Differential of the first kind

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, differential of the first kind is a traditional term used in the theories of <a href="/facts/Riemann_surface/qoj8ogv8">Riemann surfaces</a> (more generally, <a href="/facts/Complex_manifold/DlvPnpzm">complex manifolds</a>) and <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curves</a> (more generally, <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic varieties</a>), for everywhere-regular <a href="/facts/Differential_form/T9gyHtMv">differential 1-forms</a>. Given a complex manifold M, a differential of the first kind ω is therefore the same thing as a 1-form that is everywhere <a href="/facts/Holomorphic_form/307vM8pd">holomorphic</a>; on an <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a> V that is <a href="/facts/Algebraic_curve/LnWsPmDP">non-singular</a> it would be a <a href="/facts/Global_section/MM3hcRw4">global section</a> of the <a href="/facts/Coherent_sheaf/h2NoUN7e">coherent sheaf</a> Ω1 of <a href="/facts/K%C3%A4hler_differential/1aPfygbf">Kähler differentials</a>. In either case the definition has its origins in the theory of <a href="/facts/Abelian_integral/2oLFBNWk">abelian integrals</a>.
The dimension of the space of differentials of the first kind, by means of this identification, is the <a href="/facts/Hodge_number/b6Mju74J">Hodge number</a>

h1,0.
The differentials of the first kind, when integrated along paths, give rise to integrals that generalise the <a href="/facts/Elliptic_integral/q8EGYzQ3">elliptic integrals</a> to all curves over the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>. They include for example the hyperelliptic integrals of type

∫
        
          
            
              
                x
                
                  k
                
              
              
              d
              x
            
            
              Q
              (
              x
              )
            
          
        
      
    
    {\displaystyle \int {\frac {x^{k}\,dx}{\sqrt {Q(x)}}}}

where Q is a <a href="/facts/Square-free_polynomial/G5j2B4hM">square-free polynomial</a> of any given degree > 4. The allowable power k has to be determined by analysis of the possible pole at the <a href="/facts/Point_at_infinity/GAuS0wHQ">point at infinity</a> on the corresponding <a href="/facts/Hyperelliptic_curve/sf07gwGS">hyperelliptic curve</a>. When this is done, one finds that the condition is

k ≤ g − 1,
or in other words, k at most 1 for degree of Q 5 or 6, at most 2 for degree 7 or 8, and so on (as g = [(1+ deg Q)/2]).
Quite generally, as this example illustrates, for a <a href="/facts/Compact_Riemann_surface/qoj8ogv8">compact Riemann surface</a> or <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curve</a>, the Hodge number is the <a href="/facts/Genus_(mathematics)/o1Yhf8ty">genus</a> g. For the case of <a href="/facts/Algebraic_surface/2zHU0o9J">algebraic surfaces</a>, this is the quantity known classically as the <a href="/facts/Irregularity_of_a_surface/y3ozma2e">irregularity</a> q. It is also, in general, the dimension of the <a href="/facts/Albanese_variety/IC9tc5l0">Albanese variety</a>, which takes the place of the <a href="/facts/Jacobian_variety/HFedCxIn">Jacobian variety</a>.

Differential of the first kind open-in-new

Differential of the first kind