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

learn more… | top users | synonyms (1)

2
votes
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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 ...
1
vote
0answers
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
1answer
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
1answer
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
votes
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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 ...
1
vote
0answers
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
1answer
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
votes
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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). ...
0
votes
0answers
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
votes
0answers
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
votes
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
votes
0answers
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
2answers
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
2answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
2answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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 ...
0
votes
0answers
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
1answer
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 ...