Deje $M$ $n$- dimensiones del colector y $E$ a un rango de $m$ vector complejo paquete de más de $M$. El Chern-Weil descripción de las clases de Chern de $E$ es, como usted ha dicho, dada por la generación de la función $$\det\left( \mathbb{I}_E + \frac{it}{2\pi}\Omega_A \right) = \sum c_k(E)t^k,$$ where $\Omega_A$ is the curvature of an arbitrarily chosen connection $$ on $E$; the cohomology class of each $c_k(E)$ is independent of the choice of $$. Now $\Omega_A$ can be viewed as an $\mathrm{End}(E)$-valued $2$-form, or, as you guessed, an $m \times m$ matrix of complex-valued $2$-forms. When taking products of the entries of $\Omega_A$, we use the wedge product. Note that with this interpretation, $i\Omega_A/2\pi$ is a skew-symmetric matrix and is hence diagonalizable. Let $x_1, \dots, x_m$ be the eigenvalues of $\Omega_A$ , and note that they are complex-valued $2$-forms. Then we have $$\det\left(\mathbb{I}_E + \frac{it}{2\pi}\Omega_A\right) = \det(\mathrm{diag}(1 + tx_1, \dots, 1 + tx_m)) = \sum_{k = 0}^m S_k(x_1, \dots, x_m)t^k,$$ where $S_k(x_1, \dots, x_m)$ is the $k^\text{th}$ elementary symmetric polynomial in $x_1, \dots, x_m$, i.e. $$S_0(x_1, \dots, x_m) = 1,$$ $$S_1(x_1, \dots, x_m) = \sum_{i = 1}^m x_i,$$ $$S_2(x_1, \dots, x_m) = \sum_{i < j} x_i x_j,$$ and so on until $$S_m(x_1, \dots, x_m) = x_1 \cdots x_m.$$ Using this, it is clear that $c_k(E)$ is a form of degree $2k$. Furthermore, this permits calculation of the Chern classes: $$c_1(E) = \frac{i}{2\pi} \mathrm{Tr}(\Omega_A),$$ $$c_2(E) = \frac{1}{2} \left( \frac{i}{2\pi} \mathrm{Tr}(\Omega_A) \right)^2 - \frac{1}{2} \mathrm{Tr}\left( \frac{i\Omega_A}{2\pi}\right)^2 = \frac{1}{2}c_1(E)^2 + \frac{1}{8\pi^2} \mathrm{Tr}(\Omega_A \wedge \Omega_A),$$ and so on until $$c_m(E) = \left( \frac{i}{2\pi} \right)^m \det(\Omega_A).$$
Deje $M$ ser un complejo colector de estructura compleja $J$ en su tangente paquete. A continuación, el paquete de $S^{1,1}M$ real simétrica $J$-invariante formas en $TM$ y el bulto $\wedge^{1,1} T^\ast M$ $2$- formas de tipo $(1,1)$ $M$ se puede poner en una correspondencia uno a uno por la asociación de a $b \in S^{1,1}M$ $2$forma $\beta \in \wedge^{1,1} T^\ast M$ definido por $$\beta(X,Y) = b(JX,Y)$$ for all $X, Y\, en T_x M$ and $x \in M$; if $b$ is positive definite (resp. negative definite), then $\beta$ is called positive (resp. negative). Then a cohomology class $\alpha \H^2(M;\mathbb{R})$ is positive (resp. negative) if it can be represented by a real positive (resp. negative) $2$-form of type $(1,1)$. This notion of positivity/negativity depends only on the complex structure $J$.
El ajuste a cero en $\mathbb{C}P^n$ homogéneo de grado $k$ polinomio $p$ es un buen colector de al $p$ satisface $(\nabla p)(Z) \neq 0$ cualquier $Z \neq 0$ tal que $p(Z) = 0$. Llamamos a la puesta a cero de un polinomio de la no-singular complejo de la hipersuperficie de grado $k$ $\mathbb{C}P^n$ y se denota por a $V^n(k)$; el uso de Ehresmann del fibration teorema se puede demostrar que la diffeomorphism clase de $V^n(k)$ sólo depende de $n$ $k$ (y no en el polinomio $p$). El uso de la contigüidad de la fórmula, se encuentra que $$c_1(V^n(k)) = (n + 1 - k)\omega,$$ where $\omega$ is the pullback of the Fubini-Study form on $\mathbb{C}P^n$ to $V^n(k)$. The Fubini-Study form is of course positive. Hence one sees that $c_1(V^n(k))$ is positive for $k \leq n$, zero for $k = n + 1$, and negative for $k \geq n + 2$. In particular, $V^n(n+1)$ is a Calabi-Yau manifold (not $V^n(n)$ como se dijo).