2
votes
1answer
341 views

Two questions about the grassmannian

There are two statements about the grassmannian (of complex k-planes in n-space embedded via Plucker coordinates) that I have encountered in several places never accompanied with a proof or reference. ...
-1
votes
0answers
36 views

connected components of a real algebraic variety and its hyperplane section

Let $X$ be a smooth projective variety of dimension at least $2$ over the real numbers $\mathbb{R}$ and $H \subset X$ a smooth hyperplane section. Assume that the set of real points is non-empty for ...
2
votes
1answer
336 views

Topological degree and polynomial degree

Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ...
21
votes
6answers
1k views

Down-to-earth expositions of Hodge theory

What are nice expositions of Hodge theory not using advanced language of algebraic geometry or category theory? Of course, since I haven't found a (for me) readable introduction, I don't know what I ...
17
votes
4answers
1k views

Quillen's motivation of higher algebraic K-theory

Almost the same question was already asked on MO Motivation for algebraic K-theory? However, to my taste, the answers there consider the subject from a more modern point of view. When I open a book ...
1
vote
1answer
96 views

Is the equivariant Gysin map an $H_G^*(\text{pt})$-module morphism?

Let $G$ be a complex reductive group, $X$ a smooth projective variety on which $G$ acts algebraically, and $Y \subseteq X$ a $G$-invariant smooth closed subvariety such that $X\setminus Y$ is also ...
1
vote
2answers
221 views

The Schottky group and the fundamental group of a compact Riemann surface

I am quoting the following description from a paper, "...every compact Riemann surface can be obtained as the quotient $\mathbb{C}/\Gamma$ where $\Gamma$ is a Schottky group. The Schottky group of a ...
1
vote
1answer
218 views

Singular cohomology as a Zariski sheaf

Let $X$ be a complex algebraic variety, and consider the presheaf $U \mapsto H^i(U^{an}, \mathbb Z)$ in the Zariski topology. Is there a theorem that says this presheaf is already a sheaf, for ...
-3
votes
1answer
254 views

Lie algebraic Grassmannian

Assume that $L$ is a Lie algebra structure on $\mathbb{R}^{n}$, and $1<k<n$ is given. We define $Gr(k,n)_{L}$, the space of all $k$ dimensional Lie subalgebra of $(\mathbb{R}^{n}, L)$. For ...
9
votes
0answers
229 views

Topological type of Brieskorn manifolds

Let us consider the complex hypersurface and suppose that $n\geq 3$: $$F(d,n)=\{(z_0,\ldots,z_n)\in \mathbb{C}^{n+1}:z_0^d+z_1^d+\ldots+z_n^d=0\}$$ and the link $V(d,n)=F(d,n)\cap S^{2n+1}_{\epsilon}$ ...
2
votes
1answer
170 views

fundamental group and torus action

Let $T$ be the complex torus acting on a complex connected algebraic variety $X$ and let $p \colon X\rightarrow Y$ be a good quotient for this action. For any $y\in Y$ we have a sequence $p^{-1}(y) ...
7
votes
0answers
208 views

Characteristic Classes in Geometric Representation Theory

Geometric respectively topological methods are widely applied in representation theory. As far as I know mainly cohomological methods are used. I wonder if there are concrete applications of the ...
5
votes
0answers
178 views

Homology of stack points (from math.stackexchange)

(This question is duplicated here) This is a very basic question about how definitions in homology carry over to the easiest example of stacks. Let $G$ be a finite cyclic group. Consider the ...
4
votes
0answers
111 views

Projective representation of diffeomorphism group of $S^2$

We know that the projective representation of a group $G$ is classified by $H_{grp}^2(G,R/Z) = H^3(BG,Z)$, where $H^*_{grp}$ is the group-cohomology class. Then do we have a classification of the ...
2
votes
0answers
203 views

Why does this setting imply that a category is Grothendieck?

I came across the following Lemma in Mitsuyasu Hashimoto's Equivariant Twisted Inverses; it is Lemma 11.2 on page 107 of this pdf. Let $\mathcal{A}$ be an abelian category which satisfies the (AB3) ...
3
votes
1answer
90 views

Relative flasqueness?

It is known that a flasque sheaf on a topological space has trivial cohomology. Suppose that we are in a relative situation of a smooth fibration $\pi: X \to S$ and $F$ is a sheaf on $X$. Is there ...
2
votes
0answers
135 views

Stacks and Maurer-Cartan elements

One can associate to any deformation problem a dg Lie or $L_{\infty}$-algebra $g$. For instance, in algebraic deformation theory, let's say the deformation theory of algebras over a Koszul operad $P$, ...
20
votes
4answers
1k views

origin of spectral sequences in algebraic topology

I have the following somewhat vague question. I am not sure if it is appropriate for this forum, please feel free to close (or migrate to stackexchange). I have been "brought up" as an algebraic ...
1
vote
0answers
82 views

Action of Landweber-Novikov algebra on infinite polynomial ring

Denote $\text{Aut}(\hat{{\mathbb A}}^1)$ be the affine group over ${\mathbb Z}$ that sends some ring $R$ to the strict automorphisms of $R[[t]]$, i.e. those of the form $X\mapsto X + r_1 X^2 + r_2 X^3 ...
1
vote
0answers
100 views

When is equivariant cohomology generated by equivariant Euler classes?

Let $X$ be a smooth complex projective variety acted upon by algebraically by a complex torus $T$. Let $F_1,\ldots,F_n$ be the connected components of $X^T$ and assume that the restriction map ...
0
votes
1answer
177 views

Birkhoff decomposition vanishing of the Chern numbers

Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...
3
votes
1answer
236 views

Explicit examples presheaves associated to higher direct images which fail to be sheaves

So I would like to have a few simple examples where the presheaf associated to higher direct image of sheaf fails to be sheaf. So I'm looking for two (natural and simple) topological spaces $X$ and ...
8
votes
0answers
242 views

Lie algebras vs. graph complexes

A ribbon graph is a graph in which every vertex has valence at least three and is equipped with a cyclic ordering of its adjacent half edges. The ribbon graph complex $\mathcal{G}_*$ is the chain ...
16
votes
1answer
602 views

For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism?

My apologies if this question is too naive. Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to ...
1
vote
1answer
275 views

A $2$-torsion version of the motivic stable homotopy category?

For a field $k$ there exists the motivic stable homotopy $SH(k)$; it is compactly generated. My question: does there exist a 'reasonable' functor $p$ from $SH(k)$ to a certain triangulated category ...
6
votes
2answers
387 views

Equivariant Stratifications of a Variety

Let $X$ be a complex variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in S}$ is a finite $T$-equivariant stratification of $X$, so that the $X_{\beta}$ are ...
1
vote
0answers
72 views

Finding ideals of F_3[[X,S]] stable by group action

I will revise my question. Sorry, I had a mistake in the definition of σ_k. Let the action σ on F_3[[X,S]] be as follows: σ: X ---> X + S + X^2 σ: S ---> S + S^3 + S^4. Then, Conjecture: There ...
2
votes
1answer
261 views

Computing fundamental groups of the complement of plane curves

This paper of Zariski contains this statement: If $C$ is a curve in $\mathbb{CP}^2$, and $L$ is a generic line, then the injection $L\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces an ...
5
votes
0answers
141 views

Equivalence of Versions of the Affine Grassmannian

Let $G$ be a compact connected semisimple Lie group. The algebro-geometric definition of the affine Grassmannian is the coset space ...
7
votes
0answers
232 views

Cohomology and conifold transition for the quintic

Let $Y\subset \mathbb{C}P^4$ be the quintic threefold given by the equation $$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4+5X_0X_1X_2X_3X_4=0$$ it has 125 singular points whose links are homeomorphic to $S^2\times ...
9
votes
1answer
228 views

Do level sets always correspond to even graphs?

Suppose I have a level set of some function $f\colon\mathbb{R}^n\rightarrow\mathbb{R}^m$, say $L:=\{x:f(x)=c\}$. Let $S$ denote the points in $L$ at which $L$ is locally diffeomorphic to an open ...
1
vote
2answers
477 views

Uniqueness on square root of complex Line Bundle

Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
12
votes
2answers
639 views

Permuting collinear points on a curve

Let $C \subset {\bf CP}^2$ be an irreducible algebraic smooth (projectively) planar curve over the complex numbers of degree $d$ (we allow finitely many points to be deleted from $C$ to make it ...
1
vote
2answers
214 views

Fixed point for a self-mapping on subset of C[0,1]

Let $f_1$ and $f_2$ be arbitrary self-mappings on $C([0,1])$ with $f_2 > f_1$. Define set $F = \{f \in (C[0,1])| f_1 \leq f \leq f_2 \mbox{ and } f \mbox{ is increasing}\}$. Is it true that every ...
0
votes
2answers
140 views

Segre class of smooth vector bundles over smooth manifolds?

Before you read the following question, please assume I have no knowledge in algebraic geometry. Is it possible to define Segre class of a smooth complex vector bundle over a smooth manifold by using ...
4
votes
1answer
477 views

Can motivic E_∞-ring spectra be strictified to commutative motivic symmetric ring spectra?

Theorem 4.4.4.7 in Lurie's Higher Algebra (or Theorem 4.3.22 in DAG III) states (roughly speaking) that under certain conditions the ∞-category of commutative ∞-monoids in a given symmetric monoidal ...
5
votes
1answer
330 views

Category of motivic spectra

When the survey Axiomatic Stable Homotopy, Neil Strickland, 2004 was written the category of motivic spectra was not investigated from the point of view of axiomatic stable homotopy, as considered ...
3
votes
1answer
181 views

$GL_k$-equivariant cohomology of $k\times n$ matrices

I'm having a surprisingly hard time finding references for some facts about $GL_k$-equivariant cohomology of the space of $k\times n$ matrices. Specifically, I believe the following things to be true: ...
6
votes
0answers
220 views

Galois action on $E_n$-operads

Let $E_n$ be the little $n$-cubes operad which acts on $n$-fold loop spaces (up to group completion, an $E_n$-action is precisely the data needed to perform an $n$-fold delooping). I am looking for ...
2
votes
1answer
216 views

Covering seifert manifolds

Let $M$ be a 3-manifold with boundary. If $M$ has an orientable finite cover that is a Seifert fiber space, then is $M$ also a Seifert fiber space?
3
votes
0answers
117 views

Can one use the equivariant Thom-Gysin sequence on a singular affine variety?

Let $X$ be a smooth complex affine variety. Suppose that $\{X_{\beta}\}_{\beta\in B}$ is a finite stratification of $X$ into smooth locally closed subvarieties. Let $T$ be a complex algebraic torus ...
5
votes
2answers
317 views

The fundamental group of a complex, quasi-affine variety

Can the fundamental group of a quasi-affine variety over $\mathbb{C}$ be a torsion group?
4
votes
1answer
231 views

word problem for the fundamental group of complements

It is well known that the finite type (pure) Artin groups have solvable word problem. This was proved by Deligne in 1972. His aim was to show that the complement of a simplicial hyperplane arrangement ...
0
votes
0answers
187 views

is this a simplicial model category?

A basic result in the theory of model categories is that simplicial sets form a simplicial model category. The same is true for simplicial $k$-algebras. I have two questions related to this. One ...
3
votes
2answers
437 views

Is there a “by hand” proof on the symmetry of the Atiyah class of $TX$?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
7
votes
3answers
631 views

smooth manifolds as real algebraic set (continued)

There are several ways of producing manifolds,say: 1.orbits space of group action 2.connected sum of manifolds 3.underlying topological space of nonsingular algebraic set .... here,i am ...
7
votes
2answers
397 views

Open Torelli problems

I just finished studying the proof of the Torelli Theorem for K3 surfaces made by Daniel Huybrechts (following the approach of Misha Verbitsky). This theorem states that two K3 surfaces $X$ and $Y$ ...
3
votes
0answers
261 views

Analysis of Eilenberg-MacLane Stacks

In a series of three papers from the fifties, Eilenberg and MacLane did a pretty exhaustive study of what we now call "Eilenberg-MacLane spaces" and used a lot of machinery to do it, e.g. Whitehead's ...
6
votes
3answers
399 views

Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$

Let $X$ be a smooth projective complex algebraic variety of general type. Suppose that the (topological) fundamental group of $X$ is an infinite abelian group and that $\pi_2(X^{an})$ is finite. What ...
2
votes
1answer
346 views

Question about a Lefschetz hyperplane type theorem

Let $X$ be a simply connected projective manifold of dimension $n$ over $\mathbb{C}$ and $D = \cup D_i$ be a divisor with normal crossings such that its all components $D_i$ are smooth and ample. I ...