TITLE: Intuition behind the Kodaira Vanishing Theorem?
QUESTION [17 upvotes]: As the question suggests, what is the intuition behind the Kodaira Vanishing Theorem? The Kodaira Vanishing Theorem says that the cohomology groups $H^q(M, L \otimes K_M)$ vanish for $q \ge 1$ when $L$ is a holomoprhic line bundle over a compact complex manifold $M$ and $L$ admits a Hermitian metric whose curvature form is positive definite. Here, $K_M$ denotes the canonical line bundle.

REPLY [10 votes]: It is well known that a cohomology class can be uniquely expressed by a harmonic form. To show that a certain cohomology group vanishes, it is equivalent to show that every harmonic form form that cohomology group is identically zero.
How does one show that a harmonic form vanishes? Let us look at the case of a real-valued harmonic function. We know that on a compact manifold, every real-valued harmonic function is constant. There are two ways of showing this. One is to use the maximum principle, and another way is to use integration by parts. Instead of considering a harmonic function, we consider a real-valued function $f$ which satisfies the equation$$\Delta f = cf,$$for some positive constant $c$. Such a function $f$ must vanish identically. Here, one can use either the maximum principle or integration by parts.
For the maximum principle, we consider a point where the maximum of $f$ is achieved. At that point, $\Delta f$ is nonpositive. Because of the positivity of $c$, it forces $f$ to be nonpositive. So $f$ is nonpositive everywhere. Applying the same argument to $-f$ instead of $f$, we conclude that $f$ is nonnegative everywhere. Hence, $f$ is identically zero.
For integration by parts, we simply multiply$$\Delta f = cf$$ by $f$, integrate over $M$, and get$$-(df, df)_M = c(f, f)_M,$$ which forces $f$ to be identically zero by the positivity of $c$. The argument of integration by parts is the same as saying the operator $-\Delta + c$ is positive definite.
Let us go back to the consideration of harmonic forms $\varphi$. Instead of $\Delta$, we have the operator$$\square = \overline{\partial}^* \overline{\partial}  + \overline{\partial}\overline{\partial}^*.$$When we compute $\square \varphi$, we will get$$\square\varphi = -(\text{Tr}\, \nabla \overline{\nabla} \varphi) + A\varphi,$$ where $A$ is a zero-order linear operator. The term $-(\text{Tr}\,\nabla \overline{\nabla}\varphi)$ corresponds to $-\Delta f$, and the term $A\varphi$ corresponds to $c \varphi$. When the curvature form of $L$ is positive, the operator $A$ is a positive operator, it corresponds to the positivity of $c$, and we get the vanishing of the harmonic form $\varphi$.
In order to rigorously prove the Kodaira Vanishing Theorem, we would derive the formula for $\square\varphi$.