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In a holomorphic chart $z^j=x^j+i{y}^j$, defineThe derivative of every holomorphic transition map is complex linear and therefore commutes with this $J$, so the chartwise definitions glue to the intrinsic almost complex structure of the complex manifold.

For a complex manifold, Bott-Chern cohomology isIts classes are represented by $d$-closed forms of pure type. A natural map sends a Bott-Chern class to its Dolbeault cohomology class.

Using the bundle-valued conjugate-linear Hodge star, the formal adjoint of the holomorphic bundle's Dolbeault operator is ${\bar{\partial }}_E^{\ast }=-{\ast }_{E^{\ast }}{\bar{\partial }}_{E^{\ast }}{\ast }_E$. It lowers antiholomorphic degree. Stokes applied to the coefficient-contracted form $\psi \wedge {\ast }_E\eta$ proves the integrated adjoint identity on a compact complex manifold.

On a complex manifold, a sheaf of modules over the sheaf of holomorphic functions is coherent analytic if it is locally finitely generated and the kernel of every morphism ${\mathcal{O}}^r\to \mathcal{F}$ is locally finitely generated. The structure sheaf and locally free finite-rank sheaves are examples. Cartan theorem B gives vanishing of their higher sheaf cohomology on a Stein manifold.

Every scalar holomorphic function on a compact connected complex manifold is constant by the maximum modulus principle. Apply this to each entry of the holomorphic Adjoint representation of a Lie group; its value is constantly the identity. Differentiation makes the complex Lie algebra abelian, hence connectedness makes the group abelian. The holomorphic Lie exponential map identifies it with ${\mathbb{C}}^n$ modulo a discrete subgroup. Compactness forces that subgroup to span the underlying real vector space, giving a full-rank Euclidean lattice and a complex torus.

A complex algebraic variety is an algebraic variety over the complex numbers. In the scheme convention used here, it is a reduced separated scheme of finite type over $\mathbb{C}$; irreducibility is an additional condition when needed. Affine examples are zero sets of complex polynomials equipped with their regular functions.

For example, $xy=0$ is the union of two complex coordinate lines, while $y^2=x^3$ is an irreducible singular curve. The smooth locus of a complex variety has the structure of a complex manifold, but a singular point need not be smooth as a complex analytic space.

One must also distinguish topologies. The complex affine line is $\mathbb{C}$ as a point set; its proper Zariski-closed subsets are finite, whereas its usual analytic topology has many more closed and open subsets. Passing from an algebraic variety to its analytic space retains geometric information while changing which local functions and open sets are allowed.

A complex analytic hypersurface in an $n$-dimensional complex manifold is a closed analytic subset of pure complex dimension $n-1$. Locally it is the zero set of one reduced holomorphic function $f$. This local single-equation property follows from the factoriality of the ambient regular local ring.^[ref]

A complex analytic hypersurface need not be a complex submanifold. In ${\mathbb{C}}^2$, $z_1=0$ defines a smooth example, whereas $z_1{z}_2=0$ is the union of two complex axes meeting at a singular point. For a reduced local equation, smoothness at a point is equivalent to $df\ne 0$ there, by the holomorphic implicit function theorem.

Irreducibility means that the hypersurface is not the union of two proper closed analytic subsets. It is an additional condition, not another name for every complex analytic hypersurface; the two-axis example is reducible.

A complex analytic space is a locally ringed space locally isomorphic to the common zero set of finitely many holomorphic functions on an open subset of ${\mathbb{C}}^n$, equipped with the quotient sheaf of holomorphic functions by a coherent ideal sheaf. This sheaf is part of the structure: the same underlying set can carry different analytic structures.

The cusp $y^2=x^3$ gives a reduced singular example. Its local holomorphic functions are represented by ambient holomorphic functions modulo the equation $y^2-x^3$. A complex manifold is the special smooth local model, with no singular equations needed after choosing coordinates.

General complex analytic spaces can also be nonreduced. The quotient $\mathbb{C}\{z\}/(z^2)$ has underlying set consisting of one point, but retains a nonzero nilpotent class of $z$. This distinguishes an analytic space from only a zero set. The notion extends complex geometry to singularities and infinitesimal structure that ordinary manifold charts cannot describe.

Complex geometry studies spaces locally described by complex coordinates with holomorphic transition maps. A complex $n$-dimensional complex manifold has real dimension $2n$, but its structure is more restrictive than a smooth atlas: the transition maps satisfy the Cauchy-Riemann equations. The complex projective line is a compact example, described by two complex charts related by $w=1/z$. Hermitian metrics and Kähler manifolds connect this analytic structure to Riemannian geometry, while complex algebraic varieties connect it to algebraic geometry.

A complex Lie group is a complex manifold whose multiplication and inversion are holomorphic maps between complex manifolds. Its tangent space at the identity is a complex Lie algebra; the Adjoint representation of a Lie group and the Lie exponential map are holomorphic. Compatibility with a complex manifold structure is essential: an arbitrary even-dimensional real Lie group need not have holomorphic multiplication.

A complex surface is a complex manifold of complex dimension two, hence real dimension four. Compact complex surfaces need not admit a Kähler metric; a Hopf surface is an example. Every holomorphic one-form on a compact complex surface is closed, and the Betti and Hodge bounds for compact complex surfaces still apply without Kähler assumptions.

The conjugate Dolbeault operator is the type-$(1,0)$ component of the exterior derivative: $\partial :{\mathcal{A}}^{p,q}\to {\mathcal{A}}^{p+1,q}$. Complex conjugation interchanges $\partial$ and $\bar{\partial }$. On a complex manifold, $d=\partial +\bar{\partial }$, ${\partial }^2={\bar{\partial }}^2=0$, and $\partial \bar{\partial }+\bar{\partial }\partial =0$.

The conjugate bundle has the same underlying real bundle and the opposite complex scalar action. Its transitions are the complex conjugates of the original transitions. For a holomorphic bundle on $X$, these transitions are smooth and generally antiholomorphic on $X$; it is not automatically a holomorphic bundle there. It is naturally holomorphic over the conjugate complex manifold instead.

The complexified cotangent bundle of a complex manifold splits asA differential form has type $(p,q)$ when it is locally a sum of terms containing $p$ factors $d{z}^j$ and $q$ factors $d{\bar{z}}^k$.

A divisor on a complex manifold is a locally finite formal sumwhere the $Y$ are irreducible complex analytic hypersurfaces and $n_Y\in \mathbb{Z}$. Products of powers of local defining functions give local meromorphic equations $f_{\alpha }$ for $D$ whose ratios $f_{\alpha }/f_{\beta }$ are nowhere-vanishing holomorphic functions.

On a sufficiently small polydisc in a complex manifold, every $\bar{\partial }$-closed $(p,q)$-form with $q>0$ is locally $\bar{\partial }$-exact. A coefficientwise one-variable integral operator and induction over the coordinates prove the result.

For the sheaf ${\Omega }^p$ of holomorphic $p$-forms on a complex manifold, the Dolbeault-Poincaré lemma makesa resolution by fine sheaves. Taking global sections therefore gives the natural isomorphism

Let a compact Lie group $G$ act on a closed manifold and on the vector bundles of a $G$-equivariant elliptic differential operator $D$. Its finite-dimensional kernel and cokernel define the virtual representationAn equivariant index theorem computes this character of a representation from local principal symbol and curvature data on the fixed-point set of $g$. At the identity element it recovers the ordinary Atiyah-Singer index theorem; for isolated fixed points the fixed-set integral becomes a finite sum.

For example, the holomorphic fixed-point formula for the trivial line bundle on a compact complex manifold giveswhen the fixed points are nondegenerate. On the Riemann sphere, $f(z)=az$ with $|a|=1$ and $a\ne 1$ has fixed points zero and infinity, with tangent multipliers $a$ and $a^{-1}$. Their contributions areThe global answer is also one: only the constant holomorphic functions contribute. Local fixed-point terms can thus be nonintegral complex numbers even though their sum is the character of an actual virtual representation.^[ref]

A Hermitian manifold is a complex manifold whose tangent bundle carries a smoothly varying positive-definite Hermitian form. Its real part is a Riemannian metric invariant under the complex structure $J$.

A holomorphic coordinate chart on a complex manifold is a biholomorphism from an open subset of the manifold to an open subset of ${\mathbb{C}}^n$. Its component functions are holomorphic coordinates.