1
vote
0answers
16 views

Verlinde formula, moduli space vector bundle on genus 2,3 curves.

I'd like to prove "by hands" the Verlinde formula for moduli space of rank two semistable vector bundles with fixed determinant on a curve of genus two and three. For a curve of genus two and even ...
5
votes
1answer
61 views

Definition of multiplication in Grothendieck ring

Let $X$ be a smooth variety over an algebraically closed field $k$ of dimension $n$. Consider the Grothendieck Group $K(X)$ of coherent sheaves on $X$, i.e. the free abelian group generated by ...
0
votes
1answer
42 views

Manifolds as homology classes

I have found that a k-dimensional submanifold of a manifold M can be considered as a class in the homology group $H_{k}(M)$. Why ?
4
votes
0answers
113 views

Standard spaces with non-standard topology and around

The following series of questions gives me no rest. Let $\mathbb{\widetilde{R}}^n$ be $\mathbb{R}^n$ with Zariski topology, i.e. we say that $A\subset \mathbb{R}^n$ is closed if it is given by the ...
1
vote
0answers
22 views

References for the moduli space of conformal structures on a disk minus boundary points

In an article I'm reading, it is said that the moduli space of conformal structures on a disk minus $k+1$ boundary points is a ($k-2$)-dimensional manifold. I want to understand why and have done some ...
1
vote
3answers
114 views

Some practical questions on cohomology and the ring $\mathbf{Z}[x]/(x^2)$

So I know that the cohomology ring of $S^n$ is $\mathbf{Z}[x]/(x^2)$ with "$x$ in degree $n$"; if we ignore the grading then this ring fails to distinguish the spheres. What is actually meant by "in ...
3
votes
0answers
31 views

About the Chern class of infinite complex Grassmannian

I learned that any characteristic class of rank-$k$ complex vector bundles on paracompact spaces is determined bijectively by a cohomology class in $H^*(Gr_k^\infty(\mathbb C))$, the cohomology ring ...
0
votes
0answers
37 views

What is $H_1(A_\mathbb{C}^{top},\mathbb{Q})$

Let $A$ be an abelian variety defined over a number field. I have seen in a few papers the singular homology $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ being used. I read up on the singular homology but it ...
3
votes
0answers
32 views

Homology of non-singular projective algebraic variety

I am unsure whether or not the following claim is true or false and whether or not my proof works or not: Claim: Let $V \subset \mathbb{C}P^n$ be a complex $k$-dimensional, non-singular, projective ...
5
votes
0answers
41 views

Definition of Hodge structure: is torsion allowed?

I am trying to understand the definition of an integral Hodge structure. Apparently, for $X$ a compact Kahler manifold, $H^n(X,\mathbb R)$, the lattice $H^n(X,\mathbb Z)$ and the Hodge filtration give ...
1
vote
2answers
84 views

What would be the equivalent of the “gluing axiom” for a cosheaf

A sheaf is a presheaf $F$ such that for all $U$ and for all covering $\{U_i\}_{i\in I} $ of $U$, $F(U)$ is the equalizer $$ F(U) \overset{f}{\longrightarrow}\prod_{i\in I} F(U_i) ...
0
votes
1answer
57 views

Singular Chain of a Hyperplane.

I refer to the definitions of Hatcher's Algebraic Topology. Is it possible to model a hyperplane $H$ (or half of it) of $\mathbb{R}^n$ with a singular chain? And if - how would its boundary look like? ...
1
vote
2answers
102 views

“Equivalent” definitions of the gluing axioms

I tried to convince myself that the two caracterizations of a presheaf that is a sheaf given in wikipedia are equivalent but I couldn't. (F presheaf and notations from wiki) Let's take a simple ...
3
votes
1answer
54 views

Cohomology Calculation

A couple of days ago I asked this Question on calculating hypercohomology I tried a similar example for $(\mathbb{C}^*)^2$, and I have a couple of questions. Here is my calculation: We have a ...
3
votes
1answer
48 views

Homotopy groups relating to toric varieties

It is known that the toric variety $X_\Sigma$ of a simplicial fan $\Sigma$ can be constructed as a quotient $$X_\Sigma = \bigl(\mathbb C^N \setminus V(B)\bigr)/G.$$ Here $N$ is the number of rays, ...
0
votes
0answers
36 views

How to describe a homomorphism from a fundamental group to a finite group?

Let $S$ be a connected locally noetherian scheme with $s$ a geometric point of $S$. I read something like this: to give a surjective continuous homomorphism from $\pi_1(S, \bar{s})$ to a finite group ...
1
vote
1answer
45 views

Why notion of fundamental group is defined only over a connected scheme?

I went to different references on fundamental group on schemes. It is quite strange for me that the notion of fundamental group is only defined on connected scheme. Does anybody know why?
1
vote
0answers
45 views

A problem I met when reading Griffiths'Periods of Integrals on Algebraic Manifolds I

I am reading Griffiths' paper Periods of Integrals on Algebraic Manifolds I, and in section 2 I met some problems. I wish that I could get some help here. My problem is that I cannot understand ...
2
votes
0answers
66 views

pullback is injective on picard groups?

Let $E \rightarrow X$ be a rank two holomorphic vector bundle over a complex manifold $X$. I was recently asked on exam to prove an assertion that I believe boils down to showing that the pullback map ...
5
votes
1answer
156 views

Relations of Characteristic classes: Chern, Stiefel-Whitney, Pontryagin, Euler, Wu class. [closed]

We have a quite some characteristic classes: Chern class, Stiefel-Whitney class, Pontryagin class, Euler class, Wu class, etc. I wonder whether some math experts can use simple words and basic ...
4
votes
1answer
69 views

Degree of map using Poincare Duality

I have a very basic question. I want to compute the topological degree of the map $\phi: \mathbb{C}P^n \rightarrow \mathbb{C}P^n$ mapping $(z_0:\dots:z_n)$ to $(z_0^d:\dots:z_n^d)$, which, if life is ...
0
votes
0answers
42 views

Intuition of higher push-forward constant sheaves.

Let us consider the higher phsh-forward sheaves $R^if_*\mathbb{R}$ of a map $f:X\rightarrow Y$ between two compact manifolds. We assume that the fibers has a constant dimension, say $n$. I think ...
0
votes
1answer
69 views

The fundamental group of a scheme / variety

On Wikipedia (http://en.wikipedia.org/wiki/%C3%89tale_fundamental_group) it's been written In algebraic topology, the fundamental group π1(X,x) of a pointed topological space (X,x) is defined as ...
0
votes
0answers
64 views

Two books defined two Chern(Euler) classes yet differed by a negative sign, what's wrong?

In the book 'Principles of Algebraic Geometry' P141 and the book 'Differential Forms in Algebraic Topology' P72-73, they defined the Chern(Euler) classes of line bundles using the patching data ...
2
votes
0answers
31 views

How do I prove that $U(r) \to S(r,n) \to G(r,n)$ is a fibration?

$U(r)$ here is unitary group of $\mathbb{C}^n$, $S(r,n)$ is the Stiefel manifold of $r$-frames in $\mathbb{C}^n$ and $G(r,n)$ is the Grassmannian manifold of $r$-planes in $\mathbb{C}^n$. I've tried a ...
1
vote
0answers
62 views

Stereographic projection for a link/knot

I've been trying to understand the topological "link" between algebraic varieties and their associated knots/links and to this end I've been reading F. Kirwan's book, "Complex algebraic Curves". The ...
3
votes
0answers
45 views

The vanishing (?) cohomology of the Milnor fiber

Setup. Say we have a germ of a holomorphic function $f:(\mathbb C^{n+1},0)\to (\mathbb C,0)$ with a critical point at the origin. There is an $\epsilon>0$ small enough so that $f$ becomes a ...
4
votes
0answers
30 views

Isomorphism of the etale fundamental group

Given a birational proper morphism $f\colon X \rightarrow Y$ of complex algebraic varieties. It is always true that $f^* \colon \pi^{et}_1 (X)\rightarrow \pi^{et}_1(Y)$ is an isomorphism? I think ...
2
votes
1answer
67 views

A question on the classifying space $BG$, its universal property (?), and the stack $[\bullet/G]$

I am learning about the classifying space $BG$ of a topological group $G$. I know the definition $$BG=EG/G,$$ where $EG$ is any contractible space on which $G$ acts freely. If I am not mistaken, with ...
4
votes
1answer
95 views

Topological invariance of chern classes

Are Chern classes topological invariants? To be more precise: Given two complex manifolds $M$ and $N$. Does a homeomophism $f:M\to N$ map Chern classes to Chern classes?
5
votes
0answers
187 views

Lefschetz duality for non-compact relative manifolds

I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me. The exact statement in ...
2
votes
0answers
37 views

Surjectivity and exactness on higher homotopy groups

If $X$ is a normal algebraic variety over $\mathbb{C}$ and $U$ an open set of $X$ then we have surjectivity on the fundamental groups $ \pi_1(U)\rightarrow \pi_1(X)$. If $f\colon X \rightarrow Y$ is ...
1
vote
0answers
37 views

Real Spectrum of a Commutative Ring

Could anyone explain briefly what is a real spectrum of a commutative ring? If it possible please give me some easy example. Many thanks in advance. $\text{Spec}_{r}(A)$
3
votes
1answer
49 views

functoriality of $K(G,1)$ spaces in a particular situation involving complex elliptic curves

I apologize if the subject doesn't accurately describe my question. Let $F_2$ denote the free group on two generators. Suppose you have some group homomorphism $A : ...
5
votes
1answer
71 views

Working out an example of a Chern class

I'm trying to understand page 161 of Fulton's "Young tableaux" in an explicit example. I'm looking at flags in $\mathbb{C}^4$, which I think of as flags in $\mathbb{CP}^3$ (and I'm really just able ...
0
votes
0answers
54 views

Can Vanishing Cycles be Described as Fibers over Critical Points in a Lefschetz Fibrations?

I'm trying to see if it makes sense to see vanishing cycles in a Lefschetz fibration as the fibers over critical points. A Lefschetz fibration $f: M^4 \rightarrow X$ , where $M^4$ is a smooth ...
3
votes
0answers
39 views

Why is this Milnor fiber homeomorphic to a cylinder?

Let $f:(\mathbb C^2,0)\to (\mathbb C,0)$ be a holomorphic function with a critical point at the origin. Let us denote by $X_0$ the fiber $f^{-1}(0)$, in which, as we said, the point $0\in X_0$ is a ...
2
votes
1answer
28 views

Density, Irreducible Topological Space

Let X is a irreducible noetherian topological space, and $U \subseteq X$ is a nonempty open subset. If $B \subseteq U$ is dense, then so is $B \subseteq X$. Is this true or false, and why? Here ...
0
votes
0answers
13 views

Orthogonal Array Planes

In an OA-plane (Orthogonal Array-Plane) how we define the touching of two lines? If we consider OA of strength of 3 with unit index , that is the plane OA-plane of rank 3 (we call this Laguerre ...
0
votes
1answer
42 views

Correspondence of Grassmannian cells

I have shown that the correspondence $f: X \rightarrow \mathbb{R} \oplus X$ defines an embedding of the Grassmannian $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ into $\mathrm{Gr}_{k+1}(\mathbb{R} \oplus ...
3
votes
1answer
44 views

Isomorphism of Grassmannians

I want to prove that two CW complexes $\mathrm{Gr}_{n}(\mathbb{R}^{n+k})$ and $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ are isomorphic to one another. I'm pretty sure I can just show that the number of ...
6
votes
1answer
61 views

Relation between different definitions of degree in complex geometry

Consider a holomoprhic map from a Riemann surface $$ f: \Sigma_g \to \mathbb{CP}^n. $$ This is given by some homogeneous polynomials in some variables. How can we show that the homogeneous degree $d$ ...
2
votes
1answer
75 views

Computing the Todd class of projective space.

As an exercise I'm trying to verify that for $X=\Bbb{P}_k^n$, where $k$ is an algebraically closed field, we have $$\operatorname{td}(X)=\left(\frac{\epsilon}{1-e^{-\epsilon}}\right)^{n+1},$$ where ...
4
votes
1answer
146 views

Application of Lefschetz duality to prove Lefschetz hyperplane theorem

I'm trying to understand the proof of the Lefschetz hyperplane theorem in Milnor's book "Morse Theory", page 41 but I can't understand his use of Lefschetz duality. At this point it has been proven ...
3
votes
1answer
73 views

Why are there cubics in a Calabi-Yau manifold?

I heard from a recent talk that the "number of $n$-degree curves" in a Calabi-Yau manifold is an invariant of the space. But what does that mean? (Specifically I would like to ask the following.) ...
0
votes
1answer
84 views

compact open topology on M(X,Y) continuous function space

I'm really confused with the statement. let $X$ be a set endowed with the discrete topology and let $Y$ be any topological space. Show that $M(X,Y)$ with the compact open topology is homeomorphic to ...
2
votes
1answer
113 views

Is there a “geometric” interpretation of inert primes?

I am currently learning about étale morphisms and how they behave a lot like covering maps. Now if I'm in the example of the ring of integers of a number field, it is possible to think of it as a ...
6
votes
3answers
149 views

Are the fibers of a flat map homotopy equivalent?

At the end of the Wikipedia article on Deformation Retract, there is the following sentence: Two spaces are homotopy equivalent if and only if they are both deformation retracts of a single larger ...
2
votes
0answers
32 views

Cech cohomology [duplicate]

There are 2 complexes computing Cech cohomology. The difference between them is that in the second one we require skew symmetry when you change the order of indices. How to show that they are ...
5
votes
0answers
66 views

Serre Fibration long exact sequence

I want to know if there exists something like the long exact sequence on the following case: Let $p : E\rightarrow B$ a continuous surjective map such that there exists an open dense subset $U$ of ...