Question

Let $X$ be a projective algebraic variety over a algebraic closed field $k$. Let $\mathcal{F}$ be a coherent sheaf on $X$. We know that $H^0(X, \mathcal{F})$ is the vector space of global sections of $\mathcal{F}$. This gives us a geometric illustration of $H^0$. For example, let $I_D$ be the ideal sheaf of a hypersurface $D$ of degree > 1 in a projective space $\mathbb{P}^n$, then it is easy to see that $$H^0(\mathbb{P}^n,I_D\otimes\mathcal{O}_{\mathbb{P}^n}(1))=0.$$

In fact, there is no hyperplane containing $D$, which means that there is no global section of $\mathcal{O}_{\mathbb{P}^n}(1)$, which are hyperplanes, containing $D$. Hence $H^0(\mathbb{P}^n, I_D\otimes\mathcal{O}_{\mathbb{P}^n}(1))=0$. However, my first question is how to understand higher cohomologies of sheaves in geometric ways. The following questions then come out:

1) How to understand Serre's vanishing theorem, i.e., is there a geometric way to think about the vanishing of $H^q(X, \mathcal{F}\otimes A^n)$ for $n>>1$, where $\mathcal{F}$ is coherent and $A$ is ample.

2) How to understan Kodaira's vanishing theorem geometrically.

Maybe a concrete question will help, say $D$ a subvariety of $\mathbb{P}^n$, how to determine geometrically whether $H^0(\mathbb{P}^n, I_D\otimes\mathcal{O}_{\mathbb{P}^n}(1))$ vanishes or not.

Answer 1

I) The most elementary, but yet quite useful, 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 $X$, $h( )=dim_k H( )$]

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.

Answer 2

One way to think about higher cohomology groups of, say, holomorphic vector bundles is via the Dolbeault isomorphism $$ H^q(X, \mathcal O(E)) \cong H^{0,q}(X,E) $$ (and also the more general $H^{q}(\mathcal O( \Lambda^pT^*X\otimes E)) \cong H^{p,q}(E)$.)

If we also choose a Kähler metric on X and Hermitian metric on E, then Hodge theory says that cohomology is represented by harmonic forms. So we can think of the q^th cohomology group of the sheaf of sections of E as the space of harmonic (0,q)-forms with values in E.

One can interpret Kodaira vanishing from this point of view. Pick a holomorphic line bundle L with a Hermitian metric over a Kähler manifold X. Then one has a connection in L and so an induced connection on L-valued forms. This gives a "rough" Laplacian $\nabla^* \nabla$. The Weitzenbock formula tells us how this differs from the standard Laplacian. On a (p,q)-form with values in L, $$ \Delta = \nabla^*\nabla + F $$ where F is an endomorphism of the bundle $\Lambda^{p,q}\otimes L$ where the L-valued forms take their values. F depends on (p,q) and on the metrics of both L and X.

The hypothesis for Kodaira vanishing is that there is a Hermitian metric on L whose curvature is a Kahler form. If we use this metric on L and also this Kahler form on X then the operator F has a certain sign: when p+q>n, the dimension of X, F is a positive definite endomorphism of the bundle $\Lambda^{p,q}\otimes L$. From here we can prove Kodaira vanishing. A harmonic (p,q) form $\alpha$ with values in L has $\Delta \alpha = 0$ so, by integrating the Weitzenbock formula against $\alpha$, we see $$ \int |\nabla \alpha|^2 + \int (F(\alpha), \alpha) = 0 $$ Now positivity of $F$ means both terms here are non-negative and so must each vanish. This forces $\alpha$ to vanish and so $H^{p,q}(L)=0$ when $p+q >n$.