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

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### $p$-adic representations of the fundamental group of a smooth proper curve over a finite field

This question is very general. Let $C$ be a smooth and proper curve over a finite field ${\bf F}_p$. Are there any general results or conjectures on continuous non abelian representations
$$
\pi_1(C)\...

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**0**answers

93 views

### A generalization of Karamata inequality [on hold]

Fist I observe function $f(x)=x^2$ in the figure as following
I found that when $x_1 \ge y_1$ and $x_2 \le y_2$ $\Rightarrow$ $AB \ge CD$
$\Rightarrow$ $$\frac{f(x_1)+f(x_2)}{2}-f(\frac{x_1+...

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**0**answers

34 views

### Properties of Grothendieck ring for field of characterictic $p$

In this article there is a proof that for field $k$ of characteristic zero Grothendieck ring $K(\mathbf{Var}_k)$ is not an integral domain. In many articles I found statement that similar theorem for ...

**18**

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**8**answers

2k views

### Bimodules in geometry

Grothendieck's approach to algebraic geometry in particular tells us to treat all rings as rings of functions
on some sort of space. This can also be applied outside of scheme theory (e.g., Gelfand-...

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vote

**0**answers

62 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\...

**3**

votes

**1**answer

123 views

### Euler characteristic of a surface in $\mathbb{R}^3.$

Suppose I have a (smooth) surface in $\mathbb{R}^3,$ given as (a component of) a real algebraic hypersurface. Is there a good algorithm (assuming, for example, we can compute intersections with lines ...

**15**

votes

**2**answers

1k views

### What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?

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vote

**1**answer

104 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 ...

**7**

votes

**1**answer

1k views

### How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?

Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern abstract algebraic geometry is there in modern complex geometry?
What do I mean by complex geometry? ...

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votes

**0**answers

56 views

### Integral basis of an extension of complete fields

Let $\mathcal{O}_K$ be a complete discrete valuation ring with quotient field $K = \text{Quot}(A)$.
Let $L | K$ be an arbitrary finite field extension. Because $K$ is henselian, the integral closure $\...

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votes

**12**answers

4k views

### Textbook for undergraduate course in geometry

I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry (...

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votes

**1**answer

212 views

### No cuspidal character sheaves on GL(n)

We need a reference for the fact that there are no cuspidal character sheaves on $GL_n$ unless $n=1$.
See page 11 of http://www.kurims.kyoto-u.ac.jp/~arakawa/Henderson_mgsctalk2.pdf.

**1**

vote

**1**answer

96 views

### Cluster algebra structure on the coordinate ring of $Mat_3$

Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$.
We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the ...

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votes

**1**answer

390 views

### Can monodromy be described by the same matrix for chosen generators in case of the same singularity type?

Let $X$ be a surface in $\mathbb{P}^3$. We have a fibration $f: X \longrightarrow \mathbb{P}^1$, and $f^{-1}(s_1)$ and $f^{-1}(s_2)$ have the same singularity type. Let $\gamma_1$ and $\gamma_2$ be ...

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votes

**0**answers

73 views

### How can I describe explicitely a nonsingular model of the elliptic surface?

Consider the surface $\mathcal{E} = Q_1 \cap Q_2 \subset \mathbb{P}_k^3\times \mathbb{P}_k^1$ with homogenous coordinates $x$, $x^\prime$, $y$, $z$ and $t$, $s$ respectively and a field $k$ of even ...

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votes

**0**answers

115 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 ...

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votes

**0**answers

56 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 ...

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votes

**0**answers

30 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 ...

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votes

**1**answer

243 views

### Question on period map, Gauss-Manin connection and complex coordinates of $\mathcal{H}^1(k)$

Let $\mathcal{L}_g$ be the space of abelian differentials on Riemann surfaces of genus $g\ge 2$ and $\mathcal{TH}_g:=\mathcal{L}_g/Diff_0^+(S_g)$ be the Teichmuller space of abelian differentials on ...

**3**

votes

**0**answers

59 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 \...

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votes

**0**answers

70 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}{\...

**9**

votes

**1**answer

304 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 ...

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vote

**0**answers

43 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 ...

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votes

**0**answers

77 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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votes

**1**answer

125 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 $\...

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votes

**2**answers

634 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 ...

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votes

**1**answer

108 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 ...

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votes

**1**answer

363 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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**0**answers

870 views

### Big list - Equivalent descriptions of Hodge conjecture?

I would like to know equivalent descriptions of the Hodge conjecture (with references).
Dan Freed's Version:
Consider a topological cycle (boundary less chains that are free to deform) on a ...

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votes

**1**answer

357 views

### Irreducible variety

I asked a similar question at MSE, as the question seemed quite basic to me, but did not get any hint in 24 hours, except for one upvote for the question itself. I still think I am stuck with some ...

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votes

**1**answer

92 views

### Reference request: Existence of plurisubharmonic potential for positive (1,1)-current on Stein (affine) manifold

I am looking for a reference for the following fact, which I believe should be true.
Let $X$ be a Stein manifold (or smooth affine variety over $\mathbb{C}$).
If $\omega$ is a positive closed $(1,...

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votes

**1**answer

91 views

### Lifting sections to completion of closed subschemes

Let $R$ be a reduced finite type $\bar{k}$-algebra, a projective morphism $\pi \colon V \rightarrow \mathrm{Spec}(R)$ and ideals $I, J \subseteq R$. Assume there is a split $s_{IJ} \colon \mathrm{Spec}...

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votes

**4**answers

2k views

### When are GIT quotients projective?

Some background on GIT
Suppose G is a reductive group acting on a scheme X. We often want to understand the quotient X/G. For example, X might be some parameter space (like the space of possible ...

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votes

**0**answers

54 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:
...

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votes

**0**answers

70 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 ...

**22**

votes

**4**answers

3k views

### Real algebraic geometry vs. algebraic geometry

This question is predicated on my understanding that real algebraic geometry (henceforth RAG) is the version of algebraic geometry (AG) one gets when replacing (esp. algebraically closed) fields with ...

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votes

**1**answer

171 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.

**3**

votes

**1**answer

130 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

**1**answer

102 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})...

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**1**answer

183 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 ...

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votes

**1**answer

172 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 $E$...

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votes

**5**answers

2k views

### Heuristic explanation of why we lose projectives in sheaves.

We know that presheaves of any category have enough projectives and that sheaves do not, why is this, and how does it effect our thinking?
This question was asked(and I found it very helpful) but I ...

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votes

**0**answers

97 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-...

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votes

**0**answers

135 views

### Does anyone have Delzell's Thesis on Bad Points of Forms?

Since a number of papers (e.g. this one) treating denominators in Hilbert's 17th problem point to E.G. Strauss's unpublished letter to G. Kriesel or to Chapter 5 of Delzell's Thesis, which contains an ...

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votes

**1**answer

545 views

### Aysmptotic comparison of $L^2$ sections versus generating sections

Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$.
For $m$ large, let $\{e_1,\...

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votes

**0**answers

65 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 ...

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vote

**1**answer

130 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?

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votes

**1**answer

335 views

### Higgs fields whose determinant have only simple zeros

Is the following property true for every stable holomorphic bundle of rank 2 with trivial determinant on a compact Riemann surface:
The space of trace-free Higgs fields, whose determinant have only ...

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votes

**0**answers

121 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,...

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votes

**0**answers

76 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). ...