In mathematics, and especially complex geometry, the holomorphic tangent bundle of a complex manifold ${\displaystyle M}$ is the holomorphic analogue of the tangent bundle of a smooth manifold. The fibre of the holomorphic tangent bundle over a point is the holomorphic tangent space, which is the tangent space of the underlying smooth manifold, given the structure of a complex vector space via the almost complex structure ${\displaystyle J}$ of the complex manifold ${\displaystyle M}$.

Given a complex manifold ${\displaystyle M}$ of complex dimension ${\displaystyle n}$, its tangent bundle as a smooth vector bundle is a real rank ${\displaystyle 2n}$ vector bundle ${\displaystyle TM}$ on ${\displaystyle M}$. The integrable almost complex structure ${\displaystyle J}$ corresponding to the complex structure on the manifold ${\displaystyle M}$ is an endomorphism ${\displaystyle J:TM\to TM}$ with the property that ${\displaystyle J^{2}=-\operatorname {Id} }$. After complexifying the real tangent bundle to ${\displaystyle TM\otimes \mathbb {C} \to M}$, the endomorphism ${\displaystyle J}$ may be extended complex-linearly to an endomorphism ${\displaystyle J:TM\otimes \mathbb {C} \to TM\otimes \mathbb {C} }$ defined by ${\displaystyle J(X+iY)=J(X)+iJ(Y)}$ for vectors ${\displaystyle X,Y}$ in ${\displaystyle TM}$.

Since ${\displaystyle J^{2}=-\operatorname {Id} }$, ${\displaystyle J}$ has eigenvalues ${\displaystyle i,-i}$ on the complexified tangent bundle, and ${\displaystyle TM\otimes \mathbb {C} }$ therefore splits as a direct sum

where ${\displaystyle T^{1,0}M}$ is the ${\displaystyle i}$-eigenbundle, and ${\displaystyle T^{0,1}M}$ the ${\displaystyle -i}$-eigenbundle. The holomorphic tangent bundle of ${\displaystyle M}$ is the vector bundle ${\displaystyle T^{1,0}M}$, and the anti-holomorphic tangent bundle is the vector bundle ${\displaystyle T^{0,1}M}$.

The vector bundles ${\displaystyle T^{1,0}M}$ and ${\displaystyle T^{0,1}M}$ are naturally complex vector subbundles of the complex vector bundle ${\displaystyle TM\otimes \mathbb {C} }$, and their duals may be taken. The holomorphic cotangent bundle is the dual of the holomorphic tangent bundle, and is written ${\displaystyle T_{1,0}^{*}M}$. Similarly the anti-holomorphic cotangent bundle is the dual of the anti-holomorphic tangent bundle, and is written ${\displaystyle T_{0,1}^{*}M}$. The holomorphic and anti-holomorphic (co)tangent bundles are interchanged by conjugation, which gives a real-linear (but not complex linear!) isomorphism ${\displaystyle T^{1,0}M\to T^{0,1}M}$.

The holomorphic tangent bundle ${\displaystyle T^{1,0}M}$ is isomorphic as a real vector bundle of rank ${\displaystyle 2n}$ to the regular tangent bundle ${\displaystyle TM}$. The isomorphism is given by the composition ${\displaystyle TM\hookrightarrow TM\otimes \mathbb {C} {\xrightarrow {\operatorname {pr} _{1,0}}}T^{1,0}M}$ of inclusion into the complexified tangent bundle, and then projection onto the ${\displaystyle i}$-eigenbundle.

The canonical bundle is defined by ${\displaystyle K_{M}=\Lambda ^{n}T_{1,0}^{*}M}$.

In a local holomorphic chart ${\displaystyle \varphi =(z^{1},\dots ,z^{n}):U\to \mathbb {C} ^{n}}$ of ${\displaystyle M}$, one has distinguished real coordinates ${\displaystyle (x^{1},\dots ,x^{n},y^{1},\dots ,y^{n})}$ defined by ${\displaystyle z^{j}=x^{j}+iy^{j}}$ for each ${\displaystyle j=1,\dots ,n}$. These give distinguished complex-valued one-forms ${\displaystyle dz^{j}=dx^{j}+idy^{j},d{\bar {z}}^{j}=dx^{j}-idy^{j}}$ on ${\displaystyle U}$. Dual to these complex-valued one-forms are the complex-valued vector fields (that is, sections of the complexified tangent bundle),