Slope field

In algebraic geometry, the Kodaira dimension κ(X) measures the size of the canonical model of a projective variety X.

Kodaira dimension is named for Kunihiko Kodaira. The name and the notation κ were introduced by Igor Shafarevich in the seminar Shafarevich 1965.

The plurigenera

The canonical bundle of a smooth algebraic variety X of dimension n over a field is the line bundle of n-forms,

$\,\!K_{X}=\bigwedge ^{n}\Omega _{X},$

which is the nth exterior power of the cotangent bundle of X. For an integer d, the dth tensor power of K_X is again a line bundle. For d ≥ 0, the vector space of global sections H⁰(X,K_X^d) has the remarkable property that it is a birational invariant of smooth projective varieties X. That is, this vector space is canonically identified with the corresponding space for any smooth projective variety which is isomorphic to X outside lower-dimensional subsets.

For d ≥ 0, the dth plurigenus of X is defined as the dimension of the vector space of global sections of K_X^d:

$P_{d}=h^{0}(X,K_{X}^{d})=\operatorname {dim} \ H^{0}(X,K_{X}^{d}).$

The plurigenera are important birational invariants of an algebraic variety. In particular, the simplest way to prove that a variety is not rational (that is, not birational to projective space) is to show that some plurigenus P_d with d > 0 is not zero. If the space of sections of K_X^d is nonzero, then there is a natural rational map from X to the projective space

$\mathbf {P} (H^{0}(X,K_{X}^{d}))=\mathbf {P} ^{P_{d}-1}$,

called the d-canonical map. The canonical ring R(K_X) of a variety X is the graded ring

$R(K_{X}):=\bigoplus _{d\geq 0}H^{0}(X,K_{X}^{d}).$

Also see geometric genus and arithmetic genus.

The Kodaira dimension of X is defined to be −∞ if the plurigenera P_d are zero for all d > 0; otherwise, it is the minimum κ such that P_d/d^κ is bounded. The Kodaira dimension of an n-dimensional variety is either −∞ or an integer in the range from 0 to n.

Interpretations of the Kodaira dimension

The following integers are equal. A good reference is Template:Harvtxt, Theorem 2.1.33.

• The dimension of the Proj construction Proj R(K_X) (this variety is called the canonical model of X; it only depends on the birational equivalence class of X).

• The dimension of the image of the d-canonical mapping for all positive multiples d of some positive integer d₀.

• The transcendence degree of R, minus one, i.e. t − 1, where t is the number of algebraically independent generators one can find.

• The rate of growth of the plurigenera: that is, the smallest number κ such that P_d/d^κ is bounded. In Big O notation, it is the minimal κ such that P_d = O(d^κ).

When the plurigenera P_d are zero for all positive d (so the canonical ring R(K_X) is equal to the base field R₀), we have to define the Kodaira dimension to be −∞ rather than −1, in order to make the formula κ(X × Y) = κ(X) + κ(Y) true in all cases. For example, the Kodaira dimension of P¹ × X is −∞ for all varieties X. That convention is also essential in the statement of the Iitaka conjecture.

Application

The Kodaira dimension gives a useful rough division of all algebraic varieties into several classes.

Varieties with low Kodaira dimension can be considered special, while varieties of maximal Kodaira dimension are said to be of general type.

Geometrically, there is a very rough correspondence between Kodaira dimension and curvature: negative Kodaira dimension corresponds to positive curvature, zero Kodaira dimension corresponds to flatness, and maximum Kodaira dimension (general type) corresponds to negative curvature.

The specialness of varieties of low Kodaira dimension is analogous to the specialness of Riemannian manifolds of positive curvature (and general type corresponds to the genericity of non-positive curvature); see classical theorems, especially on Pinched sectional curvature and Positive curvature.

These statements are made more precise below.

Dimension 1

Smooth projective curves are discretely classified by genus, which can be any natural number g = 0, 1, ....

By "discretely classified", we mean that for a given genus, there is a connected, irreducible moduli space of curves of that genus.

The Kodaira dimension of a curve X is:

• κ = −∞: genus 0 (the projective line P¹): K_X is not effective, P_d = 0 for all d > 0.
• κ = 0: genus 1 (elliptic curves): K_X is a trivial bundle, P_d = 1 for all d ≥ 0.
• κ = 1: genus g ≥ 2: K_X is ample, P_d=(2d−1)(g−1) for all d ≥ 2.

Compare with the Uniformization theorem for surfaces (real surfaces, since a complex curve has real dimension 2): Kodaira dimension −∞ corresponds to positive curvature, Kodaira dimension 0 corresponds to flatness, Kodaira dimension 1 corresponds to negative curvature. Note that most algebraic curves are of general type: in the moduli space of curves, two connected components correspond to curves not of general type, while all the other components correspond to curves of general type. Further, the space of curves of genus 0 is a point, the space of curves of genus 1 has (complex) dimension 1, and the space of curves of genus g ≥ 2 has dimension 3g−3.

the classification table of algebraic curves
Kodaira dimension 　κ(C)
	genus of C : g(C)	structure
$1$	$\geq 2$	curve of general type
$0$	$1$	elliptic curve
$-\infty$	$0$	the projective line $\mathbb {P} ^{1}$

Dimension 2

The Enriques-Kodaira classification classifies algebraic surfaces: coarsely by Kodaira dimension, then in more detail within a given Kodaira dimension. To give some simple examples: the product P¹ × X has Kodaira dimension −∞ for any curve X; the product of two curves of genus 1 (an abelian surface) has Kodaira dimension 0; the product of a curve of genus 1 with a curve of genus at least 2 (an elliptic surface) has Kodaira dimension 1; and the product of two curves of genus at least 2 has Kodaira dimension 2 and hence is of general type.

the classification table of algebraic surfaces
Kodaira dimension 　κ(C)
	geometric genus 　pg	irregularity q	structure
$2$			surface of general type
$1$			elliptic surface
$0$	$1$	$2$	abelian surface
	$0$	$1$	hyperelliptic surface
	$1$	$0$	K3 surface
	$0$	$0$	Enriques surface
$-\infty$	$0$	$\geq 1$	ruled surface
	$0$	$0$	rational surface

For a surface X of general type, the image of the d-canonical map is birational to X if d ≥ 5.

Any dimension

Rational varieties (varieties birational to projective space) have Kodaira dimension −∞. Abelian varieties and Calabi-Yau manifolds (in dimension 1, elliptic curves; in dimension 2, complex tori and K3 surfaces) have Kodaira dimension zero (corresponding to admitting flat metrics and Ricci flat metrics, respectively).

Any variety covered by rational curves (nonconstant maps from P¹), called a uniruled variety, has Kodaira dimension −∞. Conversely, the main conjectures of minimal model theory (notably the abundance conjecture) would imply that every variety of Kodaira dimension −∞ is uniruled. This converse is known for varieties of dimension at most 3.

Template:Harvtxt proved the invariance of plurigenera under deformations for all smooth complex projective varieties. In particular, the Kodaira dimension does not change when the complex structure of the manifold is changed continuously.

the classification table of algebraic three-folds
Kodaira dimension 　κ(C)
	geometric genus 　pg	irregularity q	examples
$3$			three-fold of general type
$2$			fibration over a surface with general fiber an elliptic curve
$1$			fibration over a curve with general fiber a surface with κ = 0
$0$	$1$	$3$	abelian variety
	$0$	$2$	fiber bundle over an abelian surface whose fibers are elliptic curves
	$0$　or $1$	$1$	fiber bundle over an elliptic curve whose fibers are surfaces with κ = 0
	$0$　or $1$	$0$	Calabi-Yau 3-fold
$-\infty$	$0$	$\geq 1$	uniruled 3-folds
	$0$	$0$	rational 3-folds, Fano 3-folds, and others

A fibration of normal projective varieties X → Y means a surjective morphism with connected fibers.

For a 3-fold X of general type, the image of the d-canonical map is birational to X if d ≥ 61.

General type

A variety of general type X is one of maximal Kodaira dimension (Kodaira dimension equal to its dimension):

$\kappa (X)=\operatorname {dim} \ X.$

Equivalent conditions are that the line bundle K_X is big, or that the d-canonical map is generically injective (that is, a birational map to its image) for d sufficiently large.

For example, a variety with ample canonical bundle is of general type.

In some sense, most algebraic varieties are of general type. For example, a smooth hypersurface of degree d in the n-dimensional projective space is of general type if and only if d > n+1. So we can say that most smooth hypersurfaces in projective space are of general type.

Varieties of general type seem too complicated to classify explicitly, even for surfaces. Nonetheless, there are some strong positive results about varieties of general type. For example, Bombieri showed in 1973 that the d-canonical map of any complex surface of general type is birational for every d ≥ 5. More generally, Hacon-McKernan, Takayama, and Tsuji showed in 2006 that for every positive integer n, there is a constant c(n) such that the d-canonical map of any complex n-dimensional variety of general type is birational when d ≥ c(n).

Furthermore, varieties of general type have a only finite group as their automorphism.

Application to classification

The Iitaka conjecture states that the Kodaira dimension of a fibration is at least the sum of the Kodaira dimension of the base and the Kodaira dimension of a general fiber; see Template:Harvtxt for a survey. The Iitaka conjecture helped to inspire the development of minimal model theory in the 1970s and 1980s. It is now known in many cases, and would follow in general from the main conjectures of minimal model theory, notably the abundance conjecture.

Minimal model theory also predicts that every algebraic variety is either uniruled (covered by rational curves) or birational to a family of varieties over the canonical model, with general fiber a Calabi-Yau variety. To some extent, this would reduce the study of arbitrary varieties to the cases of Calabi-Yau varieties and varieties of general type.

The relationship to Moishezon manifolds

Nakamura and Ueno proved the following additivity formula for complex manifolds (Template:Harvtxt). Although the base space is not required to be algebraic, the assumption that all the fibers are isomorphic is very special. Even with this assumption, the formula can fail when the fiber is not Moishezon.

Let π: V → W be an analytic fiber bundle of compact complex manifolds, meaning that π is locally a product (and so all fibers are isomorphic as complex manifolds). Suppose that the fiber F is a Moishezon manifold. Then

$\kappa (V)=\kappa (F)+\kappa (W).$

See also

• Birational geometry
• Enriques-Kodaira classification
• Iitaka dimension
• Minimal model program
• Moishezon manifold

References

• Template:Eom
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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010