Projective threefold coherent comparison and holomorphic bundle algebraization

Choose a projective embedding $X\subset {\mathbb{P}}^N$. Four general linear forms $l_0,\dots ,l_3$ have no common zero on $X$: successive hyperplane sections reduce dimension unless already empty, and a fourth general hyperplane avoids the remaining finite third intersection. Equivalently the incidence of centers meeting $X$ has smaller dimension in the Grassmannian of codimension-four centers. Put $R=\mathbb{C}[z_0,\dots ,z_N]\mathrm{/}{I}_X$. Since $\mathrm{Proj}(R\mathrm{/}(l_0,\dots ,l_3))$ is empty, a power of the irrelevant ideal lies in $(l_0,\dots ,l_3)$; thus $R\mathrm{/}(l_0,\dots ,l_3)$ is finite-dimensional. Choose homogeneous lifts of a vector-space basis. Recursively reducing a homogeneous element modulo $(l_0,\dots ,l_3)$ expresses it as a combination of those finitely many lifts with coefficients in $S=\mathbb{C}[l_0,\dots ,l_3]$: each residual coefficient has smaller degree. Consequently $R$ is finite over $S$. The $l_i$ are algebraically independent, since otherwise the finite extension would give $\dim R\le 3$, whereas $\dim R=\dim X+1=4$. On the charts $l_i\mathrm{\ne }0$, degree-zero localization preserves module finiteness: a homogeneous generator $e$ of degree $d$ becomes the degree-zero generator $e\mathrm{/}{l}_i^d$. Hence these four forms define an actual finite morphism $f:X\to P$, with $f^{\ast }{\mathcal{O}}_P(1)={\mathcal{O}}_X(1)$. This construction uses no GAGA in ${\mathbb{P}}^N$.

Here are the finite analytic algebra facts used below, including their local justification. If $A\to B$ is a finite complex affine algebra map, choose finitely many algebra generators $z_j$ integral over $A$ and their monic equations. First extend the finite algebra itself to the base analytic ring: $D=B{\otimes }_A{\mathcal{O}}_{P^{an},y}$ is finite, by its finite module presentation. In a small analytic neighborhood of a base point, the finite fiber splits into disjoint neighborhoods of its points. The splitting is obtained by factoring a monic equation for a generic linear combination of the $z_j$ into factors with disjoint root sets: division with remainder and the coprimality of the factors solve the coefficient equations by the holomorphic implicit-function theorem. The resulting idempotents split $D$ (Bezout identities for the coprime polynomial factors supply the orthogonal projectors), before identifying it with any analytic direct image. In each factor, translate $z_j$ by its value at the relevant fiber point. Its monic equation can be chosen with all non-leading coefficients vanishing at the base point, as follows. In the corresponding factor of $D$, the residue of the translated generator is nilpotent, so some power lies in ${\mathfrak{m}}_{y}D$. Write multiplication by that power on a finite generating set using a matrix with entries in ${\mathfrak{m}}_y$. The adjugate determinant identity gives a monic annihilating polynomial whose lower coefficients lie in ${\mathfrak{m}}_y$; substituting that power gives the required equation for the generator. The roots therefore lie in an arbitrarily small fixed circle after shrinking the base. Division of a convergent germ $h(t,z_j)$ by this monic equation leaves a polynomial remainder of bounded degree in $z_j$ with coefficients holomorphic in $t$. For a monic polynomial $p(t,z)$ of degree $d$, that remainder is $R(t,z)=(2\pi i{)}^{-1}{\int }_{\mathrm{\mid }\zeta \mathrm{\mid }=ϵ}h(t,\zeta )(p(t,\zeta )-p(t,z))\mathrm{/}((\zeta -z)p(t,\zeta ))d\zeta$. The divided difference is polynomial in $z$ of degree less than $d$; the circle contains every root and no root lies on it after shrinking. Cauchy's formula gives $h-R=pQ$, where $Q=(2\pi i{)}^{-1}{\int }_{\mathrm{\mid }\zeta \mathrm{\mid }=ϵ}h(t,\zeta )\mathrm{/}((\zeta -z)p(t,\zeta ))d\zeta$ is holomorphic for $\mathrm{\mid }z\mathrm{\mid }<ϵ$. Repeating for the finitely many $z_j$ gives bounded-degree monomials as generators over the base analytic local ring. The Cauchy integrals converge uniformly on a smaller polydisc, so this reduction applies to convergent series, not only formal series. Thus each analytic source local ring over the finite fiber is finite over the analytic base local ring. Its finite relations are coherent by [F3].

Write $g=f^{an}$. This map is proper, because $X^{an}$ is a closed subset of compact complex projective space and its target is Hausdorff. For every abelian sheaf $M$, $(g_{\ast }M{)}_y={\prod }_{x\mid y}{M}_x$: representatives on disjoint neighborhoods of the finite fiber extend to a section on an inverse-image neighborhood, since properness excludes the closed complement after shrinking the base. The same shrinking proves that a section with zero fiber germs vanishes near that fiber. This proves both surjectivity and injectivity of the stalk map. More explicitly, for $y\in P^{an}$, the analytic algebra supplied by a finite presentation of $B$ over $A$ has a natural map to ${\prod }_{x\in f^{-1}(y)}{\mathcal{O}}_{X^{an},x}$. Both sides are finite over ${\mathcal{O}}_{P^{an},y}$ by step 2.1. After completion this is the canonical isomorphism

To verify the product decomposition, the finite semilocal algebra $B{\otimes }_A{A}_y$ has pairwise comaximal maximal ideals over $y$; sufficiently high powers of their intersection and of ${\mathfrak{m}}_{y}B$ are cofinal, since their radicals agree. The Chinese remainder theorem at every order followed by inverse limit gives the displayed product. [F2] identifies each algebraic completion with its analytic completion. Flat completion and its finite-module detection property in [F2] detect the kernel and cokernel of the finite-module comparison, so the uncompleted analytic algebra map is an isomorphism. The same argument applied to a finite $B$-module presentation gives

for every algebraic coherent $F$. All maps are the canonical coefficient-extension maps, so they respect multiplication, tensor action and morphisms.

Put $g=f^{an}$ and $C=g_{\ast }{\mathcal{O}}_{X^{an}}$, identified by step 3.1 with $(f_{\ast }{\mathcal{O}}_X{)}^{an}$ and hence coherent on $P^{an}$. The finite-fiber stalk formula of step 3.1 applies to every abelian sheaf. Consequently $g_{\ast }$ is exact, since finite products of exact stalk sequences are exact. Now let $M$ be coherent analytic. Choose finite free presentations of $M$ on disjoint neighborhoods of the fiber points. Their finite ranks may be padded by zero generators to common bounds; proper shrinking puts the whole inverse image of a base neighborhood inside their union. Thus on that inverse image there is a presentation ${\mathcal{O}}^a\to {\mathcal{O}}^b\to M\to 0$. Exact pushforward gives $C^a\to C^b\to g_{\ast }M\to 0$, so [F3] proves coherence of $g_{\ast }M$. Its $C$-action remembers the full finite algebra. Conversely a coherent $C$-module $N$ defines $g^{-1}N{\otimes }_{g^{-1}C}{\mathcal{O}}_{X^{an}}$. At $y$, step 3.1 identifies $C_y={\prod }_{x\mid y}{\mathcal{O}}_{X^{an},x}$; its idempotents split $N_y$ into the corresponding factors, and the displayed tensor takes exactly the $x$ factor at $x$. Finite $C$-module presentations pull back to finite analytic presentations, so the inverse sheaf is coherent by [F3]. The canonical unit and counit maps are isomorphisms on these stalk factors and hence on sheaves. This proves the finite-algebra module equivalence, including ramified and nonreduced fibers.

Finite analytic pushforward is exact even as a functor on underlying abelian sheaves. For any sheaf $M$, its pushforward stalk at $y$ is ${\prod }_{x\mid y}{M}_x$: disjoint neighborhoods of the finitely many fiber points and the proper shrinking argument of step 4.1 are cofinal among inverse-image neighborhoods. Finite products of exact stalk sequences are exact. Pushforward preserves flasque sheaves, since its restriction maps are those on inverse-image opens. Pushing a flasque resolution therefore gives canonically

For algebraic coherent $F$, on an affine base chart $U=\mathrm{Spec}A$ the finite morphism has $f^{-1}U=\mathrm{Spec}B$ with $B$ finite over $A$. By [F6], $F{\mathrm{\mid }}_{f^{-1}U}$ is a finite $B$-module and $(f_{\ast }F){\mathrm{\mid }}_U$ is that module restricted to $A$; restriction of scalars is exact, and the module is finite over $A$. Thus $f_{\ast }F$ is coherent and finite pushforward is exact on coherent modules. Take the standard finite affine cover of $P={\mathbb{P}}^3$; its finite intersections and their inverse images under $f$ are affine. The Čech complexes of $F$ on the inverse-image cover and of $f_{\ast }F$ on the base cover identify term by term. By [F7] they compute the respective cohomology, so $H^q(X,F)=H^q(P,f_{\ast }F)$. Combining these identities with step 3.1 and the P³ comparison [F1] proves the claimed canonical cohomology comparison on $X$ for all $q$. Naturality follows because each comparison is induced by the same analytification map on sheaves and resolutions.

Set $\mathcal{B}=f_{\ast }{\mathcal{O}}_X$; this is a coherent algebraic ${\mathcal{O}}_P$-algebra by [F6], and step 3.1 identifies ${\mathcal{B}}^{an}=f_{\ast }^{an}{\mathcal{O}}_{X^{an}}$. For an arbitrary coherent analytic $M$ on $X^{an}$, put $N=f_{\ast }^{an}M$, coherent by step 4.1. Algebraize $N$ to $N_0$ by [F1]. Its analytic action ${\mathcal{B}}^{an}\otimes N\to N$ uniquely algebraizes to $\mu :\mathcal{B}\otimes N_0\to N_0$, since [F1] is tensor compatible and fully faithful. The unit and associativity equalities for $\mu$ hold algebraically because their analytifications hold and [F1] is faithful. Thus $N_0$ is a coherent $\mathcal{B}$-module. On every base chart $\mathrm{Spec}A$, [F6] identifies this module with a finite module over $B=\mathrm{\Gamma }(f^{-1}\mathrm{Spec}A,{\mathcal{O}}_X)$; its associated sheaf on $f^{-1}\mathrm{Spec}A=\mathrm{Spec}B$ localizes compatibly on overlaps, so these sheaves glue to a coherent algebraic $E$ on $X$. The analytic finite-algebra equivalence of step 4.1 identifies $E^{an}$ with $M$. This is essential surjectivity for all coherent analytic $M$, rather than an argument requiring $M$ already algebraic.

A morphism between two analytic sheaves on $X^{an}$ is exactly a morphism between their finite pushforwards commuting with the ${\mathcal{B}}^{an}$-actions, by the same local finite-algebra construction. [F1] uniquely algebraizes the underlying morphism, and faithfulness algebraizes the commuting equality. This proves full faithfulness on $X$. [F2] proves exactness stalkwise at complex closed points; by [F8], any nonzero algebraic coherent kernel or cokernel has a closed point in its support, so these stalk tests detect exactness globally. Associativity of tensor product proves its tensor compatibility.

Now let $M=\mathcal{O}(V)$ be analytically locally free of rank $r$, and take the coherent algebraization $E$ of step 5.2. At a closed point $x$, [F8] gives residue field $\mathbb{C}$; write $A={\mathcal{O}}_{X,x}$, $B={\mathcal{O}}_{X^{an},x}$ and $L=E_x$. Then $L{\otimes }_{A}B\cong B^r$ and $B$ is faithfully flat over $A$ by [F2]. The residue fields are both $\mathbb{C}$, so $L\mathrm{/}{\mathfrak{m}}_{A}L$ has dimension $r$. Lift a residue basis to obtain $A^r\to L$, surjective by [F4]. After tensoring with $B$, this is a surjection $B^r\to B^r$ whose residue matrix is invertible, hence whose determinant is a unit and whose kernel is zero. Flatness and faithfulness give that the original kernel is zero. Thus $L\cong A^r$. A presentation of the coherent $E$ extends this isomorphism to an algebraic neighborhood of $x$; the nonfree locus, detected by its finite presentation minors, is closed. By [F8], if it were nonempty it would contain a closed point, contrary to freeness at every closed point. Hence $E$ is finite locally free everywhere.

For finite locally free $E$, local bases identify its dual with the dual free module. Coefficient extension then gives the canonical isomorphism $(E^{\vee }{)}^{an}\cong (E^{an}{)}^{\vee }$, compatible with evaluation and changes of basis. Full faithfulness of step 6.1 uniquely algebraizes any isomorphism between two algebraizations and its inverse; the identity composites follow by faithfulness. Together with steps 5.1 and 6.2 this proves the complete Statement. Smoothness, the canonical bundle and any Calabi–Yau assumptions have not been used to restrict the holomorphic input.