The Riemann-Roch Theorem

TheRiemann-Roch theorem is a vital component in complex geometry,calculation of the space dimension of functions in meromorphic formwith set zeroes, and certified poles. Riemann-Roch theorem relatesthe intricate analysis of a compact Riemann surface to itstopological group. Bernhard Riemann proved as Riemann`s inequality in1857,

Theorem1.1

LetRiemann surface of genus g be X, subsequently

dim(L(D)) ≥ deg(D) + 1- g

where,L(D) equals the meromorphic functions space of with poles enclosed bya divisor D.

Riemann-Rochtheorem attained its definitive form after the integration of theerror term computed by Gustav Roch, i.e,

Theorem1.2

Let X represent Riemannsurface of genus g, subsequently for every divisor D and everycanonical divisor K, we obtain,

dim(L(D))- dim(L(K &shy&shy- D)) = deg(D) + 1 – g.

Initiallycomputed in complex function theory, the theorem was proven in acontext that was purely analytical by both Riemann and Roch. Theauthority of the proof was questioned by Weierstrass, who designed acounter-example to a core tool in the attestation referred to as theDirichlet`s principle. F. K. Schmidt developed a proof for algebraiccurves in relation to Riemann-Roch’s Theorem. The algebraic curvesproof was later on developed into a broad spectrum to incorporatehigher-dimensional varieties by employing the notion of line bundle,or divisor. The theorem’s broad-spectrum formulation is thereforedependent on division of the theorem into dual components. The SerreDuality component, construes the L(K – D) component as the firstsheaf cohomology’s group dimension L(D) depicts the zerothdimension of the cohomology sections space, or group, the theorem’sleft-hand side establishes a Euler characteristic, and the theorem’sright-hand side defines a degree of the theorem’s computation withthe a degree definitive in relation to Riemann surface’s topology.

TheGrothendieck-Riemann-Roch Theorem

Laurenttail divisors

Laurenttail divisors are related to ordinary divisors using an operationreferred to as the truncation operation. The truncation operationaids in the definition of Τ [D](X) group, X defines an algebraiccurve, established through divisions of Laurent tail divisors andbounded by a divisor D. Similarly, Laurent tail divisors can also beemployed in relating meromorphic functions. For each meromorphicfunction there exists corresponding element in T [D] through a grouphomomorphism, i.e., αD: M(X) — Ʈ [D] (X). This mapping is vitalin developing the Riemann-Roch Theorem’s proof.

Leta compact Riemann surface be defined by X, subsequently for everypoint PX,select a local coordinate with a centre p. Then A Laurent tail divisor on X refers to a formalset defined as

p.p,where the Laurent polynomial is defined by()is in the coordinate.

Theset of finite Laurent tail divisors are defined by Ʈ (X).

Inexceptional subgroups of Ʈ (X) for any D (divisor), describes Ʈ[D] (X) as the set of all finite formal ∑p.p i.e. for all p with ≠ 0, the upper term of has degree strictly less than –D (p). In this case Ʈ [D] (X) is asubgroup of Ʈ (X)

Grouphomomorphisms are constructed by considering a Laurent tail divisor∑p.pand D. At every p, we obtain

()=

Byletting bethe least integer from npto mpi.e., +1-D (p),the truncation of ()can be denoted by

Theabove mapping designates a group homomorphism

:Ʈ (X)Ʈ[D] (X).

Each∑p.pis sent to ∑p.p,defines truncation with the upper term being the largest integer in the spacetoand strictly smaller than -.If is inexistent, then is mapped to zero. The resultant truncation defines a grouphomomorphism designated by

Thisgroup homomorphism is described by truncating terms of with degreesequal or greater than – D2(P)from each ().

Theorem1.3 (Properties of)

(1)Commutes with the truncation maps: If and are divisors with subsequently the following triangle commutes:

M(X) T [1](X)

T [](X)

(2)scompatible with the multiplication operators: If on X , f and g aremeromorphic functions, subsequently

((g)) = -()(.g)

i.e.,for any divisor D

(3)L (D) = Ker()

Proof1:

Letp Xand let be a coordinate centred at p.

(1)Let bethe Laurent series of f in the coordinate andlet be the largest integer i.e. (p). Then (f) = ∑p.pwhere (=i.e. -(p)(p). Let be the largest integer such that + 1 (p). Note that

&lt .Then ==

(2)Consider the Laurent series (=ofand in coordinate.We obtain ( . )(= .Similarly, we have ()= ,k is the largest integer meeting the condition k &lt – . The series, .()= can be mapped on ().Similarly, let + be the largest integer i.e,

&lt -D(p) + (f)(p) = -D(p) +()

Subsequently,

((g))==().

(3)Let L(D). Then () D.Consider the Laurent series (= .Because = ()-D (p), the series.()= is mapped to 0 by

Letthe Ker (),hence every = 0 and therefore, -D (p), such that, ()(p) -D (p).

Riemann-RochFirst Form Theorem

Recall,L (D) = Ker (),then we need to define CoKer ()such that

(D):CoKer ()= Ʈ [D] (X)/Im ).

Thereforewe need to prove that there exists a finite dimension in (D) space. From proof 1, components i, ii, and iii, we have thesequence

0

Byconsidering the M(X)/ L(D) quotient space we find that become extinct on L(D)hence we only account for the mapping

:M(X)/ L(D)Ʈ [D] (X), provided that, (+L(D))= defining a group monomorphism.

Again,(D)),hence we have a short exact sequence of the form

0

&amp refer to two divisors meeting the condition:,from where we obtain the truncation map ,hence.

Thereforethere exists a short sequence of the form

0for every

Constructionof two homomormiphsm

  1. Let the map given by F (f +L()) = f + L() be defined by F: M(X)/ L() M(X)/ L(). The map is exemplarily defined, i.e, if f –g L(), subsequently (f) –( g) hence f –g L(). Therefore F is designated as a group homomorphism

  2. , defines the map we investigate if G is well defined by letting Z – Hence Z – Z’= (f) for any meromorphic function f. Applying the first proof of the above theorem, we obtain. Hence G is defined as a group homomorphism.

Theorem1.4

Theabove diagram commutes and possesses exact rows. &ampdefineprojection maps.

Proof

, applying part 1of proof 1:

=

=,hence the left square commutes.

Similarly,, Hence the right square commutes.

References

Harada,M., &amp Johns Hopkins University (Baltimore, Md). (1987).&nbspAproof of the riemann-Roch Theorem.S.l.: s.n..