ரெய்மான் ரோஸ் தேற்றம்

 

ரெய்மான் ரோஸ் தேற்றம் 

---------------------------------------------------------------------------------------------இ பரமசிவன் 

சிக்கல் கணிதத்தில் அல்ஜிப்ரைக் வடிவ கணிதத்தை நிறுவுவதே இந்த தேற்றம் இங்கு மேரோமார்ஃபிக் இயங்கியங்களில் உள்ள சுழிகளும் துருவங்களும் 

அந்த வெளியின் பரிமாணங்களை நிறுவனம் செய்யும். ரெய்மானின் "காம்பேக்ட் பரப்பு "என்பது "டோப்பாலஜியின்"ஜெனஸ் ஜி " எனும் பரப்பு பற்றிய கணிதம் ஆகும்.இது முழுக்க  முழுக்க வடிவ கணித அமைப்பில் வருவது.1857ல் ரெய்மான் "ரெய்மான் சமப்பாடின்மை"என்பது தான் "ரெய்மான் பரப்புகள்"பற்றிய கணிதம் வழக்குக்கு வந்தது.அவரது மாணவரான "கஸ்டாவ் ரோஸ் "(1865) ன் ஆய்வில் தான் அது முடிந்தது.இவர் சிறு வயதில் இறந்தது ஒரு 

துயரமான நிகழ்வு.இதற்குப்பிறகே "அல்ஜிப்ரைக் வளைவுகள்" அதிக பரிமாணங்களிலும் வகைகளிலும் பயனுக்கு வந்தன.

Preliminary notions

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A Riemann surface of genus 3.

A Riemann surface ${\displaystyle X}$ is a topological space that is locally homeomorphic to an open subset of ${\displaystyle \mathbb{C}}$, the set of complex numbers. In addition, the transition maps between these open subsets are required to be holomorphic. The latter condition allows one to transfer the notions and methods of complex analysis dealing with holomorphic and meromorphic functions on ${\displaystyle \mathbb{C}}$ to the surface ${\displaystyle X}$. For the purposes of the Riemann–Roch theorem, the surface ${\displaystyle X}$ is always assumed to be compact. Colloquially speaking, the genus ${\displaystyle g}$ of a Riemann surface is its number of handles; for example the genus of the Riemann surface shown at the right is three. More precisely, the genus is defined as half of the first Betti number, i.e., half of the ${\displaystyle \mathbb{C}}$-dimension of the first singular homology group ${\displaystyle H_1(X,\mathbb{C})}$ with complex coefficients. The genus classifies compact Riemann surfaces up to homeomorphism, i.e., two such surfaces are homeomorphic if and only if their genus is the same. Therefore, the genus is an important topological invariant of a Riemann surface. On the other hand, Hodge theory shows that the genus coincides with the ${\displaystyle \mathbb{C}}$-dimension of the space of holomorphic one-forms on ${\displaystyle X}$, so the genus also encodes complex-analytic information about the Riemann surface.^[1]

A divisor ${\displaystyle D}$ is an element of the free abelian group on the points of the surface. Equivalently, a divisor is a finite linear combination of points of the surface with integer coefficients.

Any meromorphic function ${\displaystyle f}$ gives rise to a divisor denoted ${\displaystyle (f)}$ defined as

${\displaystyle (f):=\sum _{z_{\nu }\in R(f)}{s}_{\nu }{z}_{\nu }}$

where ${\displaystyle R(f)}$ is the set of all zeroes and poles of ${\displaystyle f}$, and ${\displaystyle s_{\nu }}$ is given by

.

The set ${\displaystyle R(f)}$ is known to be finite; this is a consequence of ${\displaystyle X}$ being compact and the fact that the zeros of a (non-zero) holomorphic function do not have an accumulation point. Therefore, ${\displaystyle (f)}$ is well-defined. Any divisor of this form is called a principal divisor. Two divisors that differ by a principal divisor are called linearly equivalent. The divisor of a meromorphic 1-form is defined similarly. A divisor of a global meromorphic 1-form is called the canonical divisor (usually denoted ${\displaystyle K}$). Any two meromorphic 1-forms will yield linearly equivalent divisors, so the canonical divisor is uniquely determined up to linear equivalence (hence "the" canonical divisor).

The symbol ${\displaystyle \mathrm{deg}(D)}$ denotes the degree (occasionally also called index) of the divisor ${\displaystyle D}$, i.e. the sum of the coefficients occurring in ${\displaystyle D}$. It can be shown that the divisor of a global meromorphic function always has degree 0, so the degree of a divisor depends only on its linear equivalence class.

The number ${\displaystyle \ell (D)}$ is the quantity that is of primary interest: the dimension (over ${\displaystyle \mathbb{C}}$) of the vector space of meromorphic functions ${\displaystyle h}$ on the surface, such that all the coefficients of ${\displaystyle (h)+D}$ are non-negative. Intuitively, we can think of this as being all meromorphic functions whose poles at every point are no worse than the corresponding coefficient in ${\displaystyle D}$; if the coefficient in ${\displaystyle D}$ at ${\displaystyle z}$ is negative, then we require that ${\displaystyle h}$ has a zero of at least that multiplicity at ${\displaystyle z}$ – if the coefficient in ${\displaystyle D}$ is positive, ${\displaystyle h}$ can have a pole of at most that order. The vector spaces for linearly equivalent divisors are naturally isomorphic through multiplication with the global meromorphic function (which is well-defined up to a scalar).

Statement of the theorem

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The Riemann–Roch theorem for a compact Riemann surface of genus ${\displaystyle g}$ with canonical divisor ${\displaystyle K}$ states

${\displaystyle \ell (D)-\ell (K-D)=\mathrm{deg}(D)-g+1}$.

Typically, the number ${\displaystyle \ell (D)}$ is the one of interest, while ${\displaystyle \ell (K-D)}$ is thought of as a correction term (also called index of speciality^[2][3]) so the theorem may be roughly paraphrased by saying

dimension − correction = degree − genus + 1.

Because it is the dimension of a vector space, the correction term ${\displaystyle \ell (K-D)}$ is always non-negative, so that

${\displaystyle \ell (D)\ge \mathrm{deg}(D)-g+1}$.

This is called Riemann's inequality. Roch's part of the statement is the description of the possible difference between the sides of the inequality. On a general Riemann surface of genus ${\displaystyle g}$, ${\displaystyle K}$ has degree ${\displaystyle 2g-2}$, independently of the meromorphic form chosen to represent the divisor. This follows from putting ${\displaystyle D=K}$ in the theorem. In particular, as long as ${\displaystyle D}$ has degree at least ${\displaystyle 2g-1}$, the correction term is 0, so that

${\displaystyle \ell (D)=\mathrm{deg}(D)-g+1}$.

The theorem will now be illustrated for surfaces of low genus. There are also a number other closely related theorems: an equivalent formulation of this theorem using line bundles and a generalization of the theorem to algebraic curves.

Examples

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The theorem will be illustrated by picking a point ${\displaystyle P}$ on the surface in question and regarding the sequence of numbers

${\displaystyle \ell (n\cdot P),n\ge 0}$

i.e., the dimension of the space of functions that are holomorphic everywhere except at ${\displaystyle P}$ where the function is allowed to have a pole of order at most ${\displaystyle n}$. For ${\displaystyle n=0}$, the functions are thus required to be entire, i.e., holomorphic on the whole surface ${\displaystyle X}$. By Liouville's theorem, such a function is necessarily constant. Therefore, ${\displaystyle \ell (0)=1}$. In general, the sequence ${\displaystyle \ell (n\cdot P)}$ is an increasing sequence.

Genus zero

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The Riemann sphere (also called complex projective line) is simply connected and hence its first singular homology is zero. In particular its genus is zero. The sphere can be covered by two copies of ${\displaystyle \mathbb{C}}$, with transition map being given by

${\displaystyle \mathbb{C}\setminus \{0\}\ni z\mapsto \frac{1}{z}\in \mathbb{C}\setminus \{0\}}$.

Therefore, the form ${\displaystyle \omega =dz}$ on one copy of ${\displaystyle \mathbb{C}}$ extends to a meromorphic form on the Riemann sphere: it has a double pole at infinity, since

${\displaystyle d\left(\frac{1}{z}\right)=-\frac{1}{z^2}dz}$

Thus, its canonical divisor is ${\displaystyle K:=\mathrm{div}(\omega )=-2P}$ (where ${\displaystyle P}$ is the point at infinity).

Therefore, the theorem says that the sequence ${\displaystyle \ell (n\cdot P)}$ reads

1, 2, 3, ... .

This sequence can also be read off from the theory of partial fractions. Conversely if this sequence starts this way, then ${\displaystyle g}$ must be zero.

Genus one

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A torus

The next case is a Riemann surface of genus ${\displaystyle g=1}$, such as a torus ${\displaystyle \mathbb{C}/\mathrm{\Lambda }}$, where ${\displaystyle \mathrm{\Lambda }}$ is a two-dimensional lattice (a group isomorphic to ${\displaystyle {\mathbb{Z}}^2}$). Its genus is one: its first singular homology group is freely generated by two loops, as shown in the illustration at the right. The standard complex coordinate ${\displaystyle z}$ on ${\displaystyle C}$ yields a one-form ${\displaystyle \omega =dz}$ on ${\displaystyle X}$ that is everywhere holomorphic, i.e., has no poles at all. Therefore, ${\displaystyle K}$, the divisor of ${\displaystyle \omega }$ is zero.

On this surface, this sequence is

1, 1, 2, 3, 4, 5 ... ;

and this characterises the case ${\displaystyle g=1}$. Indeed, for ${\displaystyle D=0}$, ${\displaystyle \ell (K-D)=\ell (0)=1}$, as was mentioned above. For ${\displaystyle D=n\cdot P}$ with , the degree of ${\displaystyle K-D}$ is strictly negative, so that the correction term is 0. The sequence of dimensions can also be derived from the theory of elliptic functions.

Genus two and beyond

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For ${\displaystyle g=2}$, the sequence mentioned above is

1, 1, ?, 2, 3, ... .

It is shown from this that the ? term of degree 2 is either 1 or 2, depending on the point. It can be proven that in any genus 2 curve there are exactly six points whose sequences are 1, 1, 2, 2, ... and the rest of the points have the generic sequence 1, 1, 1, 2, ... In particular, a genus 2 curve is a hyperelliptic curve. For  it is always true that at most points the sequence starts with ${\displaystyle g+1}$ ones and there are finitely many points with other sequences (see Weierstrass points).

Riemann–Roch for line bundles

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Using the close correspondence between divisors and holomorphic line bundles on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: let L be a holomorphic line bundle on X. Let ${\displaystyle H^0(X,L)}$ denote the space of holomorphic sections of L. This space will be finite-dimensional; its dimension is denoted ${\displaystyle h^0(X,L)}$. Let K denote the canonical bundle on X. Then, the Riemann–Roch theorem states that

${\displaystyle h^0(X,L)-h^0(X,L^{-1}\otimes K)=\mathrm{deg}(L)+1-g}$.

The theorem of the previous section is the special case of when L is a point bundle.

The theorem can be applied to show that there are g linearly independent holomorphic sections of K, or one-forms on X, as follows. Taking L to be the trivial bundle, ${\displaystyle h^0(X,L)=1}$ since the only holomorphic functions on X are constants. The degree of L is zero, and ${\displaystyle L^{-1}}$ is the trivial bundle. Thus,

${\displaystyle 1-h^0(X,K)=1-g}$.

Therefore, ${\displaystyle h^0(X,K)=g}$, proving that there are g holomorphic one-forms.

Degree of canonical bundle

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Since the canonical bundle ${\displaystyle K}$ has ${\displaystyle h^0(X,K)=g}$, applying Riemann–Roch to ${\displaystyle L=K}$ gives

${\displaystyle h^0(X,K)-h^0(X,K^{-1}\otimes K)=\mathrm{deg}(K)+1-g}$

which can be rewritten as

${\displaystyle g-1=\mathrm{deg}(K)+1-g}$

hence the degree of the canonical bundle is ${\displaystyle \mathrm{deg}(K)=2g-2}$.

Riemann–Roch theorem for algebraic curves

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Every item in the above formulation of the Riemann–Roch theorem for divisors on Riemann surfaces has an analogue in algebraic geometry. The analogue of a Riemann surface is a non-singular algebraic curve C over a field k. The difference in terminology (curve vs. surface) is because the dimension of a Riemann surface as a real manifold is two, but one as a complex manifold. The compactness of a Riemann surface is paralleled by the condition that the algebraic curve be complete, which is equivalent to being projective. Over a general field k, there is no good notion of singular (co)homology. The so-called geometric genus is defined as

${\displaystyle g(C):={\dim }_k\mathrm{\Gamma }(C,{\mathrm{\Omega }}_C^1)}$

i.e., as the dimension of the space of globally defined (algebraic) one-forms (see Kähler differential). Finally, meromorphic functions on a Riemann surface are locally represented as fractions of holomorphic functions. Hence they are replaced by rational functions which are locally fractions of regular functions. Thus, writing ${\displaystyle \ell (D)}$ for the dimension (over k) of the space of rational functions on the curve whose poles at every point are not worse than the corresponding coefficient in D, the very same formula as above holds:

${\displaystyle \ell (D)-\ell (K-D)=\mathrm{deg}(D)-g+1}$.

where C is a projective non-singular algebraic curve over an algebraically closed field k. In fact, the same formula holds for projective curves over any field, except that the degree of a divisor needs to take into account multiplicities coming from the possible extensions of the base field and the residue fields of the points supporting the divisor.^[4] Finally, for a proper curve over an Artinian ring, the Euler characteristic of the line bundle associated to a divisor is given by the degree of the divisor (appropriately defined) plus the Euler characteristic of the structural sheaf ${\displaystyle \mathcal{O}}$.^[5]

The smoothness assumption in the theorem can be relaxed, as well: for a (projective) curve over an algebraically closed field, all of whose local rings are Gorenstein rings, the same statement as above holds, provided that the geometric genus as defined above is replaced by the arithmetic genus g_a, defined as

${\displaystyle g_a:={\dim }_k{H}^1(C,{\mathcal{O}}_C)}$.^[6]

(For smooth curves, the geometric genus agrees with the arithmetic one.) The theorem has also been extended to general singular curves (and higher-dimensional varieties).^[7]

Applications

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Hilbert polynomial

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One of the important consequences of Riemann–Roch is it gives a formula for computing the Hilbert polynomial of line bundles on a curve. If a line bundle ${\displaystyle \mathcal{L}}$ is ample, then the Hilbert polynomial will give the first degree ${\displaystyle {\mathcal{L}}^{\otimes n}}$ giving an embedding into projective space. For example, the canonical sheaf ${\displaystyle {\omega }_C}$ has degree ${\displaystyle 2g-2}$, which gives an ample line bundle for genus ${\displaystyle g\ge 2}$.^[8] If we set ${\displaystyle {\omega }_C(n)={\omega }_C^{\otimes n}}$ then the Riemann–Roch formula reads

Giving the degree ${\displaystyle 1}$ Hilbert polynomial of ${\displaystyle {\omega }_C}$

${\displaystyle H_{{\omega }_C}(t)=2(g-1)t-g+1}$.

Because the tri-canonical sheaf ${\displaystyle {\omega }_C^{\otimes 3}}$ is used to embed the curve, the Hilbert polynomial

${\displaystyle H_C(t)=H_{{\omega }_C^{\otimes 3}}(t)}$

is generally considered while constructing the Hilbert scheme of curves (and the moduli space of algebraic curves). This polynomial is

and is called the Hilbert polynomial of a genus g curve.