Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 (Complex Manifolds)

A complex manifold of complex dimension $n$ is a Hausdorff, second-countable topological manifold with charts to open subsets of ${\mathbb{C}}^n$ whose transition maps are biholomorphic. A rank-$r$ holomorphic vector bundle is a complex vector bundle with local trivializations $E{|}_U\cong U\times {\mathbb{C}}^r$ whose transition matrices are holomorphic maps to ${\mathrm{GL}}_r(\mathbb{C})$.

In a holomorphic coordinate chart, $T^{1,0}X$ has frame $\partial /\partial z^1,\dots ,\partial /\partial z^n$. Under a holomorphic coordinate change $w=w(z)$, the chain rule givesThe Jacobian is an invertible matrix of holomorphic functions. These frames therefore make $T^{1,0}X$ the holomorphic tangent bundle.

On each member of a cover, choose a reduced local defining function of a complex analytic hypersurface $f_i$ for $Y$. On overlaps $f_i=g_{ij}{f}_j$ for $g_{ij}\in {\mathcal{O}}_X^{\ast }$. Since $f_j{|}_Y=0$,Each $d{f}_i$ annihilates $TY$ and is nonzero in the normal direction because $Y$ is smooth. If $e_i=g_{ij}^{-1}{e}_j$ are the frames of the holomorphic line bundle associated to a divisor $\mathcal{O}(Y)$, thenis a well-defined nowhere-zero holomorphic map of line bundles. Thus the normal bundle of a smooth analytic hypersurface is

The tangent-normal short exact sequencegives $\det (TX{|}_Y)\cong \det (TY)\otimes N_{Y/X}$. Dualizing and using the canonical bundle,Since $N_{Y/X}\cong \mathcal{O}(Y){|}_Y$, rearrangement proves the Adjunction formula

For an ordered open cover $\mathcal{U}=(U_i)$ and a sheaf of sets $\mathcal{F}$, the Čech cochain group isIts Čech coboundary isTerms in ${\delta }^2$ cancel in pairs. Hence

Take pairwise disjoint coordinate discs $U_m$ around $x_m$ and put $U_0=S\setminus \{x_1,\dots ,x_N\}$. A prescribed principal part of a meromorphic function $P_m$ is holomorphic on $U_0\cap U_m$. Definewith all other components zero. No three distinct cover members meet, so the Čech cocycle condition is automatic. Thus $(P_{ij})$ is a Čech one-cocycle for $\mathcal{O}$.

If ${\check{H}}^1(S,\mathcal{O})=0$, that cocycle is a Čech coboundary: there are $f_i\in \mathcal{O}(U_i)$ withSet $F=-f_0$ on $U_0$ and $F=P_m-f_m$ on $U_m$. These formulas agree on overlaps, so the sheaf gluing axiom gives a global meromorphic function. Moreover $F-P_m=-f_m$ is holomorphic near $x_m$. Thus $F$ solves the Mittag-Leffler problem on a Riemann surface with precisely the prescribed principal parts.

Let ${\Omega }^p$ be the sheaf of holomorphic $p$-forms and ${\mathcal{A}}^{p,q}$ the sheaf of smooth $(p,q)$-forms. The Dolbeault-Poincaré lemma makesan exact resolution by fine sheaves, which are acyclic for global sections. Its global cochain complex computes Dolbeault cohomology, proving the Dolbeault theorem

Let ${\mathcal{K}}_X^{\ast }$ be the sheaf of nonzero meromorphic functions. A local equation for a divisor on a complex manifold is determined modulo a nowhere-zero holomorphic factor and hence defines a section of the divisor sheaf on a complex manifold ${\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$. Conversely, local representatives of such a section have quotients in ${\mathcal{O}}_X^{\ast }$, so their zero and pole orders agree on overlaps and define a divisor. The constructions are inverse:

The exact sequenceinduces a long exact sequence in sheaf cohomology. Under $H^0({\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast })=\mathrm{Div}(X)$ and $H^1({\mathcal{O}}_X^{\ast })=\mathrm{Pic}(X)$, its connecting homomorphism is the divisor-to-Picard map $D\mapsto \mathcal{O}(D)$. Exactness identifies its kernel with divisors of global nonzero meromorphic functions, namely principal divisors:

A holomorphic line bundle is ample when some positive tensor power is very ample, so its sections define an embedding into Complex projective space. It is a positive holomorphic line bundle when it has a Hermitian metric with positive Chern curvature $i{F}_{\nabla }$. The Kodaira embedding theorem says that a compact complex manifold carrying a positive holomorphic line bundle is projective; sufficiently high tensor powers give a holomorphic embedding into projective space.

Choose a very ample line bundle $A$ on $X$. By the theorem that a high ample twist is very ample, for sufficiently large $m$ bothare very ample. The dual bundle cancels the power of $A$, giving

Write $L\cong H_1\otimes H_2^{\ast }$ as in part (d). Each very ample line bundle is the pullback of $\mathcal{O}(1)$ under a projective embedding. A hyperplane section therefore supplies an effective divisor on a complex manifold $D_i$ with $H_i\cong \mathcal{O}(D_i)$. HenceEvery Picard group class is therefore in the image of the divisor-to-Picard map:

A connection on a vector bundle is a complex-linear map $D:{\Omega }^0(E)\to {\Omega }^1(E)$ satisfying the Leibniz rule $D(fs)=df\otimes s+fDs$. It is compatible with the Hermitian metric on a holomorphic vector bundle $h$ whenand compatible with the holomorphic structure when its type-$(0,1)$ part is the Dolbeault partial connection:

In a unitary frame the metric matrix is $I$. If $A$ is the connection one-form, metric compatibility gives $0=dI=A^{\ast }+A$, sothe matrix is Skew-Hermitian. In a holomorphic local frame, every frame vector is annihilated by ${\bar{\partial }}_E$, and compatibility with the holomorphic structure gives

In a holomorphic local frame $e$, let $H=(h(e_i,e_j))$ and write $De=eA$. Holomorphic compatibility forces $A^{0,1}=0$, while metric compatibility forces the local formula for the Chern connection on a vector bundleThis proves uniqueness. The formula transforms by the connection one-form law under a holomorphic frame change, so the local definitions glue and satisfy both conditions. This proves existence of the unique Chern connection.

Let $D_1$ and $D_2$ each be a connection on a vector bundle. Their difference is tensorial, so $D_1=D_2+a$ with $a\in {\Omega }^1(\mathrm{End}E)$. Extend $D_2$ to the endomorphism bundle connection. Expanding $(D_2+a{)}^2$ with the supplied graded Leibniz rule gives the curvature difference formula

Because the two Chern connections have the same $(0,1)$ part, $a=D_1-D_2$ has type $(1,0)$. The curvature difference formula has only $(2,0)$ and $(1,1)$ parts. Both Chern curvatures have type $(1,1)$, so the total $(2,0)$ part vanishes. The remaining part is obtained from $D_2^{0,1}={\bar{\partial }}_{\mathrm{End}E}$, proving the curvature difference of two Chern connections

For the formal adjoints $d^{\ast }$, ${\partial }^{\ast }$ and ${\bar{\partial }}^{\ast }$ defined by the Kähler metric,andThese are respectively the Hodge Laplacian and the two Dolbeault Laplacians.

For the Lefschetz operator of a Kähler manifold and its adjoint $\Lambda$, the Kähler identities includeThey make the mixed anticommutators in the expansion of ${\Delta }_d$ vanish and imply ${\Delta }_{\partial }={\Delta }_{\bar{\partial }}$. Expanding $d=\partial +\bar{\partial }$ therefore proves the Kähler Laplacian identityThus ${\Delta }_d\alpha =0$ exactly when ${\Delta }_{\bar{\partial }}\alpha =0$.

The Hodge decomposition theorem for compact Kähler manifolds gives a unique harmonic differential form representative of every complex de Rham cohomology class, with a decomposition into harmonic pure-type components. ConsequentlyEquivalently, the Dolbeault Hodge decomposition on a compact Hermitian manifold is

If $\alpha =\partial \bar{\partial }\beta$ and $h\in {\mathcal{H}}^{p,q}$, the Kähler Laplacian identity makes $h$ both $\partial$- and $\bar{\partial }$-harmonic. Hence ${\partial }^{\ast }h={\bar{\partial }}^{\ast }h=0$, and integration by parts gives $⟨\alpha ,h⟩=0$.

Conversely, let $d\alpha =0$ and $\alpha \perp {\mathcal{H}}^{p,q}$. Pure type gives $\partial \alpha =\bar{\partial }\alpha =0$. Dolbeault Hodge decomposition removes the harmonic and coexact components, soNow $\bar{\partial }(\partial \theta )=-\partial \alpha =0$, and the Kähler anticommutation identity gives ${\bar{\partial }}^{\ast }(\partial \theta )=-\partial ({\bar{\partial }}^{\ast }\theta )=0$. Thus $\partial \theta$ is $\bar{\partial }$-harmonic and therefore $\partial$-harmonic; being $\partial$-exact, it vanishes. Moreover $\theta \in \mathrm{im}{\bar{\partial }}^{\ast }$ is orthogonal to the common $\bar{\partial }$- and $\partial$-harmonic space. Its $\partial$-Hodge decomposition therefore gives $\theta =\partial \varphi$. HenceTaking $\beta =-\varphi$ proves the harmonic orthogonality criterion for ddbar exactness