{
  "id": "enwiki-001335879561",
  "slug": "arithmetic-genus",
  "title": "Arithmetic genus",
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  "lead": "In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface.",
  "body": "In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface. Let X be a projective scheme of dimension r over a field k, the arithmetic genus p_a of X is defined as p_a(X)=(-1)^r (\\chi(\\mathcal{O}_X)-1). Here \\chi(\\mathcal{O}_X) is the Euler characteristic of the structure sheaf \\mathcal{O}_X . ==Complex projective manifolds== The arithmetic genus of a complex projective manifold of dimension n can be defined as a combination of Hodge numbers, namely p_a=\\sum_{j=0}^{n-1} (-1)^j h^{n-j,0}. When n=1, the formula becomes p_a=h^{1,0} . According to the Hodge theorem, h^{0,1}=h^{1,0} . Consequently h^{0,1}=h^1(X)/2=g , where g is the usual (topological) meaning of genus of a surface, so the definitions are compatible. When X is a compact Kähler manifold, applying h p,q = h q,p recovers the earlier definition for projective varieties. ==Kähler manifolds== By using h p,q = h q,p for compact Kähler manifolds this can be reformulated as the Euler characteristic in coherent cohomology for the structure sheaf \\mathcal{O}_M : p_a=(-1)^n(\\chi(\\mathcal{O}_M)-1).\\, This definition therefore can be applied to some other locally ringed spaces. ==See also== Genus (mathematics) Geometric genus ==References== ==Further reading==",
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    "title": "Arithmetic genus",
    "author": [
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        "family": "Anticloud FZ LLE",
        "literal": "Anticloud FZ LLE"
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    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
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          15
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