INTRODUCTION TO ANALYTIC AND FORMAL GEOMETRY · INTRODUCTION TO ANALYTIC AND FORMAL GEOMETRY 3: by this we mean that there is a scheme the completion of which is the formal scheme

INTRODUCTION TO ANALYTIC AND FORMAL GEOMETRY

CARLO GASBARRI

Abstract. These are a preliminary draft of my notes for the lectures in Grenoble. Theymay be incomplete or they may contain some mistakes. Please fell free to comment or helpto improve them.

Contents

1. Introduction 12. The theorem of Siegel on the field of meromorphic functions 32.1. The field of meromorphic functions of a variety 32.2. Meromorphic functions and line bundles 42.3. Siegel Theorem 63. The Theorem of Chow on analytic subvarieties of the projective space 84. The Andreotti – Frankel approach to Lefschetz’s Theorem of hyperplane sections 95. Introduction to formal geometry 135.1. The Grothendieck algebraization Theorem 175.2. A formal scheme which is not algebraizable 21References 24

1. Introduction

These are notes of my five hours lectures held at the Summer school Foliations and AlgebraicGeometry held at the Institut Fourier in Grenoble (F) in June 2019.

In the first part of the notes we explain some cornerstone theorems of algebraic geometryover the complex numbers. The proofs - and the statements - of these theorems use in aessential way methods coming from complex analysis and differential topology.

These theorems are:– The Siegel Theorem on the field of meromorphic functions of a compact complex variety.This theorem tells us that the field of meromorphic functions of a compact complex variety

has transcendence degree at most the dimension of the variety. Observe that the compactness

Date: June 14, 2019.Research supported by the project ANR-16-CE40-0008 FOLIAGE.

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2 CARLO GASBARRI

property is necessary: even the field of meromorphic function of a one dimensional disk or anaffine variety has infinite transcendence degree over the complex number.

The proof uses some complex analysis (the maximum modulus property of analytic func-tions) and the interaction between meromorphic functions and line bundles.

The result is very classical and its proof may be found for instance in [1].– The Chow Theorem on analytic sub varieties of the projective spaces.In this Theorem, one shows that a smooth analytic subvariety of a projective space is

actually algebraic (it is the zero set of a system of polynomial equations). This Theorem hasmany proofs, the one proposed here is a simple adaptation of the proof of the Siegel Theorem.It actually uses the same tools of it.

Both the theorems above use only tools from complex analysis and algebraic geometry. Ifwe allow to use also tools from differential topology - actually basic Morse theory - we cangive a reasonably self contained proof of:

– The Lefschetz Theorem on the fundamental group of hyperplane sections.This Theorem tell us that, if D is a smooth hyperplane section of a smooth projective

variety of dimension at least three, then the natural map between the fundamental group ofD and the fundamental group of X is an isomorphism. The proof we present here - due toAndreotti and Frankel - is a classical application of the standard Morse theory thus it usestools from differential geometry and from algebraic topology.

One can read a proof of this theorem for instance in [6]

When one look to technics and the tools used to prove these theorems, even if some of themmay be stated in a quite general contest, one realizes that they cannot be easily translated inmore general contests. For instance, one can state the Lefschetz Theorem over arbitrary fieldor even over arbitrary schemes but Morse Theory is no more available.

Formal Geometry is a tool which allows to use technics which look like analysis and topol-ogy in the contest of general algebraic geometry. It essentially uses the fact that an analyticfunction is known when we know its Taylor expansion. Following this principle, formal ge-ometry may be seen as an avatar of the germ of a analytic neighbourhood of a subscheme ofa scheme.

After a brief introduction to the general machinery of formal geometry -one can find amore detailed introduction in [5] or [4] - we will explain a theorem, due to Grothendieck,which plays the role of the Chow Theorem in this contest. We will end the lectures withthe description of a formal scheme which is not ”algebrizable” (which means that it not theformal neighbourhood of a closed subscheme of a scheme).

In these lectures we do not mention possible applications of formal geometry. We wouldlike to quote some of them:

– One of the main application is in deformation theory: Usually one want to deform ageometric structure (a scheme, a sheaf, etc) defined over a field, over a more general base (forinstance over a discrete valuation ring). In order to do this, one first deform it over artinianrings and builds up a formal deformation of the object. Grothendieck algebraization theoremallows, under some conditions, to prove that the formal deformation is actually ”algebraic”

INTRODUCTION TO ANALYTIC AND FORMAL GEOMETRY 3

: by this we mean that there is a scheme the completion of which is the formal scheme weconstructed.

– An important application of formal geometry is an algebraic proof of an analogue of theLefschetz hyperplane sections theorem for the fundamental group and for the Picard groups.One can give a proof of these theorems in the general theory of schemes (which for instanceholds in positive characteristic). A posteriori one can see some similarities with the Andreotti– Frankel proof of it.

– Another application of formal geometry is the ”algebraic study” of differential equations.A differential equation on a general scheme is usually called a ”foliation”. Over analyticvarieties, regular differential equations always have local solution. In the Zariski topologythis is not anymore true. Nevertheless one can see that, at least in characteristic zero, thereis always a solution in the formal geometry. In positive characteristic this is still true undersome hypothesis (called p– closure condition).

These lectures are introductory to the lectures by J.- B. Bost. His lectures will cover moreadvanced topics in formal geometry. In particular he will develop some of the themes quotedabove.

2. The theorem of Siegel on the field of meromorphic functions

2.1. The field of meromorphic functions of a variety. Let U be an open set of Cn.Denote by Γ(U,O) the ring of holomorphic functions over it.

A meromorphic function F on U is a function f on the complement of a nowhere densesubset S of U with the property that there exists a covering U = ∪Ui and holomorphicfunctions fi and gi in Γ(Ui,O) such that gi · F |Ui\S = fi|Ui\S.

The set of meromorphic functions on U is a field which will be denoted by MU .By construction, given a meromorphic function F on U and a point z ∈ U , we can find

a open disk ∆ ⊆ U centred in z and two relatively prime holomorphic functions f and gon ∆ with the property that F |∆ = f

g. The functions f and g are well defined up to units.

The analytic subset Z(f) and Z(g) depend only on F and not on the choice of f and g.Consequently, given a meromorphic function on U , we may define the zero locus of F as theanalytic subset given locally by Z(f) and the pole locus of F as the analytic subset locallygiven by Z(g).

Remark 2.1. On a U.F.D. A, two elements a and b are said to be relatively prime if the classof a is not a zero divisor in A/(b) (the definition do not depend on the order or a and b).

Observe that a point z ∈ U may be at the same time a zero and a pole of a meromorphicfunction: for instance, (0, 0) is a zero and a pole of the function F = x

yon C2.

Let X be a smooth analytic variety defined over the field of the complex numbers.The sheaf MX of meromorphic functions of X is the sheaf which associates to every open

set U of X the field MU .

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We define MX to be the field H0(X,MX).

2.2. Meromorphic functions and line bundles. We prove now a theorem which showshow a meromorphic function may be interpreted as a holomorphic function on a line bundle.

As in the case of a open set of Cn, given a meromorphic function F on an analytic varietyX we can define the set Z(F ) of zeroes of F and the set P (F ) of poles of it.

We define Ind(F ) to be the analytic subset Z(F ) ∩ P (F ).

Example 2.2. Suppose that X = Pn with homogeneous coordinates x0 . . . xn. If P (x0, . . . , xn)and Q(x0, . . . , xn) are two homogeneous non zero polynomials (with no common factors) of thesame degree d, then the function F ; = P

Qdefines a meromorphic function on X. In particular

we obtain that MPn = C(X1, . . . , Xn).

One can easily prove (exercise) that

Proposition 2.3. Let f : X → Y be an analytic dominant map between smooth analyticvarieties and F ∈MY . Then f ∗(F ) is a meromorphic function on X. In particular we havean inclusion f ∗ : MY ↪→MX .

In particular we obtain:

Proposition 2.4. Suppose that Z ⊂ X is an analytic subset of X and UZ := X \ Z. LetιUZ

: UZ ↪→ X the inclusion. Then the natural inclusion ι∗ : MUZ↪→MX is an isomorphism.

As a corollary we obtain the following, very important:

Proposition 2.5. Let f : X → Y be an analytic proper map between smooth analytic varietieswith the property that there exists proper analytic subvarieties Z ⊂ X and W ⊂ Y such thatf |UZ

: UZ → UW is an isomorphism. Then the induced map fast : MY → MX is anisomorphism.

Maps like the ones in the Proposition above are called modifications. Proposition tells usthat MX is invariant by modifications.

and

Proposition 2.6. Suppose that f : X → P1 is an analytic dominant map and let f ∗ :C(X) ↪→ MX the induced inclusion. Then f determines a meromorphic function f ∗(X)which is well defined up to a scalar.

Let’s see how the last proposition may be almost inverted:

Proposition 2.7. Let F ∈MX be a non constant meromorphic function. Then we can finda closed analytic set Z of codimension at least two and a dominant analytic map h : UZ → P1,such that h∗(X) = F .

Observe that we used Proposition 2.6 to identify MUZand MX .


Proof. Let Z := Ind(F ). We can cover UZ by open sets UZ = ∩Ui in such a way that wecan find coprime holomorphic functions fi and gi on Ui such that F |Ui

= figi

. The analytic

functions hi : Ui → P1 defined by h(z) := [fi(z) : gi(z)] glue together to an analytic functionh : UZ → P1. Since, by construction h∗i (X) = F |Ui, the conclusion follows . �

A similar proof gives

Theorem 2.8. Let F1, . . . , Fn be C linearly independent meromorphic functions on X, thenwe can find an analytic subset Z ⊂ X of codimension at least two (possibly empy) and ananalytic map h : UZ → Pn such that h∗(Xi) = Fi.

Let’s recall the statement of Hartogs Theorem

Theorem 2.9. (Hartogs) Let U be a open set in Cn and Z ⊂ U be a closed analytic set ofcodimension at least two. Then every analytic function f defined over U \Z extends uniquelyto an analytic function on U .

Hartogs Theorem can be stated in the following way: Under the hypothesis of Theorem 2.9the natural restriction map

(2.1) Γ(U,O) −→ Γ(U \ Z,O)

is an isomorphism.As a direct corollary of Hartogs Theorem we get

Corollary 2.10. Let L be a line bundle on X and let Z ⊂ X be an analytic subset ofcodimension at least two. Then the natural restriction map

(2.2) H0(X,L ) −→ H0(UZ ,L )

is an isomorphism.

Another corollary is the following:

Corollary 2.11. Suppose that Z ⊂ X is an analytic subset of codimension at least two, thenthe natural restriction map

(2.3) ι : Pic(X) −→ Pic(UZ)

is an isomorphism.

Proof. The zero locus of an analytic function is an analytic set of codimension one. Conse-quently, suppose that U is an open set of X and g is a non vanishing analytic function onU \ Z, then g extends (by Hartogs) to an analytic function on U which is non vanishing too.Indeed otherwise its zero locus would be contained in Z which is at least of codimension twoand this is not possible.

The map ι is injective: Suppose that ι(L ) is the trivial line bundle, this implies that ι(L)has a nowhere vanishing global section. This global section, by Hartogs, extends to globalsection of L which is nowhere vanishing. Thus L is isomorphic to the trivial line bundle.

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The map ι is surjective: Let M be a line bundle on UZ . We mat suppose that it is definedby a covering {Ui} of X and a cocycle {gij} with gij nowhere vanishing functions on Ui∩Uj \Z(and satisfy the usual cocycle condition). Each gij extends to a nowhere vanishing functionon Ui ∩Uj and the cocycle condition is preserved because it holds on dense subsets. Thus Mextends to a line bundle on X. �

Suppose that F0, . . . , Fn are meromorphic functions over a complex variety X, by Theorem2.8 we get an analytic map h : UZ → Pn where Z is an analytic subset of codimension at leasttwo. Thus we obtain a line bundle LUZ

:= h∗(O(1)). The natural inclusion H0(Pn, cO(1)) ⊂H0(UZ ,LUZ

) give rise to a n + 1 dimensional subspace V of H0(UZ ,LUZ) and a surjective

map

(2.4) V ⊗ OUZ−→ LUZ

The line bundle LUZcan be extended to a line bundle L on X and the subspace V extends

to a subspace of global sections of it.Observe that the restriction map from V to L is not surjective anymore.To resume we obtain

Theorem 2.12. We can associate to n C–linearly independent meromorphic functions F1, . . . , Fnon X a line bundle L and a n+ 1 dimensional subspace V of H0(X,L ). Viceversa, we canassociate to a given a line bundle L and a subspace V of dimension n + 1 of H0(X,L )a closed set Z of codimension at least two and an analytic map h : UZ → Pn such thath∗(O(1)) = LUZ

. Thus if h∗ : C(X1, . . . , Xn) → MX is the induced map, h∗(Xi) = Fiare n C–linearly independent meromorphic functions on X. These constructions are one theinverse of the other.

Remark 2.13. If F1, . . . , Fn are meromorphic functions on X and V is the space of holomorphicsections of the line bundle L associated to them, then we can find a basis s0, . . . , sn of Vsuch that, over the open set s0 6= 0, we have Fi = si

s0.

2.3. Siegel Theorem. In this paragraph we would like to prove the following

Theorem 2.14. Let X be a smooth compact analytic variety of dimension n then

(2.5) TrdegC(MX) ≤ n.

Theorem 2.14 will be consequence of the following Theorem which has its own interest

Theorem 2.15. Let X be a smooth analytic variety of dimension n and L be a line bundleover it. Then we can find a positive constant C such that, for every positive integer d we have

(2.6) h0(X,L⊗d) ≤ C · dn.

As it is costumary, we denote by h0(X,L) the dimension of the space of global sectionsΓ(X,L).


Let’s show first how Theorem 2.15 implies Theorem 2.14: It suffices to prove that, ifF1, . . . , Fn+1 are non zero meromorphic sections over X, then we can find a polynomialP (X1, . . . , Xn+1) ∈ C[X1, . . . , XN+1] such that P (F1, . . . , Fn+1) = 0 identically on X.

By Remark 2.13, we can find line bundle L on X, a vector space V ⊂ H0(X,L) of dimensionN + 2, a basis s0, ,sN+1 and a dense open set U ⊆ X where s0 6= 0, each of the Fi has nopoles and Fi|U = si

s0. Moreover, each of the Fi is holomorphic on U .

For every I := (i0, . . . , in+1) ∈ Nn+2 such that i0 + · · ·+ in+1 = d the product sI := si00 ·sin+1

n+1

is a global section of Ld. We will call sections of this forms homogeneous monomials of degreed in the Fi’s.

The number Nd of homogeneous polynomials of degree d in the Fi’s is(n+1+dn

), thus there

is a positive constant A (independent on d) such that Nd ≥ Adn+1. Since, by Theorem2.15, we have that h0(X,L⊗d) ≤ C ⊗ dn, for d sufficently big d we have Nd ≥ h0(X,L⊗d).Consequently, for d big enough, the set of homogeneous monomials of degree d in the Fi isC- linearly dependent.

Suppose that∑

I∈Nn+2 , |I|=d aIsI = 0 be linear dependence relation within the homogeneous

monomials of degree d in the Fi’s and let P (X0 . . . , Xn+1) be the polynomial∑

I∈Nn+2 , |I|=d aIXI ∈

C[X0, . . . , Xn+1]. Then P (1, F1, . . . , Fn+1) = 0 on U . Since U is dense in X, the conclusionfollows.

In order to prove Theorem 2.15 we begin by recalling the classical Schwartz Lemma:

Lemma 2.16. Let f(z1, . . . , zn) be a holomorphic function on the n- dimensional disk B(0, r+ε). Suppose that the order of vanishing of f at the origin is at least d and denote by M :=maxz∈B(0,r){|f(z)|}. Then, for every z0 the disk B(0, r) we have

(2.7) |f(z)| ≤M · rd.

We recall that the fact that f(z) has order of vanishing at least d at the origin means thatthe Taylor expansion of f at the origin is

(2.8) f(z1, . . . , zn) =∑|I|≥d

aIzI .

. The one dimensional case of Lemma 2.16 can be found in any classical text of complexanalysis and the multidimensional case can be easily reduced to it.

We can now give the proof of Theorem 2.15

Proof. (of Theorem 2.15) Since X is compact, we can find a finite set of points ai ∈ X andopen neighbourhoods Ui of them with the following properties:

– Each Ui is biholomorphic to the open ball B(0, 1) centered in ai.– Denote by Vi the open balls centered in ai and with radius exp(−1). We suppose that

the open sets Vi also cover X.– The restriction of the line bundle L to Ui is trivial and we denote by gij the transition

functions of L with respect to the covering given by Ui.

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Observe that the gij give also the transition functions of L with respect to the coveringgiven by the Vi.

Denote by B the real number maxij{log |gij|} and by A the smallest integer strictly biggerthen B.

We claim the following Lemma:

Lemma 2.17. Suppose that s ∈ H0(X,L⊗d) is a global section which vanishes at order atleast d · A+ 1 at every ai, then s = 0.

Proof. (Of the Lemma) Let s be such a section. We can write s|Ui= fi where fi are holomor-

phic functions on B(0, 1) and fi = gdij · fj. Since X is compact, there exists i0 and z0 ∈ Vi0such that M := |fi0(z0)| = maxj{|fj(z)| / z ∈ Vj}.

Observe that z0 cannot be in Vi0 \ ∪j 6=i0Vj. Indeed fi0(z) can only reach his maximum onVi0 on the border of it.

By Schwartz lemma 2.16 we obtain then:

(2.9) M = |fi0(z0)| = |gi0j|d · |fj(z0)| ≤ exp(d · A) ·M exp(−d · A− 1)

wich gives M = 0. The conclusion of the Lemma follows. �

Let’s show how Lemma 2.16 implies Theorem 2.15:Let Iai be the ideal sheaf of the points ai. Let A be as in Lemma 2.16. We have a natural

linear map

(2.10) αd : H0(X,L⊗d) −→⊕i

L⊗d ⊗ OX/Id·A+1ai

Lemma 2.16 tells us that for every d, the map αd is injective. Indeed a section in the kernelof it would be a section which vanishes at order at least d · A+ 1 at every ai.

For every positive integer d, we have that OX/Idai' C[x1, · · · , xn]/(x1, · · · , xn)d. And this

last one is a C vector space of dimension(d+nn

)∼ dn. The conclusion of the proof follows. �

3. The Theorem of Chow on analytic subvarieties of the projective space

In this section we will prove that Theorem 2.15 implies also the Chow Theorem:

Theorem 3.1. Let X be a smooth closed analytic of the projective space PN . Then X isalgebraic.

This theorem tells us that if a smooth proper analytic variety X is contained in the pro-jective space (or more generally, inside any projective variety) then it is actually defined byalgebraic equations.

As a easy corollary we obtain:

Corollary 3.2. Let f : X → Y be an holomorphic morphism between smooth projectivevarieties. Then f is actually an algebraic map.


Proof. ((Of the Corollary) It suffices to apply Theorem 3.1 to the graph of f inside X × Y .This graph will be an algebraic subvariety of X × Y thus the morphism f which is given bythe projection on Y is algebraic. �

Another consequence is:

Corollary 3.3. Let X be a smooth projective variety. Then the field of meromorphic functionsMX coincides with the field of rational functions C(X).

Proof. ((Of the Corollary) Each meromorphic function corresponds to an holomorphic mor-phism f : X → P1. Then apply the Corollary above. �

Proof. (Of Theorem 3.1) Let X be the Zariski closure of X inside PN . By construction, X isthe smallest algebraic subvariety of PN containing X. It suffies to prove that the dimensionof X is the same of the dimension of X.

Indeed X is an algebraic variety and in particular an analytic variety. And it is easy to seethat an analytic variety contains only one closed subvariety of the same dimension of it.

Let n be the dimension of X and m be the dimension of X. A priori we have m ≥ n.Let O(1) be the tautological line bundle of PN . We first have the following Lemma:

Lemma 3.4. . Let Y be a projective variety of dimension m in PN . Then there is a constantC such that, for every positive integer d, we have

(3.1) h0(Y,O(d)) ≥ C · dm.

Proof. (of the Lemma) Choose a suitable projection h : Y → Pm which is surjective andgenerically finite. Then we have an injective linear map

(3.2) h∗ : H0(Pm; O(d)) −→ H0(Y,O(d)).

Since h0(Pm; O(d)) ∼ dm, the conclusion of the Lemma follows. �

Since X is Zariski dense in X, we have a natural inclusion

(3.3) H0(X,O(d)) ↪→ H0(X,O(d)|X).

Consequently, an application of Theorem 2.15 and Lemma 3.4 gives that we can find positiveconstants C1 and C2 such that, for every d� 0, we have

(3.4) C1 · dm ≤ h0(X,O(d)) ≤ h0(X,O(d)|X) ≤ C2 · dm.

Which implies that m ≤ n. Thus m = n and the conclusion of Theorem 3.1 follows. �

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4. The Andreotti – Frankel approach to Lefschetz’s Theorem ofhyperplane sections

In this section we will explain the main steps of the proof of Lefschetz’s Theorem of fun-damental group s of Hyperplane sections of a projective variety.

Let’s begin by stating the theorem: We will state it only for the first homotopy group butthe general statement concerns also some higher homotopy groups:

Theorem 4.1. Suppose that X is a smooth projective variety of dimension n at least threeand ι : D ↪→ X is a smooth ample divisor, then the natural map

(4.1) ι∗ : π1(D) −→ π1(X)

is an isomorphism.

There exists another theorem on hyperplane sections which concerns cohomology withvalues in Z; Since it is not strictly related with the topic of these lectures, we prefer not toquote it. We would like just to mention that the proof of it follows the same lines of the proofof Theorem 4.1.

The proof rely on basic Morse theory.

Remark 4.2. Observe that the hypothesis on the dimension is necessary: If you take X = P2

and D be a smooth projective curve of degree at least three, then X is simply connected andD is not.

In order to prove Theorem 4.1 we need to recall the main Theorem of Milnor Theory.Let M be a smooth differentiable variety of dimension r and

(4.2) f : M −→ R

be a smooth function. A point p ∈M is said to be critical if dfp = 0.Let p0 ∈ M be a critical point for the map f . We may define a symmetric bilinear

Hessf (p0), called the Hessian of f form on the tangent space TP0M in the following way:Let v and w be two tangent vectors to M at P0. Extend them to vector fields v and w in

a neighbourhood of p0. Then we define

(4.3) Hessf (p0)(v, w) := v(w(f)).

It is symmetric and well defined because v(w(f))(p0) − w(v(f))(p0) = [v, w](f)(p0) = 0because p0 is degenerate.

In local coordinates: if (x1, . . . , xr) are local coordinates of M around p0 and f is givenby a smooth function f(x1, . . . , xr) then, a local computation shows that Hessf (P0) is thequadratic form associated to the symmetric matrix

(4.4)

(∂2f

∂xi∂xj

)


with respect to the basis(

∂∂x1

; . . . ; ∂∂xr

). The point p0 is said to be critical non degenerate if

the quadratic form Hessf (p0) is non singular.Suppose that p0 is a non degenerate critical point for f , Inside Tp0M there is a biggest

subspace V where the quadratic form Hessf (p0) is negative defined. We will call the dimensionof V the index of f at p0.

Remark 4.3. Observe that, by definition, the fact that a point is critical for a map f or thatit is non degenerate or its index is independent on the choice of local coordinates around it.

Given a smooth function f : M → R and a ∈ R, we will denote by Ma the closed setf−1((−∞, a]).

The main Theorem of Morse theory is the following:

Theorem 4.4. Let M be a differential variety and f : M → R be a differential map. Supposethat for every a ∈ R the subset Ma. Let c ∈ R and suppose that f−1(c) contains k nondegenerate critical points p1, . . . , pk of index λ1, . . . , λk respectively and no other critical points.Suppose that, for ε > 0, the subset Mc+ε do not contain any other critical points. Then Mc+ε

has the homotopy type of Mc−ε with a cell of dimension λi attached to each point pi.

We recall that a cell of dimension λ is just the space σλ := {(x1, . . . , xλ) ∈ Rλ /∑x2i ≤ 1}

and its border is Sλ := {(x1, . . . , xλ) ∈ Rλ /∑x2i = 1}. To attach a cell to a space means to

attach σλ ”along” Sλ.Another important Lemma we need is the following:

Theorem 4.5. Let M be a smooth closed manifold contained in Rn. For every p ∈ Rn definethe map

(4.5) dp : M // R

x // dp(x) := ‖x− p‖2

.

Then, for almost all p ∈ Rn, the map dp has only non degenerate critical points.

The first consequence of Theorems 4.4 and 4.5 is:

Corollary 4.6. Each smooth projective variety is homotopy equivalent to a CW complex.

The proof of the corollary can be done by induction on the dimension of the variety and itis left by exercise.

Let’s start the proof of Theorem 4.1.

Proof. (Of Theorem 4.1) The first part of the proof relies on the following Theorem.

Theorem 4.7. Let A ⊂ Cn be a smooth affine variety of dimension r. Identify Cn ' R2n.For p ∈ R2n generic, consider the map dp : A→ R. Suppose that x0 ∈ A is a non degeneratecritical point of fp. Then the index of x0 with respect to fp is at most r.

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Suppose thatA is a closed manifold of Rn and x0 ∈ A is a non degenerate critical pointof index λ for a map dp : A → R. Let H ∈ O(n) be an isometry, b ∈ Rn and a ∈ R>0.Then, a direct computation shows that x0 is a non degenerate critical point for the mapdp(a ·H(x) + b).

Consequently we may suppose:i) x0 = (0, . . . , 0);ii) the point p = (0, . . . , 1, 0, . . . , 0) where the 1 is at the (r + 1)’s position.iii) The variety A is given by the graph of a holomorphic function (fr+1(z), . . . , fn(z)) :

Cr → Cn−r.Moreover, the map dp is given by

(4.6) dp(z1, . . . , zr) =r∑i=1

|zi|2 + |fr+1(z)− 1|2 +n∑

j=r+2

|fj(z)|2.

And, since x0 is critical for dp, a direct computation gives that each fi will have order ofvanishing at least two at the origin. We can also rewrite the function dp(z1, . . . , zr) as

(4.7) dp(z1, . . . , zr) = (1− 2 ·Re(fr+1(z))) +r∑i=1

|zi|2 +n∑

j=r+1

|fj(z)|2.

Since each of the i(z)′s has has order of vanishing at least two at the origin, we see thatthey not contribute to the Hessian of dp at x0. Consequently, if write the Taylor expansionof fr+1(z1, . . . , zr) = F2(z) + F (z) where F2(z) is an homogeneous polynomial of degree twoin the zi’s and F (z) is a holomorphic function with order of vanishing at least three at theorigin, we obtain:

(4.8) Hessdp(x0) = 2(−HessRe(F2(z))(x0) + Idr).

From equation 4.8 above we see that if λ is a negative eigenvalue of Hessdp(x0), then 1−λ2

isa positive eigenvalue of HessRe(F2(z))(x0. Consequently Theorem 4.7 is a consequence of thefollowing Lemma:

Lemma 4.8. Let F2(z1, . . . , zr) be a homogeneous polynomial of degree two. Then HessRe(F2(z))(0)has at most r positive eigenvalues.

Proof. It suffices to remark that, after a change of complex variables, we may suppose that

(4.9) F2(z1, . . . , zr) = z21 + · · ·+ z2

s

with s ≤ r. Consequently, if we write zi = xi +√−1yi we obtain

(4.10) HessRe(F2(z))(0) = x21 − y2

1 + · · ·+ x2s − y2

s .

�

.We now come to the last part of the proof of Theorem 4.1. We may suppose that X is

embedded in a projective space, D is the hyperplane at infinity of X and A = X \ D is asmooth affine variety embedded in the corresponding affine space.


Remark that a priori, D may be only ample and not very ample, in this case the hyperplaneat infinity will be D with some multiplicity but this will not affect the fundamental groups.

Since D and X are homotopically equivalent to CW complexes, a small neighbourhood Vof D in X can be deformed to D inside it.

We consider the function f : X → R defined by

(4.11) f(x) =

{0 if x ∈ D

1dp(x)

if x ∈ A

It is easy to see that, since, by Theorem 4.7, the critical points of dp(·) have are nondegenerate with index less or equal to r, the critical points of f(x) inside A will be nondegenerate of index bigger then 2r−2 = r. Consequently, for every ε > 0 sufficiently small, Xis homotopically equivalent to Xε with cells of dimension at least r attached. Consequently,since any compact complex variety is homotopically equivalent to a CW complex and thefundamental group of a CW complex do not change if we add to it cells of dimension at leastthree, the natural map

(4.12) π1(Xε) −→ π1(X)

is an isomorphism.We may suppose that Xε ⊂ V . Observe that D ⊂ Xε. Consequently we have natural maps

(4.13) π1(D)α // π1(Xε)

β // π1(V )

and

(4.14) π1(Xε)γ // π1(V )

δ // π1(X)

Since, β ◦ α and δ ◦ γ are isomorphisms, the map

(4.15) π1(D) −→ π1(X)

being injective and surjective, is a bijection. �

5. Introduction to formal geometry

In this section all the rings and schemes will be Noetherian and all the morphisms ofschemes will be separated.

In this section we will see how we can introduce, in algebraic geometry, some tools whichmimic the use of topology and analytic geometry. In some cases they can be used to findalgebraic analogues of the tools introduced in the previous sections.

The tools we are speaking about are called ”formal geometry” and the starting point is theanalogous of the Taylor expansion of a holomorphic function.

The basics of the theory require some commutative algebra. Consequently we will justrecall them without proofs:

14 CARLO GASBARRI

a) Let A be a ring and a be an ideal of it. For every n we have a surjective map A/an+1 →A/an. Let A the projective limit lim←−A/a

n is called the a–adic completion of A. We have

a natural map A → A. The ring A is said to be a–adically complete if this map is anisomorphism.

b) More generally, given a A–module M , we can introduce the a–adic completion M of it

in the same way. If M is finitely generated then M = M ⊗A A.c) Let A be a ring and a be an ideal of it. Suppose that A is a–adically complete. We

associate to the couple (A, a) a locally ringed space Spf(A) := (X0,OX ) called the formalspectrum of it and defined in the following way:

– The underlying topological space is the topological space X0 := Spec(A/a) with its Zariskitopology.

– Denote by X the scheme Spec(A) and by Ia the sheaf of ideals associated to a. Thescheme X0 is the closed set V (a) of X. For every positive integer n the sheaf On := OX/I

na

is supported on X0. The structure sheaf OX is defined then to be the sheaf lim←−On.

Example 5.1. 1) Let A = k[t] where k is a field and a = (0). Then A = k[[t]] and Spf(A)is the ringed space whose underlying topological space is a point and the structure sheaf isk[[t]].

2) Suppose that A = Z and a = (p) is a prime ideal. Then Spf(A) is a ringed spacewhose underlying topological space is a point and the structure sheaf is Zp (the ring of p–adicnumbers).

3) Let ` ⊂ A2 be the line x = 0 (the coordinates on A2 will be (x, y)). Let (x) ⊂ k[x, y] = A

be the associated ideal. Then Spf(A) is the ringed space whose underlying topological spaceis ` and the structure sheaf is the sheaf associated to the ring k[y][[x]].

d) More generally, a locally ringed space (X ,OX ) is said to be a formal scheme if it islocally isomorphic to the formal spectrum of a couple (A, a).

e) A morphism of formal schemes is a morphism of locally ringed spaces.f) The dimension of a formal scheme is its Krull dimension. Observe that in general

it is bigger then the Krull dimension of the underlying topological space: for instance inthe examples (1) and (2) above the formal spectrum has dimension one but the underlyingtopological space has dimension zero.

g) Given a formal scheme X there exists a ideal sheaf I ⊂ OX such that:– The support of OX /I is the underlying topological X0 space of X ;– The couple (X0,OX /I ) is a noetherian scheme.Such a ideal sheaf is called ideal of definition of the formal scheme X and the scheme X0

is called support scheme of X .h) A basic example of formal scheme, which generalises the construction of the formal

spectrum is the following: Let X be a scheme and Y be a closed subscheme of it. Denote byIY the ideal sheaf of Y . The formal completion of X around Y is the formal scheme whoseunderlying topological space is Y and the structural sheaf is lim←−OX/I

nY and it is denoted by

XY .


The formal completion of X around Y can be imagined as a tubular neighbourhood of Yaround X.

i) A formal scheme X is said to be algebraizable if there exists a scheme X and a closed

subscheme Y such that X ' XY . Similarly, a morphism between algebraizable formalschemes f : XY → ZT is said do be algebraizable if there exists a morphism of schemesf : X → Z such that the formal completion of it is f .

j) If X is a scheme and we take Y = X, then the formal completion XY is just X.Consequently the category of formal schemes contains the category of schemes.

k) Let X be a formal scheme, X0 be its underlying scheme and I be an ideal of definitionof X . For every positive integer n the scheme (X0,OX /I n) is a closed subscheme of X .We will denote it by Xn. Consequently we may imagine X as ”filtered” by schemes havingall the same underlying spaces but structure sheaf containing nilpotents. We have inclusionsof schemes ιn : Xn ↪→ Xn+1.

i) Not every formal morphism is algebraizable: for instance take (X, Y ) = (A1, 0) and(Z, T ) = (A2, (0, 0)). The map associated to the morphism h : k[[z, w]] → k[[x]] given byh(z) = x and h(w) =

∑∞n=1

xn

n!is not algebraizable.

We will now describe coherent sheaves on formal schemes:

Theorem 5.2. Let X be a formal scheme and I be an ideal of definition of it.If F is a coherent sheaf on X then, for every n, the restriction Fn of it to Xn is a coherent

sheaf on Xn.Conversely, if for every n, we have a coherent sheaf Fn on Xn with exact sequences

(5.1) 0 −→ I n ·Fn+1 −→ Fn+1 −→ (ιn)∗(Fn) −→ 0.

Then lim←−Fn is a coherent sheaf on X .

Observe that, the theorem above tells us that, if F is a coherent sheaf on Xn then we havean exact sequence of coherent sheaves on X :

(5.2) 0 −→ I n ·F −→ F −→ (ιn)∗(ι∗n(F )) −→ 0.

m) A standard fact in commutative algebra is that if A is a noetherian ring and I is an

ideal, then the completion AI of A with respect to I is flat over A. Consequently we get

Proposition 5.3. Let X be a scheme and Y be a closed subscheme of it. Then the naturalmorphism ι : XY → X is a flat morphism.

In particular we obtain:

Proposition 5.4. In the hypothesis of Proposition 5.3, the functor ι∗ : Coh(X)→ Coh(XY )(Coh(·) being the category of coherent sheaves) is exact.

Another important tool in formal geometry is the so called Theorem of Formal functions:

16 CARLO GASBARRI

We recall the following general fact on higher direct images of coherent sheaves: Supposewe have a cartesian diagram of schemes:

(5.3) XTιX //

g

��

X

f��

TιY // Y.

and a coherent sheaf F on X. Then, for every positive integer i, there is a natural morphismof coherent sheaves on T

(5.4) ι∗Rif∗(F ) −→ Rig∗(ι∗X(F )).

We suppose that we have a proper morphism of schemes f : X → Y . Let S be a properclosed subset of Y and F be a coherent sheaf on X. Denote by T = f−1(S).

Let XT and YS be the completions of X (resp. Y ) with respect to T (resp. S). We have acartesian diagram

(5.5) XTιX //

f��

X

f

��YS

ιY // Y.

From Proposition 5.4 we get:

Proposition 5.5. Let F be a coherent sheaf on X, then, for every integer i, the natural map

(5.6) (ιY )∗(Rif∗(F )) −→ Rif∗(ι∗X(F ))

is an isomorphism.

For every positive integer we may consider the n–th infinitesimal neighbourhoods Sn andTn of S and T in Y and X respectively: If IS (resp. IT = IS · OX) is the ideal sheaf of S inY (resp. of T in X), then Sn (resp. Tn) is the closed subset associated to the ideal sheaf InS(resp. to the ideal InT ). Then, for every n, we have a cartesian diagram:

(5.7) Tnιn //

fn��

X

f��

Snjn // Y.

which factorises:

(5.8) Tnhn //

fn

��

XTιX //

f��

X

f

��Sn

gn // YSιY // Y.


Let F be a coherent sheaf on X. From the cartesian diagram above we get, for every i andn, morphisms:

(5.9) Rif∗(ι∗X(F ))

∼ // ι∗Y (Rif∗(F )) // (gn)∗((gn)∗(ι∗Y (Rif∗(F )))) = (gn)∗(j∗n(Rif∗(F )))

We also have natural morphisms

(5.10) (jn)∗(Rif∗(F )) // Ri(fn)∗(ι∗n(F ))

Consequently we get, for every i and n, natural maps

(5.11) Rif∗(ι∗X(F )) −→ (gn)∗(R

i(fn)∗(F ))

which eventually gives a map

(5.12) Rif∗(ι∗X(F )) −→ lim←−(gn)∗(R

i(fn)∗(F )).

The Theorem of Formal functions is the following:

Theorem 5.6. (Theorem of Formal Functions) If the morphism f : X → Y is proper, thenthe morphism in 5.12 is an isomorphism.

5.1. The Grothendieck algebraization Theorem. . We will now prove a theorem whichis the analogue of the Chow Theorem on analytic subvarieties of the projective space in thecontest of formal geometry. This theorem is usually applied in deformation theory.

We begin with a Theorem which is an analogue of Serre G.A.G.A. in formal geometry. Itis often called Formal G.A.G. A.

We wuppose that A and I is an ideal of ti. We suppose that A is complete with respectto the I–adic topology: the natural morphism A → A := lim←−A/I

n is an isomorphism. Iff : X → Spec(A) is a A–scheme, for every positive integer n, consider the cartesian diagram

(5.13) Xn := X ×A Spec(A/In) //

fn��

X

f��

Spec(A/In)jn // Spec(A).

Thus we get a formal scheme X := lim←−Xn and a cartesian diagram

(5.14) XιX //

f��

X

f��

Spf(A)j // Spec(A).

Theorem 5.7. (Grothendieck Existence Theorem) Suppose, under the hypothesis above, thatthe morphism f : X → Spec(A) is proper, then the natural functor

(5.15) ι∗X(·) : Coh(X) −→ Coh(X)

18 CARLO GASBARRI

is an equivalence of categories.

Proof. We will prove the Theorem under the hypothesis that f is projective.We begin by proving that ι∗X(·) is fully faithful: Let F and G be two coherent sheaves on

X. We want to prove that the natural morphism

(5.16) Hom(F,G) −→ Hom(ι∗X(F ); ι∗X(G))

is an isomorphism.– By flatness, we have

(5.17) j∗(f∗(H om(F,G))) ' f∗(ι∗X(H om(F,G))).

Moreover H om(ι∗X(F ), ι∗X(G)) ' ι∗X(H om(F,G)).– Since Spec(A) is affine, we have Hom(F,G) = f∗(H om(F,G)).– Since Spf(A) is an affine formal scheme, we have

(5.18) Hom(ι∗X(F ), ι∗X(G)) = f∗((H om(ι∗X(F ), ι∗X(G)))).

Consequently, gathering all together, we obtain

(5.19) j∗(Hom(F,G)) ' Hom(ι∗X(F ), ι∗X(G)).

– Since A is I–adically complete and Hom(F,G) is of finite type (because f is proper),

we have isomorphisms Hom(F,G) ' Hom(F,G) = j∗(Hom(F,G)). The fully faithfulnessfollows.

In order to conclude, we need to prove that if G is a coherent sheaf on X then there existsa coherent sheaf G on X such that G ' ι∗X(G).

Since we are supposing that f : X → Spec(S) is projective, we may suppose the existence

of a relatively ample line bundle OX(1). We will denote by O(1) its pull back to X.A key step of the proof is the following Lemma:

Lemma 5.8. Under the hypothesis of Theorem 5.7, for every coherent sheaf G on X we canfind a surjective map

(5.20)r⊕i=1

O(di) −→ G→ 0.

Let’s show how Lemma 5.8 implies Theorem 5.7: Lemma 5.8 implies that we can find anexact sequence

(5.21)s⊕j=1

O(`j)h−→

r⊕i=1

O(di) −→ G→ 0.

Since ι∗X(·) is fully faithful, we can find a morphism h :⊕s

j=1 O(`j) →⊕r

i=1 O(di) on X

such that ι∗X(h) = h.

Let G := Coker(h). We have that ι∗X(G) = G and the conclusion of the Theorem follows.�


Proof. (Of Lemma 5.8) In order to prove that such a surjective map exists, we need thefollowing general Lemma:

Lemma 5.9. Let f : X → Y be a proper morphisms. Let S = ⊕iSi be a graded OY algebragenerated by S1 and let M = ⊕j∈ZMj be a coherent f ∗(S) module of finite type. Then for allinteger q the module

(5.22) Rqf∗(M) := ⊕j∈ZRqf∗(Mj)

is a S- module of finite type and there exists n0 such that, for every n ≥ n0 we have

(5.23) Rqf∗(Mn) = Sn−n0 ·Rqf∗(Mn0).

Proof. (Sketch) Define Y := Spec(S)→ Y and X := Spec(f ∗(S))→ X. We have a cartesiandiagram

(5.24) X //

f��

X

f

��Y // Y.

and f is proper by base change. The f ∗(S) module M gives rise to a OX– module which

is coherent because M is of finite type. Since f is proper, Rqf∗(M) is a coherent OY = Smodule of finite type. Thus the first part of the proposition. The second part is a standardfact on moduli of finite type over graded algebras. �

Let’s go back to the proof of the Lemma. Denote by ιn : Xn → X the inclusion and byfn : Xn → Spec(A/In) the natural map. The morphism fn is projective and the line bundle

On(1) := ι∗n(O(1)) is ample over Xn (because Xn is also a subscheme of X).

Let G be a coherent module over X. Denote by Gn its restriction to Xn.Consider the A algebra S = ⊕iI i/I i+1. It is a graded A/I algebra of finite type generated

by S1. On X1 we define the f ∗1 (S) module of finite type M = ⊕j∈NIj · G/Ij+1 · G. Observethat M1 is just G1.

As in the proof of Lemma 5.9, consider the diagram

(5.25) X1g //

f ��

X

f1

��˜Spec(A/I) // Spec(A/I).

Since O1(1) is ample on X1, the line bundle g∗(O1(1)) is relatively ample with respect to

f . Consequently, we can find a integer d0 such that, for every d ≥ d0 and q > 0, we have

(5.26) Rqf∗(M) = 0.

20 CARLO GASBARRI

Which means that we can find a positive integer d0, such that, for every positive integerd ≥ d0, for q > 0 and for every positive integer j, we have

(5.27) Rq(f1)∗(Ij ·G/Ij+1 ·G⊗ O1(d)) = 0.

and since Spec(A/I) is affine, we get that for every positive integer d ≥ d0, for q > 0 and forevery positive integer j, we have

(5.28) Hq(X1, Ij ·G/Ij+1 ·G⊗ O1(d)) = 0.

Again by ampleness of O1(1) we may suppose that, for d ≥ d0 we have that G1(d) is generatedby global sections.

Consider now the exact sequence

(5.29) 0 −→ Ij ·G/Ij+1 ·G⊗ O1(d) −→ Gj+1 −→ Gj −→ 0

Taking global sections of it we get

(5.30) 0 // H0(X1; Ij ·G/Ij+1 ·G⊗ O1(d)) // H0(Xj+1Gj+1) //

// H0(Xj;Gj) // H1(X1; Ij ·G/Ij+1 ·G⊗ O1(d))

But, since we just proved that the last term of this exact sequence vanishes, a direct applicationof the Theorem of formal functions gives a surjective map

(5.31) H0(X, G(d)) −→ H0(X1, G1(d)) −→ 0.

Since G1(d) is generated by global sections and because the surjective morphism above, wehave a commutative diagram

Take a finitely generated A module V of global sections of G(d) which generate G1 (theyexist because G1(d) is generated by global sections) and we have the surjection above.

(5.32) V ⊗ OX

β //

α%%

G(d)

��G1(d).

The map α is surjective and consequently, by Nakayama Lemma, the map β is also surjective.This means that G(d) is generated by global sections. Consequently the map

(5.33) V ⊗ O(−d) −→ G

is surjective. Which ends the proof of the Lemma. �

Theorem 5.7 and also Lemma 5.8 have many interesting corollaries:

Corollary 5.10. Under the hypothesis of Theorem 5.7, every formal closed subscheme of Xis algebraizable.


Corollary above may be seen as an analogue of Chow Theorem in the contest of formalgeometry.

Proof. let Z be such a scheme and IZ be its ideal sheaf. Since IZ is coherent, there exists an

ideal sheaf IZ such that ιX(IZ) = IZ . Consequently Z is the formal completion of Z and theconclusion follows. �

If we apply the corollary to the graph of a formal function, we find

Corollary 5.11. Under the hypothesis of Theorem 5.7 suppose that X and Y are two schemesproper over A. Let g : X → Y be a formal map. Then f is algebraizable: there is an algebraicmap f : X → Y the completion of which is f .

The last consequence we’d like to mention is a refinement of Corollary 5.10 and it can beseen as an ”algebraization theorem”

Theorem 5.12. Let A be a I–adically complete ring. Suppose that f : X → Spf(A) is a

proper formal scheme having as ideal of definition f ∗(I). Suppose that L is a line bundle onL the restriction of which to X1 is ample. Then X is algebraizable.

Proof. As in the proof of Lemma 5.8, we can find a finitely generated free A–module V anda surjective map,

(5.34) V ⊗ OX −→ L⊗d

for d big enough. Suppose that the rank of V is N + 1 and Let PNA be the completion ofh : PNA → Spec(A) along h∗(I). For every n, the surjection above give rise to an closedinclusion of Xn inside PNn . Consequently, X can be realized as a closed formal subscheme of

PNA . The conclusion follows from Corollary 5.10. �

5.2. A formal scheme which is not algebraizable. This example is taken from [3] (butdescribed in a slightly different way). We will now describe an example of formal schemewhich is not algebraizable. We recall a formal scheme X is algebraizable if there is a schemeX and closed subscheme Y such that X ' XY .

In this part we will suppose that the underlying field is algebraically closed of characteristiczero (this hypothesis is not necessary, but it will simplify the proof).

Before we describe the example, we need to recall some facts of algebraic geometry whichwill be needed.

5.2.1. Etale covering of formal schemes. Let X be a scheme. Suppose that there exists aideal sheaf J ⊂ OX such that J2 = 0. Let X0 be the closed subscheme of X associated to J .

22 CARLO GASBARRI

Suppose that f0 : Y0 → X0 is a finite etale covering of X0. Then we can find a unique finiteetale covering f : Y → X ”extending” f . This means that there exists a cartesian diagram

(5.35) Y0//

f0��

Y

f1��

X0// X.

This easily implies

Theorem 5.13. Let X be a formal scheme with ideal of definition I and X0 support scheme.Then every finite etale covering f0 : Y0 → X0 extends to a finite etale covering f : Y → X :the diagram

(5.36) Y0//

f0��

Y

f1��

X0// X .

is cartesian. Consequently if Et(X ) (resp. Et(X0)) is the category of the finite etale coveringof X (resp. of X0), the natural restriction functor Et(X ) → Et(X0) is an equivalence ofcategories.

The second fact we need is the:

5.2.2. Nagata compactification Theorem. We know that an affine variety can be seen as anopen set of a projective variety. Nagata compactification Theorem is a generalisation of thisfact to any scheme. We state it in the particular case of schemes over a field.

Theorem 5.14. (Nagata Compactifcation Theorem) Let X be a scheme of finite type over afield. Then there exists a proper scheme X and a open inclusion X ↪→ X.

The third fact is:

5.2.3. Kawamata-Viewheg Vanishing Theorem. We will need the following:

Theorem 5.15. Let X be a smooth projective surface over a field of characteristic zero. LetL be a nef and big line bundle on it. Then

(5.37) H1(X,L−1) = {0}.

We recall that a line bundle L on a surface X is said to be nef if, for every curve C on X,we have deg(L|C) ≥ 0. The line bundle L is said to be if h0(X,Ln) ∼ n2.


5.2.4. The example. . We now describe the example:Let C ↪→ P2 be a smooth projective curve of positive genus and degree d > 1. Let g : D → C

be a non trivial etale covering.Let P2

C be the completion of P2 along C.

By Theorem 5.13 we can find a formal scheme X and an etale covening g : X → P2C

extending the morphism g.We now prove that X is not algebraizable:Suppose it is algebraizable, then there is a scheme of dimension two X with a Weil divisor

D such that X ' XD.– Since X is regular, X must be regular in a neighbourhood of D. Thus D is a Cartier

divisor on it.– Using Nagata compactification Theorem 5.14, we may suppose that X is proper.– The normal bundle of C in P2, and consequently in P2

C is O(d)|C .– Since g is etale, the normal bundle of D in X , and consequently in X, is g∗(O(d)|C).

This implies, in particular, that OX(D) is a nef and big line bundle on X.– Since X has a nef and big line bundle which is base point free near C1, we may suppose

that X is projective.Let m ≥ 2. Then we have an exact sequence

(5.38)0→ H0(X,O((1−m)D)) −→ H0(X,O(D)) −→ H0(Dm,O(D)) −→ H1(X,O((1−m)D)).

– Since O(D) is nef and big, by Theorem 5.15 we have that H0(X,O((1 − m)D)) =H1(X,O((1 − m)D)) = {0}. Consequently we have that, for every m ≥ 2 the naturalrestriction map

(5.39) H0(X,O(D)) −→ H0(Dm,O(D))

is an isomorphism.– Because of the isomorphis above, by the The theorem of formal functions 5.6, the restric-

tion map

(5.40) H0(X,O(D)) −→ H0(X ,O(D))

is an isomorphism.– The morphism X → P2

C → P2 is given, by functoriality, by a subspace of dimensionthree V of H0(X , g∗(O(1))) and a surjective map

(5.41) V ⊗ OX −→ g∗(O(1))

–The linear system Symd(V ) ↪→ H0(X ,O(D)) defines a morphism

(5.42) f : X −→ P(Symd(V ))

which factorises through the Veronese embedding vd : P2 → P(Symd(V )).– The linear system space Symd(V ) defines also, via the isomorphism 5.40 a linear system

on X which is without fixed points around D.

24 CARLO GASBARRI

– The rational map ϕ : X 99K P(Sym(V )) associated to the linear system Symd(V ) isdefined in a neighbourhood of D. Consequently, solving the indeterminacies of it, we maysuppose that it is defined everywhere.

– Consequently we have a commutative diagram

(5.43) X

%%

// X

ϕ

��P(Sym(V ))

–Since the morphism 5.42 factorisez through P2, we eventually get that the morphism ϕdefines a morphism g : X → P2 which extends f . In particular g is an algebraic morphism toP2 which is etale near C.

– The branch locus of g must be a divisor which should cut C somewhere. But, since it isetale around C, we eventually find a contradiction.

– Consequently such a X cannot exist.

References

[1] Griffiths, Phillip, Topics in algebraic and analytic geometry Written and revised by John AdamsPrinceton University Press, Princeton, NJ, [2015]. vi+219 pp

[2] Grothendieck, Alexander Cohomologie locale des faisceaux coherents et theoremes de Lefschetz lo-caux et globaux. Fasc. I: Exposes 1–8; Fasc. II: Exposes 9–13. (French) Seminaire de Geometrie Algebrique

1962. Troisieme edition, corrigee. Redige par un groupe d’auditeurs Institut des Hautes Etudes Scien-tifiques, Paris 1965 Fasc. I: iii+94 pp. (not consecutively paged); Fasc. II: i+104 pp.

[3] Hartshorne, Robin, Ample subvarieties of algebraic varieties. Notes written in collaboration with C.Musili. Lecture Notes in Mathematics, Vol. 156 Springer-Verlag, Berlin–New York 1970 xiv+256 pp.

[4] Hartshorne, Robin Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, NewYork-Heidelberg, 1977. xvi+496 pp.

[5] Illusie, Luc, Grothendieck’s existence theorem in formal geometry. With a letter (in French) of Jean-Pierre Serre. Math. Surveys Monogr., 123, Fundamental algebraic geometry, 179–233, Amer. Math. Soc.,Providence, RI, 2005.

[6] Milnor, J. Morse theory. Based on lecture notes by M. Spivak and R. Wells. Annals of MathematicsStudies, No. 51 Princeton University Press, Princeton, N.J. 1963 vi+153 pp.

Carlo Gasbarri, IRMA, UMR 7501 7 rue Rene-Descartes 67084 Strasbourg CedexEmail address: [email protected]

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INTRODUCTION TO ANALYTIC AND FORMAL GEOMETRY · INTRODUCTION TO ANALYTIC AND FORMAL GEOMETRY 3: by this we mean that there is a scheme the completion of which is the formal scheme