Genus (mathematics)

There are two related definitions of genus of any projective algebraic scheme X: the arithmetic genus and the geometric genus.[7] When X is an algebraic curve with field of definition the complex numbers, and if X has no singular points, then these definitions agree and coincide with the topological definition applied to the Riemann surface of X (its manifold of complex points). For example, the definition of elliptic curve from algebraic geometry is connected non-singular projective curve of genus 1 with a given rational point on it.

By the Riemann–Roch theorem, an irreducible plane curve of degree ${\displaystyle d}$ given by the vanishing locus of a section ${\displaystyle s\in \Gamma (\mathbb {P} ^{2},{\mathcal {O}}_{\mathbb {P} ^{2}}(d))}$ has geometric genus

${\displaystyle g={\frac {(d-1)(d-2)}{2}}-s,}$

where s is the number of singularities when properly counted.

In differential geometry, a genus of an oriented manifold ${\displaystyle M}$ may be defined as a complex number ${\displaystyle \Phi (M)}$ subject to the conditions

• ${\displaystyle \Phi (M_{1}\amalg M_{2})=\Phi (M_{1})+\Phi (M_{2})}$
• ${\displaystyle \Phi (M_{1}\times M_{2})=\Phi (M_{1})\cdot \Phi (M_{2})}$
• ${\displaystyle \Phi (M_{1})=\Phi (M_{2})}$ if ${\displaystyle M_{1}}$ and ${\displaystyle M_{2}}$ are cobordant.

In other words, ${\displaystyle \Phi }$ is a ring homomorphism ${\displaystyle R\to \mathbb {C} }$, where ${\displaystyle R}$ is Thom's oriented cobordism ring.[8]

The genus ${\displaystyle \Phi }$ is multiplicative for all bundles on spinor manifolds with a connected compact structure if ${\displaystyle \log _{\Phi }}$ is an elliptic integral such as ${\displaystyle \log _{\Phi }(x)=\int _{0}^{x}(1-2\delta t^{2}+\varepsilon t^{4})^{-1/2}dt}$ for some ${\displaystyle \delta ,\varepsilon \in \mathbb {C} .}$ This genus is called an elliptic genus.

The Euler characteristic ${\displaystyle \chi (M)}$ is not a genus in this sense since it is not invariant concerning cobordisms.

Genus can be also calculated for the graph spanned by the net of chemical interactions in nucleic acids or proteins. In particular, one may study the growth of the genus along the chain. Such a function (called the genus trace) shows the topological complexity and domain structure of biomolecules.[9]

• Group (mathematics)
• Arithmetic genus
• Geometric genus
• Genus of a multiplicative sequence
• Genus of a quadratic form
• Spinor genus