# Tagged Questions

Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

**2**

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63 views

### What techniques are available for constructing D-modules over smooth projective varieties?

I'm trying to learn about D-modules for computing intersection cohomology but I'm having trouble coming up with explicit constructions of D-modules on projective varieties. Since this is an involved ...

**0**

votes

**0**answers

46 views

### Reference request for a well-known lemma in Parabolic Vector Bundle

In the paper- "Moduli Space of parabolic vector bundles on a curve" - Usha N Bhosle, Indranil Biswas-Beitr Algebra Geom (2012), 53:437-449, DOI: 10.1007/s13366-011-0053-7, Lemma $2.1$ is being ...

**0**

votes

**0**answers

25 views

### Moduli space of Parabolic Vector Bundles with arbitrary parabolic weights

I have just posted another question related to Moduli Space of Parabolic Vector Bundles on Curve. The questions came up when I was trying to read the paper (Desingularisation of the Moduli Varieties ...

**3**

votes

**0**answers

55 views

### Intersection multiplicity of limit linear spaces

Let $X\subset\mathbb{P}^N$ be a smooth projective variety. Let us fix a general point $q \in X$, and let $C\subseteq X$ be a smooth curve passing through $q$.
Now let $\Lambda_{\xi, q}$, with $\xi \...

**2**

votes

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62 views

### Kahler Einstein metric with minimal singularities

Let $X$ be a Kahler variety with snc divisor $D$ such that $K_X+D$ is
ample. then there is a Kahler metric $\omega_E$ such that
$Ric(\omega_E)=-\omega_E$ on $E=X\setminus D$, then
$h=\frac{1}{\...

**1**

vote

**0**answers

34 views

### Finding the analytic Zariski decomposition singular hermitian metric on a relative line bundle

Let $f:X\to S$ be a proper surjective projective morphism between complex manifolds with connected fibers and let $D$ be an effective $\mathbb Q$-divisor on $X$ such that
$$S^°=\{s\in S| f\text{ is ...

**2**

votes

**0**answers

69 views

### Some queries on Moduli Space of Parabolic Vector Bundles on curve

In "Moduli of Vector Bundles on curves with Parabolic Structures"-Bulletin of the American Mathematical Society Volume 83, Number 1, January 1977 the author announces the following result on moduli ...

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vote

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36 views

### Singular canonical hermitian metric

Let $M$ be a complex manifold , take
$$K_{M,m}:=\sup \{|\sigma|^{\frac{2}{m}}; \sigma\in H^0(M,\mathcal O_M(mK_M)), |\int_M(\sigma\wedge\bar\sigma)^{\frac{1}{m}}|\leq 1\}$$
Let $$K_{M,\infty}:=\lim\...

**1**

vote

**1**answer

109 views

### Affine open subsets for algebraic group actions

Let $G$ be a reductive algebraic group over an algebraically closed field $K$ of characteristic zero.
Assume that $G$ acts on an affine variety $X$. Assume that $X$ contains an open orbit $U$ (so $\...

**6**

votes

**1**answer

98 views

### Chevalley restriction theorem for non-split Cartan

Let $G$ be a reductive group over a field $k$ with maximal torus $H$. Let $\mathfrak{g}$ and $\mathfrak{h}$ denote the corresponding Lie algebra. If $k$ is algebraically closed, we have a theorem of ...

**-1**

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**0**answers

51 views

### FInd a straight line, which goes through 2 points [on hold]

I am new to analytical geometry and excuse me for my notations. We have four lines:
...

**1**

vote

**1**answer

95 views

### reference request for homotopy exact sequence of moduli stacks of curves

Let $\mathcal{M}_{g,n}$ be the moduli stack over $\mathbb{Q}$ of smooth curves of genus $g$ with $n$ marked points. I've seen in many sources an exact sequence:
$$1\rightarrow\pi_1((\mathcal{M}_{g,n})...

**11**

votes

**2**answers

579 views

### Algebraic Geometry for Topologists

As someone who is
familiar with algebraic topology, say, at the level of Hatcher's book, and
familiar with homological algebra and categories and applications in topology
but has no idea what a ...

**7**

votes

**1**answer

172 views

### Intuition for the Lefschetz motive (Tate motive)?

Yo! Maybe this question is too dumb for mathoverflow, but I believe no one will pay attention to it in math.stackexchange, so I will post it here. If this question is not suitable, just delete it.
I ...

**6**

votes

**1**answer

354 views

### Wonderful compactification

Suppose $G$ is a semi-simple group of adjoint type over an algebraic closed field, and $X$ its wonderful compactification a la De Concini and Procesi. Let $P=MU$ be a parabolic subgroup in $G$, and ...

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vote

**0**answers

78 views

### Where should I look for computing the intersection homology of projective varieties?

I'm learning about intersection cohomology topologically through MacPherson's "New York Times Article". This is a very nice guide which gives a nice idea on how to use these methods for low-...

**9**

votes

**1**answer

290 views

### Understanding the purely formal part of the sheaf theoretic (cohomological) framework for representation theory

By now I have the impression that many statements in representation theory can be phrased a lot more elegantly using cohomological language. Therefore I'm trying to understand a bit better the sheaf ...

**0**

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60 views

### Example of bundle-mapping over $S^4$ with singularity $S^2$

Could anyone give a non-trivial example of a bundle-mapping over $S^4$, i.e. find two complex rank 2 vector bundles $E_0,E_1$ over $S^4$ and a bundle mapping
$$0\to E_0\overset{v}{\to}E_1\to0$$
such ...

**3**

votes

**1**answer

160 views

### Rank 2 vector bundles over $\mathbb CP^2$

Is there any classification of the rank 2 complex vector bundles over $\mathbb CP^2$ up to diffeomorphism?
Thank you.

**0**

votes

**0**answers

64 views

### If $X_0$ is very dense in $X$ and $A \cap X$ is closed, then what is $A$?

Let $X$ be a scheme and let $X_0$ be a very dense subset (e.g., $X$ a finite type scheme over a field and $X_0$ the closed points). If $A$ is an arbitrary subset of $X$ such that $X_0 \cap A$ is ...

**1**

vote

**1**answer

124 views

### Rank 2 complex vector bundles over $S^2\times S^2$

How could people classify all rank $2$ complex vector bundles over $S^2\times S^2$ up to isomorphism?
Could you give a rank 2 complex vector bundle which cannot be split as a sum of two line bundles?

**3**

votes

**1**answer

128 views

### Relation between conjugacy class, quotient isomorphism class, and signature of Fuchsian groups

Let $\Gamma\le SL(2,\mathbb{Z})$ be a finite index subgroup, not necessarily "congruence".
Let $c_4,c_6$ be the number of conjugacy classes of elements of order 4 and 6 respectively, let $c_{-1}$ be ...

**2**

votes

**0**answers

73 views

### Compatibility of formal completion and rigid analytic generic fiber

Let $R$ be a complete valuation ring of rank $1$ (e.g., a complete discrete valuation ring) and let $K$ be its field of fractions. Consider a proper $R$-scheme $X$ that is, say, normal (if needed). ...

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119 views

### self-intersection number of rational curves in smooth projective surfaces

Given a smooth projective surface $X/\mathbb{C}$, denote the $Hom_1(\mathbb{P}^1, X)$ to be the set of all degree 1 morphisms from $\mathbb{P}^1$ to $X$. We know that we can regard $Hom_1(\mathbb{P}^1,...

**2**

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68 views

### relative quantization on fibration

Let $\pi:X\to B$ be a holomorphic submerssion of two Kaehler varieties $X,$ $B$ and $(B,\omega)$ be quantizable and fibres $X_s$ also are quantizable, then $X$ is quantizable?. I want to define ...

**-1**

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**0**answers

64 views

### Automorphism group of a class of abelian varieties

Given two abelian varieties $V$ and $V'$ sharing the same Hasse-Weil L-function, is there a well known, 'canonical' notion of automorphism groups thereof such that $Aut(V)\cong Aut(V')$? If so, does ...

**3**

votes

**1**answer

152 views

### “theta characteristics” on general motives?

Yesterday I read in some texts on theta characteristics of algebraic curves, which are structures on their Jacobians... Do you know if someone has thought if such, but more general, things can exist ...

**4**

votes

**1**answer

344 views

### Inverse galois problem and étale homotopy

Is there any relation between étale homotopy theory (Grothendieck-Galois theory) and the inverse Galois problem?...I mean...in classical homotopy theory, every finite group $G$ realizes as a "Galois ...

**3**

votes

**1**answer

270 views

### Geometric intuition for the condition of Galois descent

Continuing in my attempts to understand bits and pieces of Borceux and Janelidze's Galois Theories, I've just realized that I don't have any geometric intuition for the most convenient ...

**3**

votes

**0**answers

45 views

### Which blow ups in the base of a conic bundle preserve the “standard” condition?

Assume we are given a nontrivial standard conic bundle $\pi: X\rightarrow S$, that is $X$ and $S$ are smooth projective varieties (say over $\mathbb{C}$), $\pi$ is flat and furthermore we have $Pic(X)=...

**3**

votes

**1**answer

141 views

### Kähler classes for surfaces of general type with $c_1^2=3c_2$

Given a smooth, compact complex surface with ample canonical bundle satisfying $c_1^2=3c_2$, is it true that every Kahler class is a multiple of $c_1$? This seems to be the case for fake projective ...

**3**

votes

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63 views

### Minimal algebraic degree of symmetric unit distance embedding of Heawood graph

I'm looking at embeddings of the Heawood graph in the plane as unit distance graph. Apparently the first such embedding was given by Gerbracht, 2009 and has algebraic (over the rationals) coordinates ...

**8**

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**0**answers

232 views

### De Rham Cohomology in positive characteristic

This question is about the possible existence of a "compactified" algebraic De Rham cohomology in positive characteristic.
Namely, one knows that, for a smooth, but not proper, variety $U$ over a ...

**7**

votes

**2**answers

523 views

### When is a functor a right derived functor?

Suppose we have Grothendieck abelian categories $\mathcal{A}, \mathcal{B}$. Suppose also we have given an exact functor of triangulated categories
$$
F \colon D(\mathcal{A}) \to D(\mathcal{B})
$$
...

**5**

votes

**2**answers

163 views

### Some examples of $\mathbb Q$-Gorenstein smoothing

I am trying to understand $\mathbb Q$-Gorenstein smoothings, and especially the third condition in the following definition.
Definition. For a normal projective surface $X$ with quotient ...

**-2**

votes

**1**answer

120 views

### Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$ [closed]

I am reading from the book Topics in Galois theory by Serre.
I have the following question ,
take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by
$$\sigma x\;=\;1/(1-x)$$
where $\sigma$ ...

**2**

votes

**1**answer

171 views

### Kummer extension of Galois modules

Let $k$ be a field of characteristic $p \geq 0$, $n$ an integer prime to $p$, and $x$ an element of $k \setminus \{0, 1\}$. I have read that the $n^{th}$ root of $1-x$ gives rise to a Galois module $E$...

**3**

votes

**0**answers

135 views

### Schemes locally of finite type

Let $k$ be a field. Does it exist an irreducible and separated $k$-scheme locally of finite type which is not of finite type?

**2**

votes

**1**answer

165 views

### Reference - Generalized Hodge conjecture for triangulated motives

GHC for triangulated motives: The Hodge conjecture holds and an object $\rm M \in Dmg$ is effective if and only if its Hodge realization is effective.
I would like to know some references on GHC ...

**0**

votes

**0**answers

94 views

### Splitting the Tits algebras of a anisotropic group

Assume we are given an anisotropic algebraic group $G$ over a field $k$, having non trivial Tits algebras (i am interested in the $E_7$ adjoint cases).
Question: Is it possible that there exists a ...

**2**

votes

**2**answers

184 views

### approaches to Apollonius circle problems

I've been looking for solutions to finding the set of circles tangent to two other circles. one circle can be inverted to a line, but two circles can be mapped to a line and a circle
or equivalently ...

**-4**

votes

**0**answers

28 views

### calculating moments in a table [closed]

I am trying to calculate the moments in a data list
position data
1 15
2 22
3 5
4 2
5 1
to find out where in the list is ...

**5**

votes

**1**answer

244 views

### Subgroup schemes of $\mathbb{A}^n$

Let $R$ be an integral $\bar{k}$-algebra of finite type. Let $V(I) \subseteq \mathbb{A}_R^n$ be a reduced (closed) subgroup scheme such that $V(I)\backslash \{0\} \neq \emptyset$ and the $\mathbb{G}_m^...

**2**

votes

**0**answers

112 views

### Chern character (form) of a Gauss-Manin connection

Consider the trivial fibration $\mathbb{T}^2\to\mathbb{S}^1$, where $\mathbb{T}^2$ is the two-torus. Denote by $\mathbb{C}\to\mathbb{T}^2$ the trivial line bundle over $\mathbb{T}^2$, and equip it ...

**4**

votes

**1**answer

95 views

### A quotient group of a self-normalizing spherical subgroup

Let $G$ be simply connected, simple algebraic group over $\mathbb{C}$.
Let $H\subset G$ be a self-normalizing spherical subgroup of $G$,
not necessarily connected or reductive.
Here "self-normalizing" ...

**32**

votes

**0**answers

849 views

+50

### History: What was the Lemma? (Grothendieck Harvard Lectures; Mumford)

In an article about the life of Grothendieck, available here:
http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf
Allyn Jackson writes about how Mumford was profoundly impressed:
Mumford ...

**4**

votes

**0**answers

78 views

### Semi-algebraicness of cells involved in integrals of semi-algebraic functions

Background: In "Stability under integration of sums of products of real globally subanalytic functions and their logarithms", by R. Cluckers and D.J. Miller, it is shown that the integral of a ...

**5**

votes

**1**answer

134 views

### Good analytic spaces over a field into locally ringed spaces is fully faithful

Let $k$ be a field which is complete with respect to a non-trivial non-archimedean rank-1 valuation, and let $X$ be scheme which is locally of finite type over $k$. In section of 3.5 of Berkovich's ...

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65 views

### On orders of stabilisers of group actions and stacks

Let $X$ be a finite type irreducible separated DM-stack over $\mathbb C$. Let $x$ be an object of $X(\mathbb C)$ with stabilizer $G_x$. Let $y$ be an object of $X(\mathbb C)$ with stabilizer $G_y$.
...

**3**

votes

**1**answer

177 views

### Surjectivity of the Kodaira-Spencer map

Let $X$ be a complex projective manifold. Let $B$ be the closed subscheme of $H^1(X,T_X)$ defined by $\mathfrak{m}^2$, where $\mathfrak{m}$ is the ideal defining the origin. In other words, $B$ is a ...