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In [[complex geometry]], a '''Hopf surface''' is a compact complex surface obtained
as a quotient of the complex [[vector space]]
(with zero deleted) '''C'''<sup>2</sup>&nbsp;\&nbsp;0
by a [[Group action|free action]] of a discrete group. If this group is the integers the Hopf surface is called '''primary''', otherwise it is called '''secondary'''. (Some authors use the term "Hopf surface" to mean "primary Hopf surface".)  The first example was found by {{harvtxt|Hopf|1948}}, with the discrete group
isomorphic to the integers, with a generator acting on '''C'''<sup>2</sup> by multiplication by 2; this was the first example of a compact complex surface with no [[Kähler metric]]. 
 
Higher dimensional analogues of Hopf surfaces are called [[Hopf manifold]]s.
==Invariants==
Hopf surfaces are [[surfaces of class VII]] and in particular all have [[Kodaira dimension]] &minus;&infin; and all their plurigenera vanish. The geometric genus is 0. The fundamental group has a normal central infinite cyclic subgroup of finite index. The Hodge diamond is
{| style="font-weight:bold"
|-
|  ||  || 1
|-
|  || 0 ||  || 1
|-
| 0 ||  || 0 ||  || 0
|-
|  || 1 ||  || 0
|-
|  ||  || 1
|}
In particular the first [[Betti number]] is 1 and  the second Betti number is 0.
Conversely {{harvtxt|Kodaira|1968}} showed that  that a compact complex surface with vanishing the second Betti number and whose  fundamental group contains an infinite cyclic subgroup of finite index is a Hopf surface.
 
==Primary Hopf surfaces==
 
In the course of [[Enriques-Kodaira classification|classification of compact complex surfaces]],
[[Kunihiko Kodaira|Kodaira]] classified the primary Hopf surfaces.
 
A primary Hopf surface is obtained as
:<math>H=\bigg({\Bbb C}^2\backslash 0\bigg)/\Gamma,</math>
where <math>\Gamma</math> is a group generated by
a polynomial contraction <math>\gamma</math>.
Kodaira has found a normal form for <math>\gamma</math>.
In appropriate coordinates,  <math>\gamma</math>
can be written as
:<math> (x, y) \mapsto (\alpha x +\lambda y^n, \beta y)</math>
where <math>\alpha, \beta\in {\Bbb C}</math> are complex numbers
satisfying <math>0<|\alpha|\leq |\beta| <1</math>, and either
<math>\;\lambda=0</math> or <math>\;\alpha=\beta^n</math>.
 
These surfaces contain an elliptic curve (the image of the ''x''-axis) and if &lambda;=0 the image of the ''y''-axis is a second elliptic curve.
When &lambda;=0, the Hopf surface is an elliptic fiber space over the projective line if
&alpha;<sup>''m''</sup> =&beta;<sup>''n''</sup> for some positive integers ''m'' and ''n'', with the map to the projective line given by ''x''<sup>''n''</sup>''y''<sup>&minus;''m''</sup>, and otherwise the only curves are the two images of the axes.  
 
The [[Picard group]] of any primary Hopf surface is isomorphic to the non-zero complex numbers '''C'''<sup>*</sup>.
 
{{harvtxt|Kodaira|1966b}} has proven that a complex surface
is diffeomorphic to '''S'''<sup>3</sup>&times;'''S'''<sup>1</sup>
if and only if it is  a primary Hopf surface.
 
==Secondary Hopf surfaces==
Any secondary Hopf surface has a finite unramified cover that is a primary Hopf surface. Equivalently, its fundamental group has a subgroup of finite index in its center that is isomorphic to the integers. {{harvtxt|Kato|1975}} classified  them by finding the finite groups acting without fixed points on primary Hopf surfaces.
 
Many examples of secondary Hopf surfaces can be constructed with underlying space  a product of a [[spherical space form]]s and a circle.
 
== References ==
*{{Citation | last1=Barth | first1=Wolf P. | last2=Hulek | first2=Klaus | last3=Peters | first3=Chris A.M. | last4=Van de Ven | first4=Antonius | title=Compact Complex Surfaces | publisher= Springer-Verlag, Berlin | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-00832-3 | id={{MathSciNet | id = 2030225}} | year=2004 | volume=4}}
*{{Citation | last1=Hopf | first1=Heinz | author1-link=Heinz Hopf | title=Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948 | publisher=Interscience Publishers, Inc., New York | id={{MathSciNet | id = 0023054}} | year=1948 | chapter=Zur Topologie der komplexen Mannigfaltigkeiten | pages=167–185}}
*{{Citation | last1=Kato | first1=Masahide | title=Topology of Hopf surfaces | url=http://www.journalarchive.jst.go.jp/english/jnlabstract_en.php?cdjournal=jmath1948&cdvol=27&noissue=2&startpage=222 | id={{MathSciNet | id = 0402128}} | year=1975 | journal=Journal of the Mathematical Society of Japan | issn=0025-5645 | volume=27 | pages=222–238 | doi=10.2969/jmsj/02720222 | issue=2}} {{Citation | last1=Kato | first1=Masahide | title=Erratum to: "Topology of Hopf surfaces" | url=http://www.journalarchive.jst.go.jp/english/jnlabstract_en.php?cdjournal=jmath1948&cdvol=41&noissue=1&startpage=173 | id={{MathSciNet | id = 972171}} | year=1989 | journal=Journal of the Mathematical Society of Japan | issn=0025-5645 | volume=41 | issue=1 | pages=173–174 | doi=10.2969/jmsj/04110173}}
*{{Citation | last1=Kodaira | first1=Kunihiko | title=On the structure of compact complex analytic surfaces. II | id={{MathSciNet | id = 0205280}} | year=1966 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=88 | pages=682–721 | doi=10.2307/2373150 | issue=3 | publisher=The Johns Hopkins University Press | jstor=2373150}}
*{{Citation | last1=Kodaira | first1=Kunihiko | title=On the structure of compact complex analytic surfaces. III | id={{MathSciNet | id = 0228019}} | year=1968 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=90 | pages=55–83 | doi=10.2307/2373426 | issue=1 | publisher=The Johns Hopkins University Press | jstor=2373426}}
*{{Citation | last1=Kodaira | first1=Kunihiko | title=Complex structures on S<sup>1</sup>×S<sup>3</sup> | url=http://www.pnas.org/content/55/2/240.full.pdf+html | id={{MathSciNet | id = 0196769}} | year=1966b | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=55 | pages=240–243 | doi=10.1073/pnas.55.2.240 | issue=2}}
*{{Citation | last1=Matumoto | first1=Takao | last2=Nakagawa | first2=Noriaki | title=Explicit description of Hopf surfaces and their automorphism groups | url=http://projecteuclid.org/euclid.ojm/1200789206 | id={{MathSciNet | id = 1772841}} | year=2000 | journal=Osaka Journal of Mathematics | issn=0030-6126 | volume=37 | issue=2 | pages=417–424}}
*{{eom|id=H/h110270|first=L. |last=Ornea|title=Hopf manifold}}
[[Category:Complex surfaces]]

Revision as of 13:22, 17 June 2013

In complex geometry, a Hopf surface is a compact complex surface obtained as a quotient of the complex vector space (with zero deleted) C2 \ 0 by a free action of a discrete group. If this group is the integers the Hopf surface is called primary, otherwise it is called secondary. (Some authors use the term "Hopf surface" to mean "primary Hopf surface".) The first example was found by Template:Harvtxt, with the discrete group isomorphic to the integers, with a generator acting on C2 by multiplication by 2; this was the first example of a compact complex surface with no Kähler metric.

Higher dimensional analogues of Hopf surfaces are called Hopf manifolds.

Invariants

Hopf surfaces are surfaces of class VII and in particular all have Kodaira dimension −∞ and all their plurigenera vanish. The geometric genus is 0. The fundamental group has a normal central infinite cyclic subgroup of finite index. The Hodge diamond is

1
0 1
0 0 0
1 0
1

In particular the first Betti number is 1 and the second Betti number is 0. Conversely Template:Harvtxt showed that that a compact complex surface with vanishing the second Betti number and whose fundamental group contains an infinite cyclic subgroup of finite index is a Hopf surface.

Primary Hopf surfaces

In the course of classification of compact complex surfaces, Kodaira classified the primary Hopf surfaces.

A primary Hopf surface is obtained as

where is a group generated by a polynomial contraction . Kodaira has found a normal form for . In appropriate coordinates, can be written as

where are complex numbers satisfying , and either or .

These surfaces contain an elliptic curve (the image of the x-axis) and if λ=0 the image of the y-axis is a second elliptic curve. When λ=0, the Hopf surface is an elliptic fiber space over the projective line if αmn for some positive integers m and n, with the map to the projective line given by xnym, and otherwise the only curves are the two images of the axes.

The Picard group of any primary Hopf surface is isomorphic to the non-zero complex numbers C*.

Template:Harvtxt has proven that a complex surface is diffeomorphic to S3×S1 if and only if it is a primary Hopf surface.

Secondary Hopf surfaces

Any secondary Hopf surface has a finite unramified cover that is a primary Hopf surface. Equivalently, its fundamental group has a subgroup of finite index in its center that is isomorphic to the integers. Template:Harvtxt classified them by finding the finite groups acting without fixed points on primary Hopf surfaces.

Many examples of secondary Hopf surfaces can be constructed with underlying space a product of a spherical space forms and a circle.

References

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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.

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  • 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.

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    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.

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    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.

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  • 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.

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    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
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