Irregularity of a surface

<h2 id="complex-surfaces">Complex surfaces</h2>
For non-singular complex projective (or <a href="/facts/K%25C3%25A4hler_manifold/JFPM6Vxg">Kähler</a>) surfaces, the following numbers are all equal:

<ul><li>The irregularity;</li>
<li>The dimension of the <a href="/facts/Albanese_variety/IC9tc5l0">Albanese variety</a>;</li>
<li>The dimension of the <a href="/facts/Picard_variety/lWH0CoWQ">Picard variety</a>;</li>
<li>The <a href="/facts/Hodge_number/b6Mju74J">Hodge number</a> 
 
 
 
 
 h
 
 0
 ,
 1
 
 
 =
 dim
 ⁡
 
 H
 
 1
 
 
 (
 
 Ω
 
 X
 
 
 0
 
 
 )
 
 
 {\displaystyle h^{0,1}=\dim H^{1}(\Omega _{X}^{0})}
 
;</li>
<li>The <a href="/facts/Hodge_number/b6Mju74J">Hodge number</a> 
 
 
 
 
 h
 
 1
 ,
 0
 
 
 =
 dim
 ⁡
 
 H
 
 0
 
 
 (
 
 Ω
 
 X
 
 
 1
 
 
 )
 
 
 {\displaystyle h^{1,0}=\dim H^{0}(\Omega _{X}^{1})}
 
;</li>
<li>The difference 
 
 
 
 
 p
 
 g
 
 
 −
 
 p
 
 a
 
 
 
 
 {\displaystyle p_{g}-p_{a}}
 
 of the <a href="/facts/Geometric_genus/S2oQVNCb">geometric genus</a> and the <a href="/facts/Arithmetic_genus/hYdDY8Su">arithmetic genus</a>.</li></ul>
For surfaces in positive characteristic, or for non-Kähler complex surfaces, the numbers above need not all be equal.
<a href="/facts/Henri_Poincar%25C3%25A9/AazL2j4X">Henri Poincaré</a> proved that for complex projective surfaces the dimension of the Picard variety is equal to the <a href="/facts/Hodge_number/b6Mju74J">Hodge number</a> h0,1, and the same is true for all compact Kähler surfaces. The irregularity of smooth compact Kähler surfaces is invariant under bimeromorphic transformations.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
For general compact complex surfaces the two Hodge numbers h1,0 and h0,1 need not be equal, but h0,1 is either h1,0 or h1,0+1, and is equal to h1,0 for compact <a href="/facts/K%25C3%25A4hler_surface/JFPM6Vxg">Kähler surfaces</a>.

<h2 id="positive-characteristic">Positive characteristic</h2>
Over fields of <a href="/facts/Positive_characteristic/D2VzcQaG">positive characteristic</a>, the relation between q (defined as the dimension of the Picard or Albanese variety), and the Hodge numbers h0,1 and h1,0 is more complicated, and any two of them can be different.
There is a canonical map from a surface F to its Albanese variety A which induces a homomorphism from the cotangent space of the Albanese variety (of dimension q) to H1,0(F).<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> <a href="/facts/Jun-Ichi_Igusa/5P56tJ0b">Jun-Ichi Igusa</a> found that this is injective, so that 
 
 
 
 q
 ≤
 
 h
 
 1
 ,
 0
 
 
 
 
 {\displaystyle q\leq h^{1,0}}
 
, but shortly after found a surface in characteristic 2 with 
 
 
 
 
 h
 
 1
 ,
 0
 
 
 =
 
 h
 
 0
 ,
 1
 
 
 =
 2
 
 
 {\displaystyle h^{1,0}=h^{0,1}=2}
 
 and <a href="/facts/Picard_variety/lWH0CoWQ">Picard variety</a> of dimension 1, so that q can be strictly less than both Hodge numbers.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> In positive characteristic neither Hodge number is always bounded by the other. Serre showed that it is possible for h1,0 to vanish while 
h0,1 is positive, while Mumford showed that for <a href="/facts/Enriques_surface/Mwx4f1uI">Enriques surfaces</a> in characteristic 2 it is possible for h0,1 to vanish while h1,0 is positive.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
<a href="/facts/Alexander_Grothendieck/jLQGPY7M">Alexander Grothendieck</a> gave a complete description of the relation of q to 
 
 
 
 
 h
 
 0
 ,
 1
 
 
 
 
 {\displaystyle h^{0,1}}
 
in all characteristics. The dimension of the tangent space to the Picard scheme (at any point) is equal to 
 
 
 
 
 h
 
 0
 ,
 1
 
 
 
 
 {\displaystyle h^{0,1}}
 
.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> In characteristic 0 a result of <a href="/facts/Pierre_Cartier_(mathematician)/Mo74aNlW">Pierre Cartier</a> showed that all groups schemes of finite type are non-singular, so the dimension of their tangent space is their dimension. On the other hand, in positive characteristic it is possible for a group scheme to be non-reduced at every point so that the dimension is less than the dimension of any tangent space, which is what happens in Igusa's example. Mumford shows that the tangent space to the Picard variety is the subspace of H0,1 annihilated by all <a href="/facts/Bockstein_operation/AGjJJfuk">Bockstein operations</a> from H0,1 to H0,2, so the irregularity q is equal to h0,1 if and only if all these Bockstein operations vanish.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225 <a href="978-3-540-00832-3" target="_blank">978-3-540-00832-3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Bombieri, Enrico; Mumford, David (1977), "Enriques' classification of surfaces in char. p. II", Complex analysis and algebraic geometry, Tokyo: Iwanami Shoten, pp. 23–42, MR 0491719 <a href="/wiki/Enrico_Bombieri" target="_blank">/wiki/Enrico_Bombieri</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Poincaré, Henri (1910), "Sur les courbes tracées sur les surfaces algébriques", Annales Scientifiques de l'École Normale Supérieure, 3, 27: 55–108, doi:10.24033/asens.617 <a href="/wiki/Henri_Poincar%C3%A9" target="_blank">/wiki/Henri_Poincar%C3%A9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Igusa, Jun-Ichi (1955), "A fundamental inequality in the theory of Picard varieties", Proceedings of the National Academy of Sciences of the United States of America, 41 (5): 317–320, Bibcode:1955PNAS...41..317I, doi:10.1073/pnas.41.5.317, ISSN 0027-8424, JSTOR 89124, MR 0071113, PMC 528086, PMID 16589672 <a href="/wiki/Jun-Ichi_Igusa" target="_blank">/wiki/Jun-Ichi_Igusa</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Igusa, Jun-Ichi (1955), "A fundamental inequality in the theory of Picard varieties", Proceedings of the National Academy of Sciences of the United States of America, 41 (5): 317–320, Bibcode:1955PNAS...41..317I, doi:10.1073/pnas.41.5.317, ISSN 0027-8424, JSTOR 89124, MR 0071113, PMC 528086, PMID 16589672 <a href="/wiki/Jun-Ichi_Igusa" target="_blank">/wiki/Jun-Ichi_Igusa</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Serre, Jean-Pierre (1958), "Sur la topologie des variétés algébriques en caractéristique p", Symposium internacional de topología algebraica, Universidad Nacional Autónoma de México and UNESCO, Mexico City, pp. 24–53, MR 0098097 <a href="/wiki/Jean-Pierre_Serre" target="_blank">/wiki/Jean-Pierre_Serre</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Mumford, David (1961), "Pathologies of modular algebraic surfaces" (PDF), American Journal of Mathematics, 83 (2), The Johns Hopkins University Press: 339–342, doi:10.2307/2372959, ISSN 0002-9327, JSTOR 2372959, MR 0124328 <a href="/wiki/David_Mumford" target="_blank">/wiki/David_Mumford</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Grothendieck, Alexander (1961), Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki 221 <a href="/wiki/Alexander_Grothendieck" target="_blank">/wiki/Alexander_Grothendieck</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Mumford, David (1961), "Pathologies of modular algebraic surfaces" (PDF), American Journal of Mathematics, 83 (2), The Johns Hopkins University Press: 339–342, doi:10.2307/2372959, ISSN 0002-9327, JSTOR 2372959, MR 0124328 <a href="/wiki/David_Mumford" target="_blank">/wiki/David_Mumford</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

Irregularity of a surface open-in-new

Irregularity of a surface