Hirzebruch–Riemann–Roch Theorem
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch,
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It re ...
on
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s to all complex
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
of higher dimensions. The result paved the way for the Grothendieck–Hirzebruch–Riemann–Roch theorem proved about three years later.


Statement of Hirzebruch–Riemann–Roch theorem

The Hirzebruch–Riemann–Roch theorem applies to any holomorphic
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
''E'' on a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
''X'', to calculate the holomorphic Euler characteristic of ''E'' in
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions (holes) to solving a geometric problem glob ...
, namely the alternating sum : \chi(X,E) = \sum_^ (-1)^ \dim_ H^(X,E) of the dimensions as complex vector spaces, where ''n'' is the complex dimension of ''X''. Hirzebruch's theorem states that χ(''X'', ''E'') is computable in terms of the
Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches ...
es ''ck''(''E'') of ''E'', and the
Todd class In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encounte ...
es \operatorname_(X) of the holomorphic
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
of ''X''. These all lie in the
cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually un ...
of ''X''; by use of the fundamental class (or, in other words, integration over ''X'') we can obtain numbers from classes in H^(X). The Hirzebruch formula asserts that : \chi(X,E) = \sum_^ \operatorname_(E) \operatorname_(X), using the Chern character ch(''E'') in cohomology. In other words, the products are formed in the cohomology ring of all the 'matching' degrees that add up to 2''n''. Formulated differently, it gives the equality : \chi(X,E) = \int_X \operatorname(E) \operatorname(X) where \operatorname(X) is the
Todd class In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encounte ...
of the tangent bundle of ''X''. Significant special cases are when ''E'' is a complex
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ...
, and when ''X'' is an algebraic surface (Noether's formula). Weil's Riemann–Roch theorem for vector bundles on curves, and the Riemann–Roch theorem for algebraic surfaces (see below), are included in its scope. The formula also expresses in a precise way the vague notion that the
Todd class In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encounte ...
es are in some sense reciprocals of the Chern Character.


Riemann Roch theorem for curves

For curves, the Hirzebruch–Riemann–Roch theorem is essentially the classical
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It re ...
. To see this, recall that for each
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
''D'' on a curve there is an
invertible sheaf In mathematics, an invertible sheaf is a sheaf on a ringed space that has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their intera ...
O(''D'') (which corresponds to a line bundle) such that the
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstractio ...
of ''D'' is more or less the space of sections of O(''D''). For curves the Todd class is 1+c_1(T(X))/2, and the Chern character of a sheaf O(''D'') is just 1+''c''1(O(''D'')), so the Hirzebruch–Riemann–Roch theorem states that : h^0(\mathcal(D)) - h^1(\mathcal(D)) = c_1(\mathcal(D)) +c_1(T(X))/2\ \ \ (integrated over ''X''). But ''h''0(O(''D'')) is just ''l''(''D''), the dimension of the linear system of ''D'', and by
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Ale ...
''h''1(O(''D'')) = ''h''0(O(''K'' − ''D'')) = ''l''(''K'' − ''D'') where ''K'' is the
canonical divisor The adjective canonical is applied in many contexts to mean 'according to the canon' the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, ''canonical examp ...
. Moreover, ''c''1(O(''D'')) integrated over ''X'' is the degree of ''D'', and ''c''1(''T''(''X'')) integrated over ''X'' is the Euler class 2 − 2''g'' of the curve ''X'', where ''g'' is the genus. So we get the classical Riemann Roch theorem : \ell(D)-\ell(K-D) = \text(D)+1-g. For vector bundles ''V'', the Chern character is rank(''V'') + ''c''1(''V''), so we get Weil's Riemann Roch theorem for vector bundles over curves: : h^0(V) - h^1(V) = c_1(V) + \operatorname(V)(1-g).


Riemann Roch theorem for surfaces

For surfaces, the Hirzebruch–Riemann–Roch theorem is essentially the Riemann–Roch theorem for surfaces : \chi(D) = \chi(\mathcal) + ((D.D)-(D.K))/2. combined with the Noether formula. If we want, we can use Serre duality to express ''h''2(O(''D'')) as ''h''0(O(''K'' − ''D'')), but unlike the case of curves there is in general no easy way to write the ''h''1(O(''D'')) term in a form not involving sheaf cohomology (although in practice it often vanishes).


Asymptotic Riemann–Roch

Let ''D'' be an ample
Cartier divisor In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumf ...
on an irreducible projective variety ''X'' of dimension ''n''. Then :h^0 \left (X,\mathcal O_X(mD) \right )=\frac.m^n+O(m^). More generally, if \mathcal F is any coherent sheaf on ''X'' then :h^0 \left (X,\mathcal F\otimes \mathcal O_X(mD) \right )=\operatorname(\mathcal F)\frac.m^n+O(m^).


See also

* Grothendieck–Riemann–Roch theorem - contains many computations and examples * Hilbert polynomial - HRR can be used to compute Hilbert polynomials


References

* Friedrich Hirzebruch,''Topological Methods in Algebraic Geometry''


External links


The Hirzebruch-Riemann-Roch Theorem
{{DEFAULTSORT:Hirzebruch-Riemann-Roch theorem Topological methods of algebraic geometry Theorems in complex geometry Theorems in algebraic geometry Bernhard Riemann