View source for Riemann-Roch theorem

A theorem expressing the [[Euler characteristic|Euler characteristic]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r0819801.png" /> of a [[Locally free sheaf|locally free sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r0819802.png" /> on an algebraic or analytic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r0819803.png" /> in terms of the characteristic Chern classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r0819804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r0819805.png" /> (cf. [[Chern class|Chern class]]). It can be used to calculate the dimension of the space of sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r0819806.png" /> (the Riemann–Roch problem).

The classical Riemann–Roch theorem relates to the case of non-singular algebraic curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r0819807.png" /> and states that for any [[Divisor|divisor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r0819808.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r0819809.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198011.png" /> is the dimension of the space of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198012.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198014.png" /> is the canonical divisor and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198015.png" /> is the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198016.png" />. In the middle of the 19th century B. Riemann used analytic methods to obtain the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198017.png" /></td> </tr></table>

The equality (1) was proved by E. Roch.

The Riemann–Roch theorem for curves is the one-dimensional case of the more general Riemann–Roch–Hirzebruch–Grothendieck theorem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198018.png" /> be a non-singular projective variety of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198019.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198020.png" /> be an appropriate [[Cohomology|cohomology]] theory: either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198021.png" /> are singular cohomology spaces when the basic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198022.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198023.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198024.png" /> is a [[Chow ring|Chow ring]], or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198025.png" /> is the ring associated to the Grothendieck ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198026.png" /> (see [[#References|[2]]], [[#References|[7]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198027.png" /> be a locally free sheaf of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198028.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198029.png" />. Universal polynomials for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198030.png" /> with rational coefficients, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198032.png" />, in the Chern classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198034.png" /> are defined in the following way. The factorization

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198035.png" /></td> </tr></table>

is examined for the Chern polynomial, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198036.png" /> are formal symbols. The exponential Chern character is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198037.png" /></td> </tr></table>

and, accordingly, the [[Todd class|Todd class]] is defined as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198038.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198040.png" /> are symmetric functions in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198041.png" /> and they can be written as polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198042.png" />.

The Riemann–Roch–Hirzebruch theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198043.png" /> is a non-singular projective variety or a compact complex variety of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198044.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198045.png" /> is a vector bundle of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198046.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198047.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198048.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198049.png" /> is the tangent sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198051.png" /> denotes the component of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198052.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198053.png" />. This theorem was proved by F. Hirzebruch in the case of the ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198054.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198055.png" /> and the invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198056.png" />, it leads to the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198057.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198058.png" /> is the second Chern class of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198060.png" /> is its canonical class. In particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198061.png" /> Noether's formula is obtained:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198062.png" /></td> </tr></table>

For three-dimensional varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198063.png" /> the theorem leads to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198064.png" /></td> </tr></table>

In particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198065.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198066.png" /></td> </tr></table>

In 1957, A. Grothendieck generalized the Riemann–Roch–Hirzebruch theorem to the case of a morphism of non-singular varieties over an arbitrary algebraically closed field (see [[#References|[1]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198068.png" /> be the Grothendieck groups of the coherent and locally free sheafs on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198069.png" />, respectively (cf. [[Grothendieck group|Grothendieck group]]). The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198070.png" /> is a covariant functor from the category of schemes and proper morphisms into the category of Abelian groups. In this case, for a [[Proper morphism|proper morphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198071.png" /> the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198072.png" /> is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198073.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198074.png" /> is an arbitrary [[Coherent sheaf|coherent sheaf]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198076.png" /> is a covariant functor into the category of rings. For regular schemes with an ample sheaf, the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198078.png" /> coincide and are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198079.png" />. The Chern character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198080.png" /> is a homomorphism of rings; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198081.png" /> is also a covariant functor: The Gizin homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198082.png" /> is defined. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198083.png" />, the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198084.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198085.png" /> for homology spaces using [[Poincaré duality|Poincaré duality]]. The theorem as generalized by Grothendieck expresses the measure of deviation from commutativity of the homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198087.png" />.

The Riemann–Roch–Hirzebruch–Grothendieck theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198088.png" /> be a smooth projective morphism of non-singular projective varieties; then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198089.png" /> the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198090.png" /></td> </tr></table>

in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198091.png" /> is true, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198092.png" /> (the relative tangent sheaf of the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198093.png" />).

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198094.png" /> is a point, this theorem reduces to the Riemann–Roch–Hirzebruch theorem. There are generalizations (see [[#References|[5]]], [[#References|[6]]], [[#References|[7]]]) when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198095.png" /> is a Noetherian scheme with an ample invertible sheaf, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198096.png" /> is a projective morphism whose fibres are locally complete intersections, and also to the case of singular quasi-projective varieties.

Several versions of the Riemann–Roch theorem are closely connected with the index problem for elliptic operators (see [[Index formulas|Index formulas]]). For example, the Riemann–Roch–Hirzebruch theorem for compact complex varieties is a particular case of the Atiyah–Singer index theorem.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, J.-P. Serre, "La théorème de Riemann–Roch" ''Bull. Soc. Math. France'' , '''86''' (1958) pp. 97–136</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Yu.I. Manin, "Lectures on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198097.png" />-functor in algebraic geometry" ''Russian Math. Surveys'' , '''24''' : 5 (1969) pp. 1–89 ''Uspekhi Mat. Nauk'' , '''24''' : 5 (1969) pp. 3–86 {{MR|265355}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German) {{MR|1335917}} {{MR|0202713}} {{ZBL|0376.14001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> P. Baum, W. Fulton, R. MacPherson, "Riemann–Roch for singular varieties" ''Publ. Math. IHES'' , '''45''' (1975) pp. 101–145</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> P. Baum, W. Fulton, R. MacPherson, "Riemann–Roch for topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081980/r08198098.png" />-theory and singular varieties" ''Acta Math.'' , '''143''' : 3–4 (1979) pp. 155–192</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> "Théorie des intersections et théorème de Riemann–Roch" P. Berthelot (ed.) et al. (ed.) , ''Sem. Geom. Alg. 6'' , ''Lect. notes in math.'' , '''225''' , Springer (1971)</TD></TR></table>



====Comments====
There are analogues of the Riemann–Roch theorem in algebraic number theory and arithmetic algebraic geometry.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Lang, "Algebraic number theory" , Addison-Wesley (1970) {{MR|0282947}} {{ZBL|0211.38404}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K. Szpiro, "Sem. sur les pinceaux arithmétiques: La conjecture de Mordell" ''Astérisque'' , '''127''' (1985)</TD></TR></table>