**39**

votes

**0**answers

594 views

### Vector bundle $L$ admits connection if and only if degree of every direct summand of $L$ divisible by $\text{char}\,k$, intuition

Consider the following theorem of Atiyah.
Let $X$ be a connected smooth projective curve over an algebraically closed field $k$. Then a vector bundle $L$ on $X$ admits a connection if and only if ...

**13**

votes

**0**answers

353 views

### Why, and how badly, does the proof of “no percolation at the critical point in half-spaces” fail for full spaces?

The proof by Barsky et. al. that there is no percolation in half-spaces proceeds by a dynamic renormalization argument. The proof couples critical percolation in the half-space $\mathbb{H}^d$ with a ...

**11**

votes

**0**answers

460 views

### High-dimensional geometry: Top-down Vs. Bottom-up

There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of ...

**11**

votes

**0**answers

444 views

### What is the intuition for $\mathbb{Q}^{ab}$ having cohomological dimension $1$?

I frequently talk to people who think of finite fields as arithmetic analogs of punctured discs. This makes some sense: the absolute Galois group of a finite field is the profinite completion of ...

**6**

votes

**0**answers

140 views

### What is the analogy between the Hilbert function and L-functions?

In his book Projective Varieties and Modular Forms, M. Eichler uses the notation $L(\lambda, M)$ for the Hilbert function of a finite graded $R=k[x_0, \dots, x_n]$-module $M$. So, $L(\lambda, M) = ...

**5**

votes

**0**answers

137 views

### Making the conceptual leap from locales to Grothendieck topologies?

I find the definition for locales and sheaves on locales to be straightforward, but I'm stumbling over the idea of a Grothendieck topology. Is there a nice way to see roughly how the latter ...

**4**

votes

**0**answers

55 views

### The metric gives the optimal element in a class

In geometry there is plenty of examples in which the following happens:
Some elements are considered equivalent, in some topological or algebraic sense
We take the quotient
The metric is usually not ...

**4**

votes

**0**answers

196 views

### What is the meaning of the cospecialization map?

This question comes from the same place as my other one. In reading SGA 4 1/2, but not SGA4 itself (at least, not the obvious sections xv + xvi), one can learn about the "cospecialization morphisms" ...

**4**

votes

**0**answers

407 views

### Relationship between R-transform and free convolution of random matrices?

I've been using the R-transform to calculate the free convolution of the eigenvalue spectra of two random matrices and I am trying to understand how it works, and in particular how it relates to ...

**3**

votes

**0**answers

89 views

### Lie group cohomology with coefficients in Lie algebra

I'm looking for a reference, and basic results, about Lie algebra as modules over a Lie group (with the adjoint representation), from the point of view of cohomology. Links with the Lie algebra ...

**3**

votes

**0**answers

421 views

### Geometric picture behind tilting sheaves

I am trying to read "Tilting exercises" and have trouble to see any geometric pictures behind the formulas.
So my questions are, how to think about tilting perverse sheaves?
Are they just formal ...

**2**

votes

**0**answers

137 views

### Description of the equalizer of $\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$

This is a crosspost of this MSE question.
I have asked several questions in an attmept to get a general version of the Chinese remainder theorem without conditions on the ideals which will ...

**2**

votes

**0**answers

205 views

### Schemes with isomorphic stalks

Fact: If $ X $ and $ Y $ are varieties and we have $ \mathcal{O}_{X,q} \cong \mathcal{O}_{Y,q} $ then there are neighborhoods $U$ of $p$ and $V$ of $q$ which are isomorphic.
I understand ...

**1**

vote

**0**answers

127 views

### Importance and intuition of global sections in sheaf cohomology

I am trying to understand why global sections of a sheaf are "important" or interesting objects of study. Perhaps I have too weak of a background to appreciate it (and that is certainly an acceptable ...

**1**

vote

**0**answers

149 views

### Intuition for hereditary torsion theories

I'm looking for intuition and references for the definition of a hereditary torsion theory and two facts found here. First, the definition and facts:
Definition. A torsion theory $(\mathcal ...

**1**

vote

**0**answers

114 views

### Role of determinant of the matrix corresponding to $i$-th Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...

**0**

votes

**0**answers

53 views

### Why Does a quadratic phase term in BNLS causes collapse?

I've heard a couple of times that in the Biharmonic Nonlinear Schrodinger Equation,
$i\psi_z + \Delta ^2 \psi + |\psi | ^{2\sigma } \psi =0 $, $\psi (x, 0) = \psi _0 (x) \in H^2( \mathbb{R} ^d ) $
...

**0**

votes

**0**answers

223 views

### Weight filtration of MHSs

This is probably a very stupid question, but could someone explain to me where the weight filtration of mixed Hodge structures come from and why we actually need it?
If the Hodge-to-de Rham spectral ...