User rex - MathOverflow

Structure theorem for etale maps

2013-02-18T21:19:06Z

I have been trying to understand the structure theorem for etale maps for rings. Let $A\to B$ be a local homomorphism of Noetherian local rings which is etale. Then the structure theorem says that $B=A[T]_g/p(T)$, where $g\in A[T]$ is such that $p'(T)$ is a unit in $B$.

This theorem is given as Theorem 3.14 in Milne's book on etale cohomology. The way the proof proceeds, it seems that the degree of the polynomial $p$ is equal to field extension degree $[k(y):k(x)]$, where $y$ denotes the maximal ideal of $B$ and $x$ denotes the maximal ideal of $A$. This seems to be wrong since if we take $A$ and $B$ to be localizations of finitely generated algebras over an algebraically closed field, then this extension degree will always be 1, which would mean that $B$ is isomorphic to $A$.

Smoothness of solution to a PDE

2012-10-03T17:48:30Z

Let $X$ be a Riemann surface and let $E$ be a smooth complex vector bundle on $X$ with a connection $D$. We can write the connection $D$ as the sum $D'+D''$ where $D'$ is the (1,0) part and $D''$ is the (0,1) part of the connection. The integrability theorem for holomorphic structures says that if $D''\circ D''=0$, then $E$ admits a holomorphic structure such that $D''=\bar{\partial}$. I am trying to understand the proof of this statement given in the book by Donaldson and Kronheimer, Geometry of 4 manifolds, section 2.2.2, and I am facing problems understanding the following.

Given a smooth function $\theta:\mathbb{C}\to \mathbb{C}$ which is compactly supported in a small neighborhood around 0, define the operator $L$ by

$$ L(\theta)(w):=-\frac{1}{2\pi i}\int_{\mathbb{C}}\frac{\theta(z)}{z-w}d\mu_z$$

Then we have

$$\frac{\partial}{\partial \bar{z}}L(\theta)=\theta$$

The authors now (after replacing $\theta$ by another function $\alpha$ which is again compactly supported and with small infinity norm) find a function $f\in L^{\infty}(\mathbb{C})$ which satisfies $f=L(\alpha + f\alpha)$ and claim that $f$ is smooth. The reason they give is that, "elliptic regularity for the $\bar{\partial}$ operator implies that any bounded solution of the equation $f=L(\alpha + f\alpha)$ is smooth".

In Warner's book on Lie Groups, in the last chapter, one has the following statement for periodic elliptic operators :

Let $T$ be a periodic elliptic operator and let $u\in H_{-\infty}$ and $v\in H_{\infty}$ such that $Tu=v$, then $u\in H_{\infty}$.

In the above, $H_n$ are the Sobolev spaces. The regularity theorem for compact manifolds is then deduced by reduction to the periodic case.

I am not able to see how to apply the regularity theorem since I do not see any equation of the type $Tu=v$ in the present setup. Could explain to me how to fill in the details.

Varieties dominated by products of curves

2012-06-04T13:41:35Z

Let $X$ be an irreducible smooth projective variety of dimension $d$. Do there exist irreducible smooth projective curves $C_1, C_2,\ldots, C_d$, an open subset $U\subset C_1\times C_2\times\ldots\times C_d$ and a dominant morphism $f:U\to X$.

Square integrable functions on $\Gamma \backslash G$

2012-01-21T11:02:10Z

I am trying to understand proposition 2.1.6 in Bump's book Automorphic forms and Representations.

Let $G=GL(2,\mathbb{R})^+$ and define $G_1=G/Z^+$, where $Z^+$ denotes the center, and define $\Gamma=SL(2,\mathbb{Z})$. Let $\chi: \Gamma \to S^1$ be a group homomorphism. He writes, "Let $L^2(\Gamma\backslash G, \chi)$ be the space of measurable functions satisfying $$f(\gamma g u)=\chi(\gamma)f(g)\qquad \gamma\in\Gamma, u\in Z^+,g\in G$$ that are square integrable with respect to Haar measure on $G_1$."

Clearly no function which is periodic with respect to $\Gamma$ can be square integrable on $G_1$.

Can someone please explain to me what the right definition should be.

Coherent Sheaves on Noetherian schemes

2012-01-15T15:46:37Z

Let $X$ be a Noetherian scheme (in particular, we assume that it has only finitely many irreducible components). Is it true that for any open set $U$, the ring $\Gamma(U, \mathscr{O}_X)$ is a Noetherian ring. Let $\mathscr{F}$ be a coherent sheaf on $X$. Is it true that for any open set $U$, $\Gamma(U,\mathscr{F})$ is a finitely generated $\Gamma(U,\mathscr{O}_X)$ module.

Lie bracket of Invariant Vector fields

2011-02-17T01:07:04Z

Let $G$ be a Lie group and let $\xi.\eta$ be left invariant vector fields. We can now construct right invariant vector fields $X_\xi$ and $X_\eta$ by defining $X_\xi(e)=\xi(e)$ and $X_\eta(e)=\eta(e)$. For $GL_n$, it is true that $[X_\xi,X_\eta]=X_{[\eta,\xi]}$. Is it true for any Lie group?

Non degenerate pairing on Neron Severi Group

2011-10-30T10:59:14Z

Let $X$ be a smooth projective variety over an algebraically closed field $k$. Let $NS(X)$ denote the group $[Pic(X)/Pic^a(X)]\otimes_{\mathbb{Z}}\mathbb{Q}$, where $Pic^a(X)$ denotes the divisors which are algebraically equivalent to 0. If $X$ is a surface, then the pairing on $NS(X)$ given by $(D,D')\mapsto D\cdot D'$ is non-degenerate.

For higher dimensions, let $\Theta$ denote an ample divisor on $X$. Let $d$ be the dimension of $X$. Is it true that the pairing on $NS(X)$ given by $(D,D')\mapsto D\cdot D'\cdot \Theta^{d-2}$ is non-degenerate.

Is there a good reference for the above result.

Reference for Numerical vs Homological equivalence

2011-10-28T11:53:32Z

I would like to know why for a smooth projective variety $X$ over an algebraically closed field $k$, numerical and homological equivalence coincide for divisors. Here by homological equivalence I mean that we have chosen a Weil cohomology theory with coefficients in a field $L$, in particular, there is no torsion.

What is a good reference for this.

Thanks.

Domain of Holomorphy

2011-09-28T11:28:26Z

I found the following definition of domain of holomorphy in several places.

Def1: A connected open set $\Omega$ in the n-dimensional complex space ${\mathbb{C}}^n$ is called a domain of holomorphy if there do not exist non-empty open sets $U \subset \Omega$ and $V \subset {\mathbb{C}}^n$ where $V$ is connected, $V \not\subset \Omega$ and $U \subset \Omega \cap V$ such that for every holomorphic function $f$ on $\Omega$ there exists a holomorphic function $g$ on $V$ with $f = g$ on $U$.

From what I understand, intuitively speaking, $\Omega$ is a domain of holomorphy if we can find a function $g$ which is holomorphic on $\Omega$ such that it cannot be extended beyond the boundary of $\Omega$. Naively thinking, I would have written down the following definition for domain of holomorphy.

Def2: A connected open set $\Omega\subset \mathbb{C}^n$ is a domain of holomorphy if there is a $g$ which is homolorphic on $\Omega$ such for that any open $V\subset \mathbb{C}^n$ with $V\cap \partial\Omega\neq\phi$ there is no holomorphic function $F$ on $V$ with $F\vert_{V\cap\Omega}=g\vert_{V\cap\Omega}$.

Could someone please explain to me the need for the more complicated definition 1.

Hilbert Polynomial using non ample line bundles

2011-09-21T10:04:47Z

In his book on abelian varieties, in the Appendix to section 6, Mumford says that if $X$ is any complete variety, $F$ is a coherent sheaf and $L$ an invertible sheaf, then the function defined as $$P(F,L,n)=\chi(F\otimes L^n)$$ is a polynomial in $n$. This is an exercise in Hartshorne, Chapter 3, 5.2, for a very ample line bundle.

I can see why the above is true for $L$ a very ample line bundle. In fact the argument is by induction on the dimension of the support of $F$. We write a short exact sequence

$$0\to R(n)\to F(n-1)\to F(n)\to F\vert_H(n)\to 0$$ Note that $R$ is supported on a proper subvariety of $X$ since at the generic point $F(-1)=F$. Taking Euler characteristic, we get the result.

The above proof does not work in the ample case since an ample line bundle may not have any global sections (what is an example of this??).

How does one prove the result in the general case?

Answer by Rex for Symmetric powers of a curve = projective bundle over Jacobian, and the relative version thereof

2011-06-24T13:31:31Z

One way to see that there is a vector bundle $E$ over $J(C)$ with $C^{(n)}\cong \mathbb{P}(E)$ is using semi continuity. Consider the closed immersion $C^{(n-1)}\hookrightarrow C^{(n)}$. This is a divisor on a smooth variety and so corresponds to a line bundle $L$. We take the push forward $u_*(L)$ where $u:C^{(n)}\to J(C)$. Now using semi-continuity and Riemann-Roch, for $n$ large this is a vector bundle $E$. In order to give a morphism $\phi:C^{(n)}\to \mathbb{P}(E)$, it suffices to check that the natural map $u^*u_*L\to L$ is surjective. This is easy to see by checking it over fibers of $u$. Also it is easy to check that $\phi$ is an isomorphism and that the pullback of $\mathscr{O}(1)$ is $C^{(n-1)}$, by looking at the fibers of $u$.

But it is not clear to me why $E=\text{some line bundle}\otimes P_n$.

Dimension of Chow groups

2011-05-18T22:13:44Z

The integral Chow groups are infinite dimensional. Can we say something about their dimension, for example, how many elements are required to generate them? Or their vector space dimension after tensoring with $\mathbb{Q}$. What is known in this direction? I recall hearing some statements whose proof uses the theory of Chow varieties.

Also, is there a good reference for Chow varieties. I am looking for something which will assume familiarity with algebraic geometry at the level of Hartshorne and which uses that language. Since I am reading Vistoli's notes on stacks, reference to some construction using the language of stacks might also be helpful.

EDIT: Removed the remarks on countable generation.

Rational Equivalence is the finest

2011-05-17T17:06:35Z

In a lot of places it is mentioned that Rational equivalence of cycles on a smooth projective variety is the finest adequate equivalence relation. Assuming that it is an adequate equivalence relation, how to see that this is the finest? Any references for this?

Involution on Hyperelliptic curves and their Jacobians

2011-05-12T21:25:29Z

Let $X$ be a hyperelliptic curve and let $i:X\to X$ denote the hyperelliptic involution. Once we fix a point $x_0\in X$ we get the Abel-Jacobi map $AJ:X\to J$ where $J$ denotes the Jacobian variety. Now the Jacobian is also equipped with an involution, namely $x\mapsto x^{-1}$. Is it possible to choose the base point $x_0$ in such a way that the restriction of the involution on the Jacobian is the involution on $X$.

The Mukai Fourier Transform and Derived Categories

2011-04-28T10:46:46Z

I was trying to understand the Mukai Fourier transform from his paper : Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves, Nagoya Math Journal, 1981.

I am not very familiar with Derived categories and even less familiar with $D^-(X)$ and so this question, as Mukai works with $D^-(X)$. My knowledge of derived categories has been obtained from Iversen's book Cohomology of Sheaves.

I usually think of the bounded below derived category $D^+(X)$ by an analogy with injective resolutions modulo quasi isomorphism/homotopy equivalence. For example, $f:X\to Y$ is a morphism of algebraic varieties, then we can define the higher direct images $R^if_*F$ for any sheaf $F$ on $X$. If we have a short exact sequence $0\to F\to G\to H\to 0$, then we can find injective resolutions which are term wise split and we can get long exact sequences of higher direct images. Now instead of thinking of each of the $R^if_*F$ individually we could just take the whole complex $f_*I^\bullet$ and call this $Rf_*I^\bullet$, where $0\to F\to I\bullet$ is an injective resolution.

Since I have mostly seen only injective resolutions in action, could someone explain to me the motivation for working with $D^-(X)$, where I guess one would have to work with projective resolutions, which may not always exist.

For my purpose, I convinced myself about Proposition 1.3 in the paper of Mukai by working with $D^+(X)$ as in the case of interest (abelian varieties) the projection morphisms are flat and we are tensoring with a flat sheaf.

Thanks in advance.

$k$ structures on $K$ vector spaces

2011-04-21T13:15:31Z

The following statement is made in Borel's Linear Algebraic groups, section 11 on $k$ structures.

Let $V$ and $W$ be $K$ vector spaces with $k$ structures. If $f:V\to W$ is $K$ linear, then $f$ is said to be defined over $k$ if $f(V_k)\subset W_k$ and these are elements of $Hom_K(V,W)_k\subset Hom_K(V,W)$. This is a $k$ structure if $W$ is finite dimensional.

The author seems to be making no assumption on the dimension of $V$ (which is the source of my problem). If we allow $V$ to be infinite dimensional then it seems to me that it is incorrect that $Hom_K(V,W)_k$ is a $k$ structure on $Hom_K(V,W)$ as is given by the following example. Let $V_k=\oplus_{i\geq0} ke_i$ and take $K$ to be an extension which is not a finite $k$ vector space, and define $f\in Hom_K(V,K)$ by choosing $f(e_i)$ which are linearly independent over $k$. We will never be able to write such an element as $\sum f_i\otimes \alpha_i$ with $f_i\in Hom_K(V,K)_k$.

So my question is : Do we need to assume both $V,W$ to be finite dimensional $K$ vector spaces for $Hom_K(V,W)$ to have a $k$ structure?

Also, what are the other references for $k$ structures and rationality properties?

Representation theory of $S_n$

2011-04-19T05:58:19Z

I need to understand the representation theory of $S_n$ (symmetric group on $n$ letters) and so could someone suggest a good reference for this. There is a similar question on mathoverflow here

http://mathoverflow.net/questions/2755/a-learning-roadmap-for-representation-theory

Most of the responses to the above question give references for representation theory of Lie groups. Also the usual reference Fulton and Harris has too many exercises (on which I don't want to spend too much time ) and I find it difficult to read.

Another reference which was suggested was Flag varieties by Lakshmibai and Brown. This seems to be a good reference, but are there any other references.

EDIT: By mistake I did not notice something in the above mentioned book and so some of my remarks are being edited. Sorry.

Extending holomorphic connections

2011-02-08T22:55:22Z

Let $D$ denote the disk $|z|<1$ in the complex plane and $U=D\0$(punctured disk). Define a holomorphic connection $\nabla$ on $\mathscr{O}_U$ by $\nabla(1)=\exp{(-1/z)}$. Does this extend to a logarithmic connection on $\mathscr{O}_D$, i.e. does this extend to $\nabla_D:\mathscr{O}_D\to \Omega_D(0)$?

More generally, suppose $\nabla$ is a holomorphic connection on $\mathscr{O}_U^{\oplus r}$, can we extend it to a logarithmic connection on $\mathscr{O}_D^{\oplus r}$.

A simple(possibly trivial) question about Grothendieck Topologies

2011-02-07T21:46:25Z

Let $C$ be the category of sets. Define coverings {$U_i\to U$} to be jointly surjective maps, i.e. $U$ is the union of the images of $U_i$. Then if $F$ is a sheaf of sets on $C$, is it clear that $F(\emptyset)$ is a set consisting of exactly one point. $\emptyset$ is the empty set.

The Dual Abelian Variety

2011-01-16T22:12:15Z

Let $A$ be an abelian variety defined over an algebraically closed field, say over $\mathbb{C}$. There is a dual abelian variety $\hat{A}$, along with a Poincare line bundle $L$ on $A\times \hat{A}$. Is there any relation between $\widehat {A\times A} $ and $\hat{A}\times \hat{A}$, for instance are they isogenous. What happens when $A$ is principally polarized, can we say relate the Poincare bundles in this case.

Geometric Realization of a Simplicial Category

2011-01-05T00:32:02Z

Let $S:\varDelta^{op}\to (cat)$ be a functor where the category on the right is the category whose objects are categories with cofibrations and morphisms are exact functors(from Waldhausen's paper, Algebraic K-Theory of spaces). Waldhausen talks of the geometric realization of such a simplicial category. In the case of a simplicial set, I know how to construct the geometric realization. However, in this case, $S_n$ is a category. It is not clear to me if he is assuming that this is a small category for each n, in which case we could proceed to construct the geometric realization as in the case of a simplicial set. If each $S_n$ is not a small category, then is it still possible to define the geometric realization of $S$?

Curvature 0 and involutive horizontal distributions

2011-01-04T20:40:24Z

I am trying to check why curvature 0 implies that the horizontal distribution is involutive.

Let $\pi:P\to U$ be a principal $G:=GL_n$ bundle. Assume that $P$ is trivial and $\pi$ admits a section. Thus, $P\cong U\times G$. A connection on $P$ is a $G$ equivariant splitting of the short exact sequence of bundles $0\to P\times\mathfrak{g} \to TU\oplus TG\to \pi^*TU\to 0$. Let us denote this splitting by $s$. If $(v,u)$ is a tangent vector at the point $(x,g)$, then $s(v,u)=w_x(v)+(R_{g^{-1}})_*(u)$, where $w\in\Gamma(X,\Omega_U\otimes\mathfrak{g})$.

Let $x_1,x_2,...x_n$ be coordinates on $U$ and consider vector fields $X_i$ corresponding to them. Use these to construct horizontal $G$ invariant vector fields on $P$ which are given by $((X_i)_x, -w_x((X_i)_x))$ at the point $(x,Id)$. This should give an involutive distribution on $P$.

Using that $[X_1,X_2]=0$, when we compute the Lie bracket, we get $[(X_1,-w(X_1)),(X_2,-w(X_2))]=-[w(X_1),X_2]-[X_1,w(X_2)]+[w(X_1),w(X_2)]$. =$X_2(w(X_1))-X_1(w(X_2))+[w(x_1),w(X_2)]$. We need to show that this is 0.

The condition that curvature is 0 is given by $dw+w\wedge w=0$, i.e. $dw_{ij}+\sum_{k=0}^{n}w_{ik}\wedge w_{kj}=0$. This means that $dw(X_1,X_2)+[w(x_1),w(X_2)]=0$, which is same as saying that $X_1(w(X_2))-X_2(w(X_1))+[w(x_1),w(X_2)]=0$

There seems to be some mismatch.

Flat connections on Bundles of degree 0 on a compact Riemann surface

2010-12-20T20:20:38Z

Let $\pi:E\to X$ be a holomorphic vector bundle of degree 0 over a compact Riemann surface $X$. Why does $E$ admit a flat connection. I could work this out in the case of line bundles, where one starts with the natural logarithmic connection on $\mathscr{O}(\sum_{i=0}^{k}n_iP_i)$ (here $\sum_{i=0}^{n}n_i=0$) and modify this by an element of $H^0(X,\Omega(\sum_{i=0}^{k}P_i))$ to get a holomorphic connection, which gives a flat connection.

How to prove this for a vector bundle of rank > 1?

EDIT: Thanks Richard. In the above let $E$ be an indecomposable vector bundle. Could you give a good reference for this result?

A question on base change

2010-12-16T12:23:32Z

Let $X$ be a regular integral projective scheme of dimension 1 over a field $k$ (not algebraically closed). Further, assume $X$ satisfies $dim_kH^0(X,\mathscr{O}_X)=1$. Let $\bar{X}$ denote the fibered product $X\times_k\bar{k}$. Then is it true that $\bar{X}$ is integral?

Section of a Ruled surfaces

2010-12-13T00:19:51Z

Hartshorne in his chapter on surfaces defines a ruled surface(over an algebraically closed field) to be a smooth projective surface $X$ together with a surjective morphism $\pi:X\to C$, $C$ a smooth curve, such that the fiber over each point $y\in C$, call it $X_y$, is isomorphic to $\mathbb{P}^1$, and such that $\pi$ admits a section.

He then says that the existence of a section follows from Tsen's theorem under the earlier hypothesis.

When he says every point, does he include the generic point also? If yes, then the section will exist by defining it to be anything at the generic point and then using that a morphism from an open subset of a curve to a proper variety extends to the whole curve.

If by points he means only closed points, then how does the existence of a section follow from Tsen's theorem?

Answer by Rex for Principal $G$-bundle and vector bundle associated to representation of $G$

2010-12-11T11:49:47Z

Do the construction usual construction (as in manifolds) to get a morphism of schemes $\mathbb{V}\to X$. Now use the exercise in Hartshorne, Chapter 2 section 5, 5.18 to get a locally free sheaf of finite rank.

Answer by Rex for Canonical bundle of compactifications

2010-12-11T02:57:46Z

Why should $K_{\bar{X}}=K_X+nD$? Consider the following counterexample.

Let $E$ be an elliptic curve and let $\bar{X}=E\times \mathbb{P}^1$. Then $K_{\bar{X}}=-2E\times 0$. Let $X$ be the open subset $\bar{X}\setminus (e\times \mathbb{P}^1\cup E\times 0)$. Then $K_X$ is trivial as we have deleted $E\times 0$. So if $K_{\bar{X}}=K_X+n(e\times \mathbb{P}^1\cup E\times 0)$, then we would have that $(-n-2)(E\times 0)$ is rationally equivalent to $n(e\times \mathbb{P}^1)$, which is not correct as the picard group of $E\times \mathbb{P}^1$ is isomorphic to $Pic(E)\oplus\mathbb{Z}\cdot p_2^*\mathscr{O}(1)$.

Dual of The Lie Bracket

2010-12-07T23:18:09Z

Given a smooth manifold U, we have a map $\wedge^2\Gamma(U,TU)\to \Gamma(U,TU)$ given by $X\wedge Y\mapsto [X,Y]$, where $TU$ denotes the tangent bundle. Is it possible to describe the map $\Gamma(U,T^*U)\to \Gamma(U,\wedge^2 T^*U)$ corresponding to this map.

Various definitions of Connections on bundles

2010-12-08T21:22:41Z

Let $X$ be a smooth manifold and suppose I have a smooth vector bundle $E\to X$ which admits a connection $D$. Then on an open set $U\subset X$ where $E$ is trivial, once I choose a frame, say $e_1,...e_n$, I get a connection matrix by the rule $D(e_i)=\sum_{j=1}^{n}e_j\otimes\omega_{ji}^U$ (a column vector of 1 forms). If $V \subset X$ is another open set where $E$ is trivial and I choose a frame over $V$, say $f_1,...f_n$, then similarly I get $\omega^V$. Let $e_i=\sum_{j=1}^{n}g_{ji}f_j$. Then we have $\omega^V=g\omega^U g^{-1}-dgg^{-1}$.

On the other hand, we can form the associated principal $GL_n$ bundle $P\stackrel{\pi}{\to} X$, with $GL_n$ acting on the right. A connection on $P$ is a $GL_n$ invariant splitting of $\pi^*\Omega_X\to \Omega_P$. Locally, after we choose trivialisations for $P$ using the frames for $E$ chosen earlier, this becomes equivalent to giving a $GL_n$ equivariant splitting of $\Omega_U\to \Omega_U\oplus\Omega_G$, which is equivalent to giving a map $\Omega_{G,e}\to\Omega_{U,x}$ for each point $x\in U$, this gives the $\omega^U$. If $x\in U\cap V$, then the point $(x,A)$ in $U\times G$ is identified with the point $(x,gA)$ in $V\times G$. The gluing gives the following relation $\omega^V=g\omega^U g^{-1}+dgg^{-1}$. This is different from the one above. Is that OK?

Various definitions of Connections on bundles-2

2010-12-09T19:52:03Z

Starting with a principal $GL_n$ bundle, we could form the vector bundle associated to a representation $GL_n\to GL(V)$. Could someone please explain how we get a connection on this vector bundle.

Comment by Rex

2013-02-19T03:38:54Z

@rghthndsd : If $k(x)$ is algebraically closed, then the degree will be 1. Even for the map $\mathbb{A}^1\to \mathbb{A}^1$, if the underlying field is algebraically closed, then the map at the level of residue fields has degree 1.

Comment by Rex

2013-02-19T03:36:14Z

Let $E\to \mathbb{P}^1$ be a degree 2 map over the complex numbers. Let $y\in E$ be a closed point which is unramified and let $x$ be the image of $y$. Then I would have thought that the ring homomorphism $\mathscr{O}_{\mathbb{P}^1,x}\to \mathscr{O}_{E,y}$ is etale. Both the local rings have residue field $\mathbb{C}$ and so the degree of the extension is 1, but clearly they are not isomorphic.

Comment by Rex

2013-02-19T03:32:28Z

@pranavk : If $A$ is local, then $A[T]_g/p(T)$ can be local for the following reason. If $A$ is local, then $A[T]/p(T)$ is semi local and then if we invert an element which is contained in all but one of the maximal ideals, then $A[T]_g/p(T)$ is local.

Comment by Rex

2012-10-03T18:25:31Z

Thanks a lot. That was quite easy, I wonder why I couldn't see that. Another proof of this theorem uses the Newlander-Nirenberg theorem. I have been trying to find an understandable and not so sophisticated source from where I could read a proof of Newlander-Nirenberg theorem, but in vain. If you know of some source, I would be grateful if you could bring it to my attention.

Comment by Rex

2012-06-05T18:55:22Z

This comment is not directly relevant to your question, but it is an interesting fact that every smooth proper surface is projective.

Comment by Rex

2012-06-05T18:51:21Z

@David: Thanks for pointing out that comment.

Comment by Rex

2012-06-05T18:48:35Z

@Ulrich: Thanks a lot

Comment by Rex

2012-01-20T05:09:34Z

Thank you all for your response and interest

Comment by Rex

2011-09-29T06:40:36Z

Very nice example!

Comment by Rex

2011-09-21T11:23:54Z

Consider any function $f:\mathbb{Z}\to \mathbb{Z}$. I can always write $f(n)=a_0(n)\cdot 1$, where 1 is the constant polynomial which only the value 1, and $a_0(n)=f(n)$. Does that mean that $f$ is a polynomial? No. In the answer you give $P(F\otimes A^n,B,i)$ is definitely a polynomial in $i$ whose coefficients depend on $n$. This is like the previous example. I hope that clarifies my point.

Comment by Rex

2011-09-21T11:18:23Z

A proof using Grothendieck-Riemann-Roch was pointed out to me some time ago. If $X$ is smooth projective, then we immediately reduce to the locally free sheaves case using a resolution by locally free sheaves. $\chi(E)=P(c_1,...,c_n)$ and $c_i(E\otimes L^n)$ can be written as a polynomial in the Chern classes of $E$ and the Chern classes of $L$.

Comment by Rex

2011-09-21T11:14:14Z

But why will the coefficients be independent of $n$. That is the problem.

Comment by Rex

2011-09-21T11:02:36Z

Ah! That's clever. But this gives that $P(F\otimes A^n,B,i)$ is a polynomial in $i$, and so it looks like $a_0(n)i^d+a_1(n)i^{d-1}+\cdots a_d(n)$. Here the coefficients depend on $n$ since we are taking $F\otimes A^n$. From this I don't see how to conclude that $P(F,L,n)$ is a polynomial.

Comment by Rex

2011-05-19T13:15:06Z

Thanks for pointing that out. It was really stupid of me.

Comment by Rex

2011-05-18T16:07:16Z

@Tom: Your comment is very useful. I think this is what I was looking for.