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The holomorphic tangent bundle of a complex manifold $X$ is locally spanned in holomorphic coordinates by $\partial /\partial z^1,\dots ,\partial /\partial z^n$. Under a holomorphic coordinate change $w=w(z)$, the chain rule giveswhose coefficient matrix is holomorphic; these frames therefore define a holomorphic vector bundle.

For a smooth form $\theta =gd{z}_1\wedge \cdots \wedge d{z}_n$ of type $(n,0)$, the complex orientation gives $i^{n^2}\theta \wedge \overline{\theta }=2^n|g{|}^{2}d{x}_1\wedge d{y}_1\wedge \cdots \wedge d{x}_n\wedge d{y}_n$. Consequently its integral on a compact complex manifold is positive unless $\theta$ vanishes identically. Applying Stokes theorem to $d(\psi \wedge \overline{d\psi })$ proves that a holomorphic differential form of degree $n-1$ on a compact complex $n$-manifold is closed, without requiring a Kähler metric.

An analytic subvariety is a closed subset locally defined by finitely many holomorphic functions. It is irreducible if it is not the union of two proper closed analytic subsets. An irreducible analytic subvariety of complex codimension $k$ in an $n$-dimensional complex manifold has pure complex dimension $n-k$. Global irreducibility does not forbid multiple local branches at a singular point.

The Kodaira embedding theorem says that a compact complex manifold carrying a positive holomorphic line bundle is projective: sufficiently large tensor powers of the bundle have global holomorphic sections defining a holomorphic embedding into Complex projective space. Conversely, the restriction of $\mathcal{O}(1)$ gives a positive line bundle on every projective complex manifold.

A closed analytic subset of a complex manifold is nonsingular when every point has a neighbourhood on which the subset is the zero set of a holomorphic map of full rank along that zero set. The holomorphic implicit function theorem then makes it a local coordinate subspace. This includes a closed complex submanifold, whereas an arbitrary open piece of a complex submanifold need not be globally closed in its ambient manifold.

On a compact complex manifold of complex dimension $n$, a holomorphic p-form on a complex manifold of degree $n$ is $d$-closed. The form $i^{n^2}\alpha \wedge \bar{\alpha }$ is a nonnegative volume density, positive wherever $\alpha \ne 0$: in holomorphic coordinates it equals $2^n|f{|}^{2}d{x}_1\wedge d{y}_1\wedge \cdots \wedge d{x}_n\wedge d{y}_n$ for $\alpha =fd{z}_1\wedge \cdots \wedge d{z}_n$. If $\alpha =d\beta$, its integral would vanish by the Stokes theorem, contradicting this positivity. No Kähler metric is needed.

At a generic smooth point of an irreducible complex analytic hypersurface $Y$, a nonzero meromorphic function has the form $f=t^{m}u$, where $t$ is a reduced transverse local equation and $u$ is a holomorphic unit. The integer $m$ is its order along $Y$: positive for a zero and negative for a pole. The unique factorization domain property of the local ring of a complex manifold makes this independent of the equation and propagates it along $Y$. These orders are the coefficients of the principal divisor on a complex manifold of $f$.

An almost complex structure is a smooth endomorphism $J:TM\to TM$ with $J^2=-I$; in particular the real dimension is even. Complexifying the tangent bundle splits it into the $i$ and $-i$ eigensubbundlesinterchanged by conjugation. On a complex manifold, these are locally spanned by ${\partial }_{z^j}$ and ${\partial }_{{\bar{z}}^j}$. An integrable almost complex structure is one arising from a holomorphic coordinate atlas. Its $T^{0,1}$ sections are closed under the Lie bracket. The obstruction is the Nijenhuis tensorExpanding this expression in the two eigentypes shows that $N_J=0$ is equivalent to involutivity of $T^{0,1}$. Holomorphic coordinates make this condition necessary; the Newlander-Nirenberg theorem gives its sufficiency for smooth $J$. Integrability also gives $d=\partial +\bar{\partial }$, with ${\partial }^2={\bar{\partial }}^2=0$ and $\partial \bar{\partial }+\bar{\partial }\partial =0$. The Dolbeault operator on $T^{1,0}M=T_M^{\prime }$ defines its holomorphic vector bundle structure.

A real connection on a vector bundle $\nabla$ on $TM$ preserves $J$ when $\nabla J=0$. Its complexification then preserves both eigensubbundles. Restricting to $T_M^{\prime }$ gives a complex connection $D$. Conversely any complex connection on $T_M^{\prime }$ gives such a real connection by adjoining its conjugate on $T^{0,1}$ and restricting to the conjugation-fixed real tangent bundle. Equivalently, transfer $D$ through the real isomorphism $v\mapsto (v-iJv)/2$. These operations are inverse. Preserving $J$ alone does not impose the holomorphic compatibility condition $D^{0,1}={\bar{\partial }}_{T^{\prime }}$.

For integrable $J$, the latter condition has a precise real-connection interpretation: the mixed-type torsion tensor vanishes. In holomorphic coordinates it says $D_{{\partial }_{{\bar{z}}^i}}{\partial }_{z^j}=0$. Reality also gives ${\nabla }_{{\partial }_{z^j}}{\partial }_{{\bar{z}}^i}=0$. Since these coordinate fields commute, their torsion is zero. Conversely a $J$-preserving real connection with zero mixed torsion has these two derivatives equal; they belong to opposite eigentypes, so both vanish, proving $D^{0,1}={\bar{\partial }}_{T^{\prime }}$. This is the holomorphic tangent connection and mixed torsion criterion.

Full torsion-freeness imposes the additional symmetry ${\Gamma }_{ij}^k={\Gamma }_{ji}^k$ in $D_{{\partial }_{z^i}}{\partial }_{z^j}={\Gamma }_{ij}^k{\partial }_{z^k}$, and its conjugate. Such a $J$-preserving torsion-free connection exists on any complex manifold: local holomorphic-coordinate flat connections have these properties, and a real partition of unity combines them globally. Preservation of $J$ and zero torsion persist because both conditions are affine in the connection. Conversely if $J$ is merely almost complex, a torsion-free connection preserving it forces $N_J=0$: replace brackets by ${\nabla }_{X}Y-{\nabla }_{Y}X$ in the displayed tensor and use $\nabla J=0$; all terms cancel. Thus this connection condition detects integrability, without imposing any metric condition.

A Hermitian metric is a real Riemannian metric $g$ with $g(JX,JY)=g(X,Y)$. It induces a Hermitian metric $h$ on $T_M^{\prime }$ and the fundamental two-form $\omega (X,Y)=g(JX,Y)$. The unique Chern connection satisfies both $D^{0,1}={\bar{\partial }}_{T^{\prime }}$ and metric compatibility. In a holomorphic frame with metric matrix $H$ and coefficient-column convention, its matrix is $H^{-1}\partial H$. It need not be torsion-free when transferred to $TM$.

In holomorphic coordinates its coefficients are ${\Gamma }_{ij}^k=h^{k\bar{l}}{\partial }_i{h}_{j\bar{l}}$. Their symmetry in $i,j$ is equivalent to ${\partial }_i{h}_{j\bar{l}}={\partial }_j{h}_{i\bar{l}}$, which says $\partial \omega =0$. The conjugate gives $\bar{\partial }\omega =0$. The result that a torsion-free Chern tangent connection characterizes a Kähler metric follows: the transferred Chern connection is torsion-free exactly when $\boxed{d\omega =0}$, the Kähler metric condition. In that case it equals the Levi-Civita connection, by uniqueness of the metric-compatible torsion-free connection. Equivalently a Hermitian metric is Kähler exactly when its Levi-Civita connection preserves $J$. In the reverse direction $\nabla J=0$ implies $\nabla \omega =0$, and zero torsion gives $d\omega =0$.

This explains the role of Kähler metrics: they make the real Riemannian and holomorphic connection theories coincide. A general complex manifold has torsion-free $J$-preserving connections and has Hermitian metrics, but need not have a connection satisfying both requirements simultaneously. On a Kähler manifold, the Kähler identities then link the Dolbeault Laplacian and Hodge Laplacian, giving ${\Delta }_d=2{\Delta }_{\bar{\partial }}=2{\Delta }_{\partial }$ and the resulting harmonic Hodge decomposition theorem for compact Kähler manifolds.

For a symmetric matrix $B$ with $Y=\mathrm{Im}B$ positive definite, the Riemann theta function isOn any compact set of $z$, the absolute value of the summand is bounded by $\exp (-c|n{|}^2+C|n|)$ for some $c>0$, so the series and all its derivatives converge uniformly there. It is therefore entire. Its constant Fourier coefficient in $\mathrm{Re}z$ is $1$, so it is not identically zero. Reindexing $n$ gives its parity and automorphy laws:Although the theta function is not itself a function on $J={\mathbb{C}}^g/\Lambda$, these factors define a holomorphic line bundle on $J$, and its zeros define a divisor $\Theta$ there.

For the period matrix of $X$, the Riemann vanishing theorem asserts that a fixed class $\kappa \in J$, the vector of Riemann constants in the convention used here, satisfiesWe prove both its support statement and its reduced divisor structure. This will make the multiplicity calculation in Solution 5 independent of any assumed multiplicity theorem.

The essential contour calculation is the theta pullback zero-divisor identity. Put $f_z(p)=\theta (u(p)-z,B)$, regarding this as a local expression for a section on $X$. If it is not identically zero, let $D_z$ be its zero divisor. We claimUse the cut polygon from Solution 1 and choose cuts avoiding the zeros. There ${\alpha }_z=d_p\log f_z$ is a meromorphic differential whose residues are the zero multiplicities. Under continuation by an $a_j$ period, $u$ changes by the integer vector $e_j$, so ${\alpha }_z$ is unchanged. Under a $b_j$ period, $u$ changes by $B{e}_j$, so automorphy gives ${\alpha }_z\mapsto {\alpha }_z-2\pi i{\omega }_j$. Pairing opposite boundary sides therefore givesFor example, the two $a_j$ sides, traversed with opposite orientations and separated by a $b_j$ translation, contribute ${\int }_{a_j}({\alpha }_z-({\alpha }_z-2\pi i{\omega }_j))$. This fixes the sign of the count.

For the first moment of the zeros, the residue theorem on the polygon gives, component by component,This is interpreted modulo the period lattice, with the lifts to the polygon fixed. Vary $z$ without letting zeros cross the cuts, and put $h=\delta \log f_z$. The coordinate $u_k$ does not vary, so integration by parts givesThere is no endpoint term: the closed polygon boundary returns both the Abel integral and $h$ to their starting values. Across an $a_j$ translation $h$ is unchanged, while across a $b_j$ translation it changes by $2\pi i\delta z_j$. Pairing the boundary edges consequently gives ${\int }_{\partial P}{\omega }_{k}h=-2\pi i{\sum }_j\delta z_j{\int }_{a_j}{\omega }_k=-2\pi i\delta z_k$. Thus $\delta u_g(D_z)=\delta z$, and $u_g(D_z)-z$ is constant. The calculation uses weighted zeros, so their collisions cause no problem. The parameters where $f_z$ is identically zero form a proper analytic subset: they are given by the vanishing of all its local Taylor coefficients, and parameters outside $\Theta$ are not in it. Its complement in the connected complex manifold $J$ is connected, so the constant is one class $\kappa$. This proves the pullback identity globally, with continuation of the zero divisors where necessary.

We must account for the parameters with identically zero pullback, rather than discard them. No irreducible component $C$ of $\Theta$ can consist entirely of such parameters. Otherwise parity would imply $C-u(p)\subset \Theta$ for every $p\in X$. The image of $C\times X$ under subtraction is irreducible, is contained in $\Theta$, and contains $C$ when $p=p_0$. Since $C$ is an irreducible component of the hypersurface, that image must be $C$. Hence $C-u(p)=C$ for all $p$. The surjectivity of the large-degree Abelian sum map, proved in Solution 3, says that these translations generate all of $J$. This would make the nonempty proper subset $C$ invariant under every translation, which is impossible.

A generic point $z$ of each component of $\Theta$ therefore has nonzero pullback. Since $\theta (-z)=\theta (z)=0$, its zero divisor contains $p_0$. Write $D_z=p_0+D^{\prime }$, with $D^{\prime }$ effective of degree $g-1$. The pullback identity gives $u_{g-1}(D^{\prime })=z+\kappa$. Each component of $\Theta$ is consequently contained in $W_{g-1}-\kappa$. By Solution 3 the latter is irreducible of dimension $g-1$, so any such component equals it. The zero locus is nonempty for $g>0$: any parameter with nonzero pullback has $g$ pullback zeros. We have proved equality of supports, in both directions.

Finally the automorphy factors make the First Chern class of the theta line bundle primitive. On the two-torus spanned by $a_i,b_j$, the logarithm of the $b_j$ factor changes by $-2\pi i{\delta }_{ij}$ after the $a_i$ translation; equivalently the transition for local frames has opposite winding. Thus $c_1(a_i,b_j)={\delta }_{ij}$, while $c_1(a_i,a_j)=c_1(b_i,b_j)=0$. In particular some integral pairing equals $1$. As its support is the single irreducible hypersurface $W_{g-1}-\kappa$, the zero divisor is $m(W_{g-1}-\kappa )$ for a positive integer $m$. Its integral First Chern class would then be divisible by $m$. Primitivity forces $m=1$, proving the displayed equality as reduced divisors.

The sign convention is $\kappa =u_g(D_z)-z$, so the translate is $W_{g-1}-\kappa$; a convention naming $-\kappa$ the Riemann constant reverses that displayed sign. For $g=1$, $\kappa =(1+B)/2$ modulo periods and the zero locus is the corresponding single point. For $g=0$, the empty-lattice theta series is $1$, its zero locus is empty, and the degree-$-1$ effective locus is empty; no negative symmetric product is required.

A complex analytic Lie group is a complex manifold whose multiplication and inversion are holomorphic maps between complex manifolds. Its identity tangent space is a complex Lie algebra, and the Adjoint representation of a Lie groupis holomorphic: differentiating the holomorphic conjugation map with respect to its second variable gives holomorphically varying matrix entries.

Every scalar holomorphic function on a compact connected complex manifold is constant. Indeed its modulus attains a maximum; the maximum modulus principle, applied in complex coordinate charts and along complex lines, makes it locally constant there, and connectedness propagates that constancy. Apply this to each entry of $\mathrm{Ad}$. Since ${\mathrm{Ad}}_e=I$, it follows that ${\mathrm{Ad}}_g=I$ for every $g$. Differentiation givesThus $\mathfrak{g}$ is abelian, and the connected group $G$ is abelian by the identity-neighbourhood argument in part (i). This proves that compact connected complex Lie groups are complex tori.

If $n={\dim }_{\mathbb{C}}G$, identify $\mathfrak{g}$ with ${\mathbb{C}}^n$. The Lie exponential map is holomorphic, additive, onto, and a local biholomorphism, by the holomorphic left-invariant differential equation and its identity derivative. Its kernel $B$ is a discrete subgroup of the additive space ${\mathbb{C}}^n$. The induced map yieldsas complex Lie groups, not merely as real groups. Compactness further forces $B$ to be a full-rank real lattice. If its real span $W$ were proper, projection would give a surjective continuous map ${\mathbb{C}}^n/B\to {\mathbb{R}}^{2n}/W$, contradicting compactness. The discrete-subgroup argument in part (i) therefore gives $B\cong {\mathbb{Z}}^{2n}$. The quotient is a complex torus; it need not split into one-dimensional complex tori.

A Riemann surface is a Hausdorff, second-countable, one-dimensional complex manifold, with charts to open subsets of $\mathbb{C}$ whose transition maps are holomorphic. Connectedness is often included in the convention. A map between Riemann surfaces is holomorphic when its expression in every compatible pair of charts is holomorphic. It is biholomorphic when it is bijective and both it and its inverse are holomorphic.

On a complex manifold, holomorphic coordinates give the decomposition $d=\partial +\bar{\partial }$, withFor $\alpha ={\sum }_{I,J}{a}_{I\bar{J}}d{z}^I\wedge d{\bar{z}}^J$, the conjugate Dolbeault operator $\partial$ and the Dolbeault operator $\bar{\partial }$ are explicitlyThese are the type components of the intrinsic exterior derivative, so are independent of coordinates. The identity $d^2=0$, separated by type, gives ${\partial }^2={\bar{\partial }}^2=0$ and $\partial \bar{\partial }+\bar{\partial }\partial =0$. A holomorphic p-form on a complex manifold is a $(p,0)$-form whose coefficients are holomorphic functions; equivalently it is a $(p,0)$-form killed by the Dolbeault operator.

A rank-$r$ holomorphic vector bundle has local trivializations $E{|}_{U_i}\cong U_i\times {\mathbb{C}}^r$ whose changes are $(x,v)\mapsto (x,g_{ij}(x)v)$, with $g_{ij}:U_i\cap U_j\to {\mathrm{GL}}_r(\mathbb{C})$ holomorphic. These transition functions of a vector bundle satisfy $g_{ii}=I$ and $g_{ij}{g}_{jk}=g_{ik}$. Equivalently, for local frames we use the convention $e_j=e_i{g}_{ij}$. A holomorphic section is a section having holomorphic coordinate functions in every such holomorphic local frame; the condition is unchanged by the holomorphic invertible transitions.

The holomorphic cotangent bundle has local frames $d{z}^1,\dots ,d{z}^n$. Under another holomorphic coordinate system $w$, the chain rule givesThe Jacobian matrix is holomorphic and invertible, with holomorphic inverse supplied by the inverse coordinate change. These are precisely holomorphic bundle transitions. A section written as ${\sum }_b{f}_b(z)d{z}^b$ is holomorphic exactly when every $f_b$ is holomorphic, sothe space of holomorphic one-forms. In particular, being a holomorphic section is not the same as being a $d$-closed one-form on an arbitrary complex manifold.

The complex tautological line bundle isOn $U_i=\{Z_i\ne 0\}$ use the nowhere-zero frame $s_i([Z])=Z/Z_i$. Its components are holomorphic affine coordinates and the constant one in position $i$. On overlapsso the frame changes are nowhere-zero holomorphic functions. This proves that $\mathcal{O}(-1)$ is a holomorphic line bundle, and its inclusion into the trivial bundle is a holomorphic bundle map.

A global holomorphic section composed with that inclusion has $n+1$ component holomorphic functions on compact connected Complex projective space. Each is constant: its absolute value attains a maximum; in a coordinate ball the one-variable maximum modulus principle along complex lines makes it locally constant, and connectedness propagates that constant globally. Thus the section is a fixed vector $v\in {\mathbb{C}}^{n+1}$ lying in every fibre, namely every line $\mathbb{C}Z$. For $n\ge 1$ the intersection of all these lines is zero. HenceThe positive-dimension convention is necessary: ${\mathbb{CP}}^0$ is one point and this section space is $\mathbb{C}$.

Choose holomorphic coordinates centered at the point $p$ and a polydisc $U\subset {\mathbb{C}}^n$ about the origin. The local blowup of a complex manifold at a point is the incidence manifoldAway from $z=0$ the direction is forced to be $[\ell ]=[z]$, giving a biholomorphism with $U\setminus \{0\}$. Glue this punctured part to $X\setminus \{p\}$. The charts below make the result a complex manifold. The projection is proper near $p$, since it is the restriction of projection from a closed subset of $U\times {\mathbb{CP}}^{n-1}$; globally the modification is proper and is an isomorphism off $p$. Its exceptional divisor is

On ${\ell }_i\ne 0$, let $t=z_i$ and $u_j={\ell }_j/{\ell }_i$ for $j\ne i$. ThenThese are coordinates on the open domain where $t$ and all $t{u}_j$ lie in the original polydisc; at $t=0$ all finite $u_j$ are allowed. Their inverse reads the direction ratios and $z_i$ from the incidence point. In this chart $E$ is the nonsingular hypersurface $t=0$. On overlap with chart $k\ne i$, the holomorphic coordinate changes areThey are holomorphic with holomorphic inverses where $u_k\ne 0$, and the $n$ charts cover every point of $E$. Intrinsically, changing the original coordinates by $F$ lifts via $[\ell ]\mapsto [A(z)\ell ]$, where $F(z)=A(z)z$, $A(0)=d{F}_0$, and $A$ is holomorphic and invertible near zero. One can take $A(z)={\int }_0^{1}d{F}_{sz}ds$ on a sufficiently small polydisc. Away from the origin the lift is uniquely forced; this proves coordinate-independence of the gluing near $E$.

The canonical bundle is $K_X={\Lambda }^n{\Omega }_X^1$. In the $i$th blowup chart the top-form Jacobian iswhere the $u_j$ occur in increasing order. Every term with a second $dt$ vanishes in the wedge product of differential forms. Thus the differential gives a map ${\sigma }^{\ast }{K}_X\to K_{\tilde{X}}$ vanishing to order $n-1$ along $E$ and nowhere else. To express this through the supplied meromorphic-section hypothesis, take a nonzero meromorphic section $s$ of $K_X$ and put $D=\mathrm{div}(s)$. Locally $s=f(z)d{z}_1\wedge \cdots \wedge d{z}_n$, soHere ${\sigma }^{\ast }D$ is the total divisor pullback, including any additional exceptional multiplicity from $f$; it is not merely the strict transform. A line bundle with a nonzero meromorphic section is isomorphic to the line bundle of that section's divisor, as its local coefficient ratios are its frame transitions. Pullback of divisor line bundles and additivity of analytic divisor line bundles therefore give the canonical bundle formula for a point blowup:For $n=1$ the blowup is already an isomorphism, consistently with the zero exponent.

For the projective-plane construction, an invertible linear change of homogeneous coordinates puts $x=[1:0:0]$. Lines through $x$ are parametrized by $[a:b]\in {\mathbb{CP}}^1$, with equation $b{Z}_1-a{Z}_2=0$. Thus the required map on the punctured plane isIt is well-defined under rescaling and holomorphic on the covering sets $Z_1\ne 0$ and $Z_2\ne 0$, where its target coordinates are $Z_2/Z_1$ and $Z_1/Z_2$. These sets cover the complement of $x$.

The graph closure isOver the complement of $x$ it is the graph of $f$. In the affine chart $Z_0\ne 0$, its equation is exactly the two-dimensional local incidence model for the blowup of a complex manifold at a point, so its first projection identifies it with the required blowup globally. The second projection is a holomorphic map $\tilde{f}:S\to {\mathbb{CP}}^1$ extending $f\circ \sigma$. Explicitly, the two blowup charts giveThese formulas agree when $v=1/u$ and remain holomorphic at $t=0$. Hence the pencil extends over the exceptional divisor, where it records the projective tangent direction. This is the projective pencil resolved by a point blowup.

Let $n={\dim }_{\mathbb{C}}X$, and use the complex orientation. Write $g$ for the real Riemannian metric underlying the Hermitian metric. Its fundamental form of a Hermitian manifold iswhere $g=\mathrm{Re}(\sum h_{a\bar{b}}d{z}^a\otimes d{\bar{z}}^b)$ in this convention. It is real and of type $(1,1)$, since $g(Ju,Jv)=g(u,v)$. Choose an adapted real orthonormal frame $E_j,F_j=J{E}_j$ and its dual coframe $e^j,f^j$. Then $\omega ={\sum }_j{e}^j\wedge f^j$, giving the Riemannian volume form

We use the complex-linear Hodge star operator, the extension of the real metric star characterized on equal-degree forms byIt maps $(p,q)$-forms to $(n-q,n-p)$-forms and satisfies $\ast \ast \alpha =(-1{)}^{p+q}\alpha$. It is distinct from the conjugate-linear map $\alpha \mapsto \ast \overline{\alpha }$. In the adapted coframe the star of $e^j\wedge f^j$ is the wedge product of all the other two-dimensional factors, with positive sign. Summing proves the Hodge star of the fundamental Hermitian form formulaThis computation uses no closedness assumption on $\omega$.

Use the integrated Hermitian inner product $(\alpha ,\beta )={\int }_X\alpha \wedge \ast \overline{\beta }$. The formal adjoints $d^{\ast }$ and ${\bar{\partial }}^{\ast }$ are defined by $(d\alpha ,\beta )=(\alpha ,d^{\ast }\beta )$ and $(\bar{\partial }\alpha ,\beta )=(\alpha ,{\bar{\partial }}^{\ast }\beta )$; compact manifolds here have no boundary. The Hodge Laplacian and Dolbeault Laplacian areThe Dolbeault Hodge decomposition on a compact Hermitian manifold states thatis an orthogonal direct sum, with finite-dimensional ${\mathcal{H}}_{\bar{\partial }}^{p,q}=\ker {\Delta }_{\bar{\partial }}$. Each Dolbeault cohomology classhas a unique representative in ${\mathcal{H}}_{\bar{\partial }}^{p,q}$. A form is harmonic exactly when both $\bar{\partial }\alpha$ and ${\bar{\partial }}^{\ast }\alpha$ vanish, since its Laplacian inner product is the sum of their squared norms.

Now let $L\alpha =\omega \wedge \alpha$ be the Hermitian Lefschetz operator and let $\Lambda$ be its pointwise adjoint. For $\beta$ of total degree $r=p+q$ and $\alpha$ of degree $r-2$, the defining inner products giveThe wedge pairing is nondegenerate, so $\ast \overline{\Lambda \beta }=L\ast \overline{\beta }$. Apply another star to the degree-$(r-2)$ form and use that the star and $L$ commute with conjugation. This yieldsFor degrees below two both sides vanish. The argument also identifies its bidegree as $(-1,-1)$ and does not require the Kähler condition.

Assume now that $d\omega =0$, so the metric is a Kähler metric. Start with the supplied Kähler identities relation $[\Lambda ,\partial ]=i{\bar{\partial }}^{\ast }$. Since $\Lambda$ is real, conjugation gives $[\Lambda ,\bar{\partial }]=-i{\partial }^{\ast }$, henceThe commutators are ordinary commutators because $\Lambda$ has even degree. Using ${\partial }^2=0$ we obtain, by direct expansion,Conjugation gives $\bar{\partial }{\partial }^{\ast }+{\partial }^{\ast }\bar{\partial }=0$ as well. Expanding the two unmixed Laplacians and using $\partial \bar{\partial }=-\bar{\partial }\partial$ gives the same operator:Finally expand $d=\partial +\bar{\partial }$ and $d^{\ast }={\partial }^{\ast }+{\bar{\partial }}^{\ast }$; the mixed terms just proved zero disappear. This derives the Kähler Laplacian identity

The Dolbeault Laplacian preserves bidegree. Thus the complex $d$-harmonic forms of total degree $r$ split into their harmonic $(p,q)$ parts. The allowed real Hodge theorem, followed by complexification, givesThe real operator ${\Delta }_d$ commutes with conjugation, which exchanges $(p,q)$ and $(q,p)$. Therefore $h^{p,q}=h^{q,p}$. For odd $r$, no summand has $p=q$, so they pair off in equal dimensions. This proves that all odd Betti numbers are even.

For $1\le k\le n$, the Kähler form power ${\omega }^k$ is closed. If it were exact, say ${\omega }^k=d\eta$, thenby Stokes theorem. But ${\omega }^n=n!{\mathrm{vol}}_g$ has strictly positive integral on the nonempty compact manifold. Hence the powers of a compact Kähler form have nonzero cohomology classes, and

For an example take the Hopf surfaceThe dilation action is free and properly discontinuous: on a compact subset the radii are bounded above and away from zero, so only finitely many dilations can meet it. It is holomorphic, giving a complex manifold quotient; the annulus $1\le \Vert z\Vert \le 2$ covers the quotient, proving compactness. The mapis a diffeomorphism $H\cong S^3\times S^1$. The Künneth theorem gives $b^1(H)=1$. This violates the necessary odd-degree parity, so this compact complex surface admits no Kähler metric.

A Riemann surface is a connected one-complex-dimensional complex manifold: it has local complex coordinates with holomorphic transition maps. These coordinates induce an orientation and a conformal structure. Fix a closed oriented topological surface $S_g$ of genus $g$. A point of Teichmüller space is a marked Riemann surface $(X,f)$, where $f:S_g\to X$ is an orientation-preserving marking. Two pairs are equivalent if there is a biholomorphism $h:X\to X^{\prime }$ with $h\circ f$ isotopic to $f^{\prime }$. Thus the marking remembers how curves on the reference surface sit in $X$; forgetting it gives the moduli space of Riemann surfaces, obtained by quotienting by the mapping class group.

For $g\ge 2$, the uniformization theorem supplies a unique curvature-$-1$ hyperbolic metric in each conformal class. A pants decomposition has $3g-3$ cuffs, and their positive lengths together with their real twists give Fenchel–Nielsen coordinates. HenceFor completeness, ${\mathcal{T}}_0$ is a point, whereas ${\mathcal{T}}_1$ is the upper half-plane, parametrizing marked complex tori $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$ with $\mathrm{Im}\tau >0$.

The Wolpert generic spectral rigidity theorem says that, for fixed $g\ge 2$, there is a closed proper real-analytic exceptional subset of ${\mathcal{T}}_g$ such that a hyperbolic surface outside that subset is determined up to isometry by its unmarked length spectrum, equivalently its Laplacian spectrum. This is an unoriented isometry statement: reversing orientation cannot be detected by either spectrum.

The Huber spectral equivalence theorem says that two closed curvature-$-1$ hyperbolic surfaces have the same Laplacian spectrum, with multiplicities, if and only if they have the same unmarked length spectrum, with multiplicities. One may consistently count unoriented closed geodesics including their iterates; alternatively, the primitive multiset determines the multiset with iterates and conversely. This equivalence comes from the Selberg trace formula.

The Buser finite length spectrum theorem is the finite-determination ingredient: for each $g\ge 2$ and $\epsilon >0$ there is a finite number $L(g,\epsilon )$ such that, for any two closed hyperbolic surfaces of genus $g$ with hyperbolic systole at least $\epsilon$, equality of their length multisets through $L(g,\epsilon )$ implies equality of their entire length spectra. The finite-length statement is used in Buser's chapter on Wolpert's theorem. It determines the full spectrum, not necessarily the isometry class.

Here is a proof of the Buser finite length spectrum theorem through finite spectral matching by polynomial invariants. We use the following subsidiary results, stating the required content. First, Mumford's compactness theorem makes the part of the genus-$g$ moduli space of Riemann surfaces with hyperbolic systole at least $\epsilon$ compact. Second, marked hyperbolic holonomy representations of a closed oriented surface lift from ${\mathrm{PSL}}_2(\mathbb{R})$ to ${\mathrm{SL}}_2(\mathbb{R})$, and normalized representatives can be chosen continuously on sufficiently small parameter charts. Their hyperbolic translation length satisfiesThird, the length spectrum of a compact hyperbolic surface is locally finite, and metrics on compact marked parameter charts are uniformly bilipschitz equivalent. Finally, the Hilbert basis theorem implies that a descending sequence of algebraic sets in a finite-dimensional affine space stabilizes. These are compactness, holonomy, elementary geometric finiteness and algebraic finiteness results; none is the finite unmarked-spectrum assertion we are proving.

By compactness, finitely many relatively compact marked charts cover the thick moduli space of Riemann surfaces. Choose normalized lifted hyperbolic holonomy representations on these charts and take their compact closures. Their finite union $K$ is a compact set of determinant-one matrix tuples representing every surface in the thick part, possibly more than once. Each tuple records the images of the $2g$ generators of the fundamental group. Thus $K$ sits in a fixed finite-dimensional real matrix space. Every group word is a polynomial in these matrix entries: multiplication is polynomial, and for a determinant-one two-by-two matrix its inverse is its adjugate, also polynomial. Consequently each $q_{\gamma }$ is a polynomial on the ambient space. Squared traces remove the sign ambiguity of the lift, and $4{\cosh }^2(\ell /2)$ is strictly increasing for positive $\ell$.

Enumerate the unoriented nontrivial free homotopy classes as ${\gamma }_1,{\gamma }_2,\dots$, including the iterates, and write ${\ell }_j(x)$ for their geodesic lengths at $x\in K$. The uniform finiteness of short geodesic classes follows here directly: on each of the finitely many compact marked charts, choose a fixed reference metric $h$ and a uniform bilipschitz constant $C$. A class with ${\ell }_j(x)\le R$ somewhere on that chart has reference length at most $CR$. Only finitely many classes have that reference length, and a finite union of these finite collections is finite. In particular, for every $j$ the candidate setis finite. If two parameters in $K$ have equal lengths for classes $j,k$, then $k\in F_j$. These candidate sets depend on the fixed compact set $K$ and the source label $j$, not on the number of labels currently being matched.

For every $N$, define $E_N$ in the full ambient space of pairs of matrix tuples $(x,y)$ as follows. Require an injective choice $\sigma :\{1,\dots ,N\}\to \mathbb{N}$ with $\sigma (j)\in F_j$ andand independently require an injective choice $\tau :\{1,\dots ,N\}\to \mathbb{N}$ with $\tau (j)\in F_j$ andThere are finitely many possible pairs of choices $(\sigma ,\tau )$ because every $F_j$ is finite. For each choice these are finitely many polynomial equations. Therefore $E_N$ is a finite union of algebraic sets, hence an algebraic set. Restricting an injective matching to its first $N$ entries provesThe Hilbert basis theorem now supplies $N_0$ such that $E_N=E_{N_0}$ for all $N\ge N_0$. This step takes place in a fixed affine space and uses polynomial equations, so it gives stabilization of the actual sets, not just stabilization of germs at individual parameters.

For $(x,y)\in E_{N_0}\cap (K\times K)$, stabilization gives injective partial length matchings in both directions for every $N$. Fix any length $a>0$. Only finitely many labels have length $a$ at $x$ or at $y$, by local finiteness. Take $N$ large enough to contain every such label at $x$. Its injective matching into the labels at $y$ preserves $q_j$, and hence preserves length, so the multiplicity of $a$ at $x$ is at most its multiplicity at $y$. The reverse matching gives the opposite inequality. Thus the two multiplicities are equal for every $a$, proving equality of the entire unmarked length spectra. No assumed global matching or unjustified exchange of infinitely many neighborhoods is needed.

Finally choose a numberSuppose the two length multisets agree through $L$. For each of the first $N_0$ labels at $x$, select a distinct equal-length label at $y$, which is possible precisely because the truncated multisets agree with multiplicity. Each selected label belongs to the corresponding $F_j$. Do the same in the reverse direction. Then $(x,y)\in E_{N_0}$, and the previous paragraph proves equality of the full length spectra. Since $K$ was selected using only the fixed thick genus-$g$ parameter space, the resulting finite $L$ depends only on $g,\epsilon$. This proveswith each intersection interpreted as a multiset. By the Huber spectral equivalence theorem, the corresponding entire Laplacian spectra also agree. This is the finite matching reduction needed before the analytic and geometric rigidity steps in the Wolpert generic spectral rigidity theorem.

The complex structure on a complex manifold splits its complexified cotangent bundle into the $+i$ and $-i$ eigenspaces, locally spanned by $d{z}^j$ and $d{\bar{z}}^j$. A differential form of type (p, q) is a smooth section of ${\bigwedge }^p(T^{1,0}X{)}^{\ast }\otimes {\bigwedge }^q(T^{0,1}X{)}^{\ast }$, so in holomorphic coordinates it has the expressionHere the increasing multi-indices avoid duplicating alternating factors. The exterior derivative has just two type components: the conjugate Dolbeault operator $\partial$ and the Dolbeault operator $\bar{\partial }$. On such a local expression they areHolomorphic changes of coordinates preserve these types, so the definitions are intrinsic. Comparing type components of $d^2=0$ gives ${\partial }^2={\bar{\partial }}^2=0$ and $\partial \bar{\partial }+\bar{\partial }\partial =0$.

A real (p, p)-form satisfies $\bar{\eta }=\eta$, where complex conjugation acts on both coefficients and differential factors. Conjugation exchanges the two Dolbeault operators, soThus $\bar{\partial }\eta =0$ forces $\partial \eta =0$ as well. Conversely, if $d\eta =0$, its distinct $(p+1,p)$ and $(p,p+1)$ components both vanish. ThereforeThe assertion remains valid in top degree, when the indicated type components are zero automatically.

For a holomorphic function $f$, $\bar{\partial }f=0$ and $\partial \bar{f}=0$. Writing its real part as $u=(f+\bar{f})/2$ givesHence the real part of a holomorphic function is a pluriharmonic function.

For the local exactness assertion, work on a smaller polydisc whose closure lies in the original one. The allowed one-variable integral is the Cauchy-Green operator $T_j$ in coordinate $z_j$. Applied coefficientwise, it satisfies ${\partial }_{{\bar{z}}_j}{T}_{j}a=a$ on a smaller disc and commutes with differentiation in all other coordinates; the smooth parameter dependence follows by applying the integral to a smooth compactly supported extension in the integration variable. This extension can be made with a cutoff equal to one on the smaller disc.

Begin with a Dolbeault operator closed form $\phi$ and decompose it uniquely as $\phi =d{\bar{z}}^1\wedge \alpha +\beta$, where neither $\alpha$ nor $\beta$ contains $d{\bar{z}}^1$. Set ${\psi }_1=T_1\alpha$. The $d{\bar{z}}^1$ component of $\bar{\partial }{\psi }_1$ is exactly $d{\bar{z}}^1\wedge \alpha$, sohas no $d{\bar{z}}^1$ factor. It is still $\bar{\partial }$-closed. The $d{\bar{z}}^1$ component of $\bar{\partial }{\phi }_1=0$ now shows that every coefficient of ${\phi }_1$ is holomorphic in $z_1$.