First Chern class ($c_1(E)$)

The first Chern class classifies complex line bundles topologically. For a Hermitian holomorphic line bundle with Chern curvature $F_{\nabla }$, Chern-Weil theory gives $c_1(E)=[i{F}_{\nabla }/(2\pi )]$.

The symplectic canonical class is the negative First Chern class of the tangent bundle for a compatible almost complex structure. The contractibility of compatible almost complex structures makes it independent of that choice. It is the class appearing in the symplectic adjunction formula.

For line-bundle transition functions $g_{ij}=e^{2\pi i{f}_{ij}}$, the integer cocycle is $c=\delta f$. Connection forms satisfy $A_j-A_i=2\pi id{f}_{ij}$. Set $B=-A/(2\pi i)$ and $F=i\Theta /(2\pi )$. In the Čech-de Rham total complex, $F-c=D_{\mathrm{tot}}(B-f)$, so the normalized vector-bundle curvature represents the image of the integral First Chern class. Curvature cannot detect integral torsion.

The First Chern class turns the tensor product of vector bundles of complex line bundles into addition in integral cohomology. This follows by computing the universal complex line bundle on the product of two Complex projective spaces and restricting to each factor. The proof retains torsion elements, which differential forms alone cannot detect.

The Chern connection is the unique connection on a Hermitian holomorphic bundle compatible with both its metric and holomorphic structure. Its curvature has type $(1,1)$.

For a Hermitian metric on a complex manifold, the Chern connection on its holomorphic tangent bundle corresponds to a real metric-compatible connection preserving the almost complex structure. Its mixed torsion vanishes, and in holomorphic coordinates its remaining coefficients are ${\Gamma }_{ij}^k=h^{k\bar{l}}{\partial }_i{h}_{j\bar{l}}$. Symmetry in $i,j$ is equivalent to $\partial \omega =0$; its conjugate gives $d\omega =0$. Thus zero torsion is exactly the Kähler metric condition, and then uniqueness identifies this connection with the Levi-Civita connection.

The vector-bundle curvature of the unique metric-compatible holomorphic Chern connection is an endomorphism-valued form of type $(1,1)$. It is Dolbeault closed and represents the curvature Dolbeault class. Its trace gives normalized first-Chern forms, while its full endomorphism action enters the Lefschetz-Dolbeault curvature commutator.

For the $(1,0)$ component of a Chern connection, $(D_E^{\prime }{)}^{\ast }=-{\ast }_{E^{\ast }}{D}_{E^{\ast }}^{\prime }{\ast }_E$, where the middle operator is the corresponding component of the dual Chern connection. This is the holomorphic-degree counterpart of the bundle-valued Dolbeault adjoint.

In a holomorphic local frame the Chern matrix is $A=H^{-1}\partial H$. The identity $\partial A+A\wedge A=0$ gives vector-bundle curvature $\Theta =\bar{\partial }A$ of type $(1,1)$. Therefore ${\bar{\partial }}_{\mathrm{End}E}\Theta =0$, allowing the vector-bundle curvature to represent a bundle-valued Dolbeault class.

In a holomorphic local frame, a Hermitian metric on a holomorphic vector bundle has matrix $h$. The trace of its Chern connection curvature is $\bar{\partial }\partial \log \det h$. A frame change multiplies $\det h$ by the squared modulus of a holomorphic unit, whose logarithm has vanishing mixed derivative. For two metrics, the quotient of their determinants is a global positive function, and the trace curvatures differ by the exterior derivative of its logarithmic $\partial$-derivative.

If $S$ is a holomorphic subbundle of a Hermitian vector bundle $E$, its induced Chern connection is $D_S={\pi }_S{D}_E$ on sections of $S$. Orthogonal projection preserves metric compatibility, and holomorphicity of $S$ preserves the required $(0,1)$ operator. Uniqueness of the Chern connection therefore identifies the projected connection.

For a holomorphic subbundle $S\subset E$ and quotient $Q=E/S$, the second fundamental form is $A(s)=(I-{\pi }_S)D_{E}s$. The Leibniz rule gives $A(fs)=fA(s)$, and holomorphicity of the subbundle makes its $(0,1)$ part vanish. Thus $A\in {\mathcal{A}}^{1,0}(\mathrm{Hom}(S,Q))$: the form factor is a cotangent bundle, not a tangent bundle.

In an adapted smooth unitary frame with input-index connection convention, the Chern connection matrix has blocks $({\theta }_F,B;-{\overline{B}}^t,{\theta }_Q)$, where the second fundamental form of a holomorphic subbundle $B$ has type $(1,0)$. The Cartan curvature equation with input indices gives the displayed formula. If the ambient curvature is zero, ${\Theta }_F(X,\overline{X})=-B(X)B(X{)}^{†}$ is negative semidefinite. Dualizing the determinant line bundle therefore gives semipositive curvature.

For a holomorphic local frame $e$ with $h_e=h(e,e)>0$, this form isIt is a closed real (1, 1)-form representing the First Chern class. A positive Hermitian metric makes it a positive real (1, 1)-form. Replacing $h$ by $e^{-u}h$ adds $(i/(2\pi ))\partial \bar{\partial }u$ to the form.

In a holomorphic local frame of a Hermitian holomorphic vector bundle, let $H$ be the matrix of its Hermitian metric on a holomorphic vector bundle. With the convention $\nabla e=eA$, the Chern connection hasThe condition ${\nabla }^{0,1}={\bar{\partial }}_E$ forces $A$ to have type $(1,0)$, while metric compatibility forces this formula, proving both existence and uniqueness.

At any point of a Hermitian holomorphic vector bundle one may choose a holomorphic local frame with $H=I$ and $\partial H=0$. First normalize $H$ by a constant frame change, then prescribe a holomorphic matrix's first jet to cancel $H^{-1}\partial H$. The Chern matrix vanishes there. This frame reduces first-order bundle identities at that point to scalar identities.

Let $D_1$ and $D_2$ be Chern connections on the same holomorphic vector bundle, possibly for different Hermitian metrics, and put $a=D_1-D_2$. Then $a$ has type $(1,0)$ andIndeed, the curvature difference formula has only types $(2,0)$ and $(1,1)$; both Chern curvatures have type $(1,1)$, so the $(2,0)$ part vanishes and the remaining part is $D_2^{0,1}a={\bar{\partial }}_{\mathrm{End}E}a$.

Let $e$ be a holomorphic local frame of a Hermitian holomorphic line bundle and put $h=h(e,e)$. The Chern connection and its curvature are locallyThus $F_{\nabla }$ has type $(1,1)$. Because the connection is unitary, ${\bar{F}}_{\nabla }=-F_{\nabla }$, so $i{F}_{\nabla }$ is a real (1, 1)-form.

The Ricci form is the real curvature form $i{F}_{\nabla }$ of the Chern connection on $T^{1,0}X$. In complex dimension one, $\rho =K\omega =\frac{1}{2}\mathrm{Scal}\omega$.