2 Added "H" in displayed formula. Replaced first "impressive" by "useful".

I) The most elementary, but yet quite impressiveuseful, use of a vanishing theorem is via Riemann-Roch for a smooth complete curve $X$ over the algebraically closed field $k$. Riemann-Roch for a divisor $D$ on $X$ says that $$h^0(X,L(D)-h^1(X, L(D)=1-g+deg (D)$$ [Here $g$=genus of $X$, $h( )=dim_k( =dim_k H( )$]

If $deg(D)>2g-2$, the $H^1$ term will vanish and the precise formula $dim L(D)=1-g+deg(D)$ drops out: it gives you the number of rational functions on $X$ with poles controlled by $D$.

II) At a more advanced level an impressive use of vanishing cohomology is Mumford's m-regularity. A coherent sheaf $\mathcal F$ on $\mathbb P_n(k)$ is said to be m-regular if $H^i(\mathbb P_n, \mathcal F(m-i)))=0$ for all $i>0$ . If you find this definition strange, don't worry: Mumford himself calls it "apparently silly" (in his book "Lectures on Curves on an Algebraic Surface", Princeton university Press, 1966). But then he shows how to use this notion to construct hilbert schemes and (re-)derive several theorems on algebraic surfaces (index theorem, completeness of characteristic linear system).

III) An application which might help you understand and appreciate Kodaira's vanishing theorem is one of its corollaries: Lefschetz's "weak" theorem [ proof in Griffiths-Harris, pages 156-157].

It says essentially that in a projective manifold $X$, a smooth hypersurface $Y$ with positive associated line bundle $\mathcal O (Y)$ (e.g. a smooth hyperplane section) has the same singular cohomology (with coefficients in $\mathbb Q$) as the ambient manifold $X$, up to degree $dim(Y)-1$ and bigger cohomology in degree $dim(Y)$ .

IV) etc.

1

I) The most elementary, but yet quite impressive, use of a vanishing theorem is via Riemann-Roch for a smooth complete curve $X$ over the algebraically closed field $k$. Riemann-Roch for a divisor $D$ on $X$ says that $$h^0(X,L(D)-h^1(X, L(D)=1-g+deg (D)$$ [Here $g$=genus of $X$, $h( )=dim_k( )$]

If $deg(D)>2g-2$, the $H^1$ term will vanish and the precise formula $dim L(D)=1-g+deg(D)$ drops out: it gives you the number of rational functions on $X$ with poles controlled by $D$.

II) At a more advanced level an impressive use of vanishing cohomology is Mumford's m-regularity. A coherent sheaf $\mathcal F$ on $\mathbb P_n(k)$ is said to be m-regular if $H^i(\mathbb P_n, \mathcal F(m-i)))=0$ for all $i>0$ . If you find this definition strange, don't worry: Mumford himself calls it "apparently silly" (in his book "Lectures on Curves on an Algebraic Surface", Princeton university Press, 1966). But then he shows how to use this notion to construct hilbert schemes and (re-)derive several theorems on algebraic surfaces (index theorem, completeness of characteristic linear system).

III) An application which might help you understand and appreciate Kodaira's vanishing theorem is one of its corollaries: Lefschetz's "weak" theorem [ proof in Griffiths-Harris, pages 156-157].

It says essentially that in a projective manifold $X$, a smooth hypersurface $Y$ with positive associated line bundle $\mathcal O (Y)$ (e.g. a smooth hyperplane section) has the same singular cohomology (with coefficients in $\mathbb Q$) as the ambient manifold $X$, up to degree $dim(Y)-1$ and bigger cohomology in degree $dim(Y)$ .

IV) etc.