Let \(X\) be a smooth projective variety of dimension \(n\) over an algebraically closed field.
The geometric genus is \(p_g(X) = h^0(X, \omega_X)\), and the \(m\)-th plurigenus is \(P_m(X) = h^0(X, \omega_X^{\otimes m})\) for \(m \geq 1\).
The irregularity is \(q(X)=h^1(X,\OO_X)\); for surfaces this is also \(p_g(X)-p_a(X)\) [@Har10a, Remark III.7.12.3]. The number of regular one-forms is \(h^{1,0}(X)=h^0(X,\Omega^1_{X/k})\). In characteristic zero, Hodge symmetry gives \(q(X)=h^{1,0}(X)\) [@Har10a, Appendix B].
\(X\) is of general type if \(P_m(X)\) grows like a positive multiple of \(m^n\), that is, \(X\) has Kodaira dimension \(n\).
Let \(X\) and \(X'\) be birational smooth projective varieties of dimension \(n\) over the same algebraically closed field \(k\). For every \(0\le p\le n\) and \(m\ge1\), birational pullback gives an isomorphism \[ H^0(X,(\Omega^p_{X/k})^{\otimes m}) \cong H^0(X',(\Omega^p_{X'/k})^{\otimes m}), \qquad\Omega^p_{X/k}=\bigwedge^p\Omega_{X/k}. \] In particular \(p_g\), every \(P_m\), and every \(h^{p,0}\) are birational invariants in any characteristic. The inverse pullbacks are constructed in P-AGH288PLURIGENUS, following the method of [@Har10a, Theorem II.8.19 and Exercise II.8.8].
(Characteristic-zero Hodge comparison)
For smooth projective varieties in characteristic zero, Hodge symmetry gives \[ h^i(X,\OO_X)=h^0(X,\Omega^i_{X/k}) \] [@Har10a, Appendix B]. For a general characteristic-zero ground field, descend the smooth projective variety to a subfield finitely generated over \(\QQ\) and embed that subfield in \(\CC\); the complex comparison gives the equality, and coherent cohomology commutes with both field extensions [@Har10a, Proposition III.9.3]. Consequently the preceding theorem also gives birational invariance of \(q(X)=h^1(X,\OO_X)\), of \(\chi(X,\OO_X)=\sum_i(-1)^i h^i(X,\OO_X)\), and of the arithmetic genus \[ p_a(X)=(-1)^n\bigl(\chi(X,\OO_X)-1\bigr) \] [@Har10a, Chapter I, §7 and Exercise III.5.2]. The Euler characteristic and arithmetic genus are distinct quantities.
For a smooth projective surface, blowing up a point gives \(K_{\widetilde X}=\pi^*K_X+E\), by P-AGH285BLOWUPCANON. Since \(E^2=-1\) and \((\pi^*K_X).E=0\), one obtains \(K_{\widetilde X}^2=K_X^2-1\) [@Har10a, Chapter V, §3]. Thus the self-intersection of the canonical divisor is not a birational invariant.
Dependencies
D-G1AEHThe genus in its several sensesuses