Now I am studying vanishing theorems in algebraic geometry, and I am so curious about why the word $\textbf{Positivity}$ is used.
So if somebody can give any intuition about that word, it will be useful.
$\endgroup$ 52 Answers
$\begingroup$To expand a little on Gunnar's answer, I'll attempt to give you some intuition as to what positivity means in the context of embeddings of complex manifolds into projective space.
To recap, a line bundle $L$ on a compact complex manifold $M$ is positive iff its first Chern class $c_1(L) \in H^{1,1}(M)$ can be represented by a positive $(1,1)$-form, i.e. by a $(1,1)$-form $\omega$ such that at any point $p \in M$, the inequality $-i\omega_P(v, \bar v) \geq 0$ holds for all holomorphic tangent vectors $v$ at $p$.
There are three useful and equivalent definitions of the first Chern class $c_1(L)$:
As Gunnar says, $c_1(L)$ is the cohomology class of the curvature form associated to any Hermitian metric on $L$.
Identitying $L$ as an element of $H^1(\mathcal O_M^\ast) \cong {\rm Pic}(M)$, $c_1(L)$ is the image of $L$ under the map $H^1(\mathcal O_M^\ast) \to H^2(\mathbb Z)$ in the long exact sequence associated to the exponential sequence.
$c_1(L)$ is the Poincare dual of the divisor associated to $L$ (the "divisor" being the formal linear combination $D$ of codimension-one submanifolds such that $L = \mathcal O_M(D)$).
Here are some important examples of positive line bundles:
If $C$ is a curve, then a line bundle $L$ on $C$ is positive iff $L$ has positive degree, i.e. if $L = \mathcal O(\sum_i a_i p_i)$, where $p_i$ are points on $C$ and $\sum_i a_i > 0$. This is because the Poincare dual of any single point is the volume form, which is certainly positive.
In $\mathbb {CP}^n$ for any $n \geq 1$, the line bundle $\mathcal O_{\mathbb {CP}^n}(1)$ is positive. This is because there exists a hermitian metric on $\mathcal O_{\mathbb {CP}^n}(1)$ such that the associated curvature form is proportional to the Fubini-Study Kahler form, and all Kahler forms are positive.
Given a complex manifold $M$ and a line bundle $L$ on $M$, a natural question you might ask is whether the global sections of $L$ define an embedding of $M$ into projective space. In other words, suppose $s_0, \dots, s_n$ form a basis of global sections of $L$. We would like to know whether the rational map $i : M \dashrightarrow \mathbb {CP}^n$ defined by $p \mapsto [s_0 (p) : \dots : s_n(p)]$ is a morphism, and furthermore, whether $i$ is a closed immersion.
There is a simple criterion we can use to test this. If $i : M \dashrightarrow \mathbb {CP}^n$ is indeed a morphism $M \to \mathbb {CP}^n$, then the sections $s_0, \dots, s_n$ are the pullbacks under $i$ of the sections $z_0, \dots , z_n$ of $O_{\mathbb {CP}^n}(1)$, so $L = i^\star O_{\mathbb {CP}^1}(1)$, and hence, $c_1(L) = i^\star \left( c_1(O_{\mathbb {CP}^1}(1)) \right)$. Furthermore, if $i$ defines a closed immersion $M \hookrightarrow \mathbb {CP}^n$, then $i^\star(c_1(O_{\mathbb {CP}^1}(1)))$ can be represented by a positive $(1,1)$-form, namely, the pullback under $i$ of the positive $(1,1)$-form $\omega$ representing $c_1(O_{\mathbb {CP}^1}(1))$; this pullback is positive because if $-i\omega_P(v, \bar v) \geq 0$ holds for all $v \in T_P(\mathbb {CP}^n)$, then certainly $-i\omega_P(v, \bar v) \geq 0$ holds for all $v$ in the vector subspace $T_P(M) \subset T_P(\mathbb {CP}^n)$. In other words, positivity of $L$ is a necessary condition for the global sections of $L$ to define a projective embedding.
What's really nice is that positivity of $L$ is very nearly also a sufficient condition for the global sections of $L$ to define a projective embedding: if $L$ is positive, then there exists some $m > 0$ such that the global sections of $L^{\otimes m}$ define a projective embedding. This is the Kodaira embedding theorem.
Let's verify that the Kodaira embedding theorem holds for line bundles on curves. As mentioned already, a line bundle on a curve is positive iff it has positive degree. And it is an easy consequence of Riemann-Roch that if ${\rm deg}(L) > 0$, then there exists an $m > 0 $ such that the global sections of $L^{\otimes m}$ define an embedding of the curve into projective space.
In my opinion, the Kodaira embedding theorem is one of the main reasons why the notion of positivity is so useful. I learned about this from Griffiths and Harris, Chapter 1.4. The proof of the Kodaira embedding theorem in Griffiths and Harris uses the Kodaira vanishing theorem alluded to by Zach, which is an example of the kinds of vanishing theorems that you are reading about.
$\endgroup$ $\begingroup$"Positivity" comes from the analytic part of things, which is where most of the early vanishing theorems were proven and then applied to the algebraic side.
If $L \to X$ is a line bundle, we can put a Hermitian metric $h$ on $L$. This is just a Hermitian inner product $h_x$ on each line $L_x$ for $x \in X$, that is, a positive real scalar. We can now get a curvature form $\frac i{2\pi}\Theta_{L,h} = -i \partial \bar \partial \log h$ from the Hermitian metric. Because of some very nice Bochner-type formulas, positivity of this form implies the vanishing of some cohomology groups.
The details form a vast area of research that's still being worked on. Demailly's book and other book will get you reasonably up to speed on the line-bundle side of things. Extending things to vector bundles is still largely work in progress.
$\endgroup$
