h_n)^n}{n!} \ |_{h_j=0}$$ which by making a substitution $y=d+h_0+\dots+h_n$ transforms into $Td(\partial/\partial y)^{n+1} (y^n/n!)\ |_{y=d}.$</p> <p>The usual Hirzebruch-Riemann-Roch, on the other hand, says that $h^0(\mathbb P^n,\mathcal O(d))$ is the coefficient of $x^n$ in the expression $Td(x)^{n+1} e^{dx}$. So why is this the same? Because $$Td(x)^{n+1} e^{dx} = Td(\partial/ \partial y)^{n+1} e^{yx}\ |_{y=d}$$ (here we used the fact that $(\partial/ \partial y)^k e^{yx} = x^k e^{yx}$) and the coefficient of $x^n$ in $e^{yx}$, expanded as a power series in $x$, is $(y^n/n!)$. QED</p> <p>Now that wasn't so hard, but why isn't this written somewhere? Or am I missing a reference? <hr> So what does this suggest conceptually about the meaning of Hirzebruch-Riemann-Roch? I think, clearly, it suggests that </p> <ol> <li><p>The pushforward $$f_!:K(X)\to K(pt)=\mathbb Z, \qquad \mathcal F\mapsto \chi(\mathcal F) = h^0(F)-h^1(F)+\dots$$ between the K-groups should be considered to be the "discrete summation" of a "function" $f=f(\mathcal F)$. Indeed, for say a toric variety $X$ and an ample line bundle $\mathcal F$ we are just counting integral points in a polytope $P$. So that fits.</p></li> <li><p>The pushforward $$f_*: A(X)_Q\to A(pt)_Q=\mathbb Q$$ between the Chow groups should be considered to be a "continuous" version, an integral. Indeed, for a cycle on $X$ its pushforward can be interpreted as, and computed by, an integral of a corresponding differential form. So this makes perfect sense as well.</p></li> </ol> <p>So now the Riemann-Roch, = the Euler-Maclaurin for this situation, transforms the integral into the sum, by multiplying it by the differential operator given by the Todd class. This also explains why in HRR the Todd class of $T_X$ appears and not, say, of $\Omega^1_X$. The tangent bundle is the place where the derivations $\partial/\partial z$ live.</p> http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch/116339#116339 Answer by Roman for Euler-Maclaurin formula and Riemann-Roch Roman 2012-12-14T02:25:23Z 2012-12-14T02:25:23Z <p>I was about to post the same question and came across yours. I wasn't aware of the "toric" direction here that other people have referred to, but I know a pretty answer in the particular case when $X$ is the flag variety of a semi-simple algebraic group. In this case RR reduces to saying that $\chi(F)={\mathrm{const}} \int ch(F\otimes L^{-1})$ where $L$ is the square root of the canonical class and the constant is explicit. So in this case at least multiplication by Todd does amount to shift (by half-forms) as in Euler-Maclaurin formula. Furthermore, in this form the formula has a very short proof via characteristic $p$, deducing it from the fact that $Fr_*(L)=L^{p^d}$, $d=\dim(X)$.</p> <p>[Well, in fact it also follows from Weyl dimension formula which of course has many other proofs, I just happen to like this char p proof.] It would be cool to have a proof of the general case along these lines. Something related has been done by Pink and Rossler, arXiv:0812.0254. </p>