Let \(X\) be an integral projective scheme of dimension \(\geq 1\) over a field \(k\), and let \(\mcl\) be an ample invertible sheaf on \(X\). Show that \[ H^0(X, \mcl^{-1}) = 0 . \]
This is a special case of Kodaira vanishing.
Solution
Tensor powers of the invertible sheaf \(\mcl\) are written \(\mcl^a\) for \(a\in\ZZ\).
1. The ring \(K_0=H^0(X,\OO_X)\) is a finite field extension of \(k\).
Finiteness of coherent cohomology on a projective scheme makes \(K_0\) finite-dimensional over \(k\) [@Har10a, Theorem III.5.2]. Restriction to the generic point is injective because \(X\) is integral, so \(K_0\) is a domain. Multiplication by any nonzero element is therefore an injective endomorphism of this finite-dimensional vector space, hence is surjective. In particular that element has an inverse in \(K_0\). Thus \(K_0\) is a field, and every nonzero global regular function is a unit at every point of \(X\).
2. One has \(\boxed{H^0(X,\mcl^{-1})=0}\).
Suppose \(s\) is a nonzero section of \(\mcl^{-1}\). Choose \(m>0\) for which \(\mcl^m\) is very ample, using [@Har10a, Theorem II.7.6]. Its immersion \(i:X\to\PP_k^N\) is closed because \(X\) is proper, and \(i^*\OO(1)=\mcl^m\). Choose a closed point \(x\in X\). Since \(X\) has positive dimension, its homogeneous ideal is strictly contained in the homogeneous ideal of the closed subscheme \(\{x\}\subseteq\PP^N\). There is consequently a positive-degree homogeneous polynomial \(h\), of some degree \(q>0\), vanishing at \(x\) but not identically on \(X\). It gives a nonzero section of \(\mcl^{mq}\) whose fibre value at \(x\) is zero.
The product \(h s^{\otimes mq}\) is a global section of \(\OO_X\). Its generic value is nonzero, since both factors have nonzero generic values on the integral scheme. Its value at \(x\) is zero. This contradicts step 1, which makes every nonzero global function a unit at \(x\). Thus no such \(s\) exists. The argument does not assume that \(x\) is \(k\)-rational or that \(k\) is infinite.
3. Q.E.D.
Step 2 establishes the required vanishing under exactly the stated hypotheses.