The Stacks project

Remark 53.11.2. Suppose that $X$ is a $d$-dimensional proper smooth variety over an algebraically closed field $k$. Then the arithmetic genus is often defined as $p_ a(X) = (-1)^ d(\chi (X, \mathcal{O}_ X) - 1)$ and the geometric genus as $p_ g(X) = \dim _ k H^0(X, \Omega ^ d_{X/k})$. In this situation the arithmetic genus and the geometric genus no longer agree even though it is still true that $\omega _ X \cong \Omega _{X/k}^ d$. For example, if $d = 2$, then we have

\begin{align*} p_ a(X) - p_ g(X) & = h^0(X, \mathcal{O}_ X) - h^1(X, \mathcal{O}_ X) + h^2(X, \mathcal{O}_ X) - 1 - h^0(X, \Omega ^2_{X/k}) \\ & = - h^1(X, \mathcal{O}_ X) + h^2(X, \mathcal{O}_ X) - h^0(X, \omega _ X) \\ & = - h^1(X, \mathcal{O}_ X) \end{align*}

where $h^ i(X, \mathcal{F}) = \dim _ k H^ i(X, \mathcal{F})$ and where the last equality follows from duality. Hence for a surface the difference $p_ g(X) - p_ a(X)$ is always nonnegative; it is sometimes called the irregularity of the surface. If $X = C_1 \times C_2$ is a product of smooth projective curves of genus $g_1$ and $g_2$, then the irregularity is $g_1 + g_2$.

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