26
votes
Residues in several complex variables
There is a gentle introduction, starting with the single variable case before cranking up the dimension: "Introduction to residues and resultants" by Cattani and Dickenstein. There are also ...
22
votes
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Complex analytic vs algebraic geometry
Though I am not an expert on this I think that the shift toward algebraic geometry is not entirely sociological. Consider the following statement which is true in both the category of schemes and ...
- 7,300
17
votes
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When do real analytic functions form a coherent sheaf?
For real-analytic manifolds, coherence of the structure sheaf always holds. The 1-sentence reason is that one can pass to real and imaginary parts on Oka's coherence theorem in several complex ...
15
votes
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Conformal mappings that preserve angles and areas but not perimeters?
No.
Either condition implies $|\,f'(z)| = 1$ for all $z$ in the domain of $\,f$,
which in turn implies that $\,f'(z)$ is locally constant
(for instance using the open mapping theorem) and thus that
$\...
- 79.9k
14
votes
The $r$-dimensional volume of the Minkowski sum of $n$ ($n\geq r$) line sets
The keyword you are looking for is "zonotope", which is defined to be the Minkowski sum of line segments. An early reference for zonotope is: P. McMullen, “On zonotopes”, Transactions of the American ...
- 1,538
14
votes
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The $r$-dimensional volume of the Minkowski sum of $n$ ($n\geq r$) line sets
Let $M$ be the matrix whose rows are the vectors $\boldsymbol{h_i}$. Then the $r$-dimensional volume of $\mathcal{S}=\mathcal{S}_1+\cdots+\mathcal{S}_n$ is equal to the sum of the absolute values of ...
- 50.8k
12
votes
Complex analytic vs algebraic geometry
I'm not sure this question is a good fit form mathoverlow, but here are a few thoughts. I'll probably delete this answer in a while.
Let me elaborate my comment concerning sociology a bit further. A (...
- 35.2k
10
votes
Accepted
Does GAGA hold over other topological fields?
If $k$ is a field that is complete with respect to some ultrametric valuation, then there is the "GAGR" (i.e. géométrie algébrique et géométrie rigide) theorem. A succinct explanation (in ...
- 4,746
10
votes
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Intersection theory in analytic geometry
You don't say what kind of space $X$ and $Y$ are subspaces of. But if they sit in an oriented manifold there's an easy way to define an intersection product in homology. Namely if $M$ is an oriented $...
- 40.2k
9
votes
Conformal mappings that preserve angles and areas but not perimeters?
See this handout where Brian Conrad talks about the mapmaker's paradox. A mapmaker would like to draw a map that is area preserving and conformal, however a $C^{\infty}$ isomorphism between Riemannian ...
- 85.6k
8
votes
Norms as Points in $C(X)$
A slight improvement to Terry Tao's answer: It is not necessary to assume a priori that the multiplicative seminorm $\|\cdot\|$ is continuous with respect to the sup-norm on $C(X)=C(X\to \mathbb R)$ ...
- 16.4k
8
votes
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Norms as Points in $C(X)$
$\newcommand\Abs[1]{\lVert{#1}\rVert}\newcommand\abs[1]{\lvert#1\rvert}$If one also enforces non-triviality and continuity, then there is a one-to-one correspondence between multiplicative seminorms ...
- 114k
7
votes
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Cotangent Complex in Analytic Category
I will attack the problem three ways, in increasing level of elaboration.
1. Here is a recent paper that proves all this. Compared to older sources, this paper uses more machinery. It uses ...
- 8,717
7
votes
Conformal map from a 7-sided polyhedron to a square pyramid
No such conformal map exists.
Conformal mapping in dimensions above 2 is very different from conformal mapping in dimension 2. In dimensions above 2, any conformal mapping is a (finite) composition ...
- 108k
7
votes
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Smooth analogue of Cartan's Theorem B
It seems like Gro-Tsen's comment pretty much answers your main question, doesn't it? If "simple" = $C^\infty$ manifold, and "nice" = sheaf of modules over $C^\infty_X$. Then such a ...
- 35.2k
6
votes
Residues in several complex variables
I believe residue currents encompass most definitions of residues in several complex variables. Residue currents as developed in the 20th century are discussed for example in the survey "Residue ...
- 1,457
6
votes
Easiest proof for showing finite etale (analytic) quotients of algebraic varieties are algebraic
Up to passing to the Galois closure $\bar{X} \to X$ and using Riemann Extension Theorem (ensuring that $\bar{X}$ is algebraic) we may assume that $X \to Y$ is a Galois cover, induced by the action of ...
- 66.3k
6
votes
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Polynomials (or analytic functions) vanishing on a real algebraic set
The more general statement that is true is the following:
Let $X \subset \mathbb{C}^d$ be an irreducible affine variety defined by real polynomials. If $X$ has a smooth real point, then $X(\mathbb{...
- 8,529
6
votes
Accepted
When is a real-analytic variety a union of non-singular subvarieties?
The general problem mentioned at the beginning of the question is extremely difficult, and, without more hypotheses, there is not that much that can be said.
The OP might be interested in this answer ...
- 108k
6
votes
Deform a divisor from a fiber in a fibration
The statement for divisors is true. Indeed, up to an étale cover, we may suppose that $Z$ is affine and that $\mathscr L$ is a lift of $L$ to $X$. Take $\mathscr M$ on $X$ very ample relative to $Z$, ...
- 28.7k
5
votes
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Regular sequence from prime ideal
It is impossible to do this if $\dim V(I)\geq 2$. Because then $g$ defines a complete intersection of dimension at least $2$. But for any Cohen-Macaulay local ring of dimension at least $2$, the ...
- 30.5k
5
votes
Accepted
5
votes
Accepted
Geometric interpretation of algebraic tangent cone
You may wish to read the discussion on pages 106-108 of Eisenbud/Harris "The Geometry of Schemes", where they explicitly compute the tangent cone of $(y^2-x^3)$ as a degeneration of the tangent cones ...
- 505
5
votes
Accepted
Can an analytic variety extend along a codimension 2 subvariety?
Yes, by Remmert-Stein's extension theorem. See e.g. Fritsche-Grauert, From holomorphic functions to complex manifolds, Theorem 6.9.
- 23.3k
5
votes
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Complete residue field of a point of type 5
A point of type 5 corresponds to a valuation of rank 2, so I am not sure if the meaning of completion is completely clear here. I think that the usual thing is to complete with respect to the ...
- 4,132
5
votes
The intersection number $C\cdot D=\deg(D_{/C})$
Let $(X, \mathcal{O}_X)$ be a scheme. Recall how one defines a closed subscheme. First we take a sheaf of ideals $\mathcal{I}$ of $\mathcal{O}_X$. The support of $\mathcal{O}_X / \mathcal{I}$ is a ...
- 363
5
votes
Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions
Below I give an example of a triangle which seems to grow exponentially when iterating the process from the question (so it satisfies Scenario 3). The example relies on the existence of a fixed point ...
- 10.6k
4
votes
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Contractible real analytic varieties
It is a consequence of Sullivan's work
Sullivan, D., Combinatorial invariants of analytic spaces, Proc. Liverpool Singularities-Sympos. I, Dept. Pure Math. Univ. Liverpool 1969-1970, Lect. Notes Math. ...
- 12.2k
4
votes
Accepted
Centroid of $\Omega$ and $\partial\Omega$ concides then $\Omega$ must be a ball
As Sean said, it can be as smooth as you like.
Depicted ... $\Big\{\big((3+\sin(8t))\cos(t),(3+\sin(8t))\sin(t)\big) : 0 \le t \le 2\pi\Big\}$
- 41.1k
3
votes
Norms as Points in $C(X)$
$\newcommand{\M}{\mathcal{M}}\newcommand{\abs}[1]{\lvert #1 \rvert}\newcommand{\blank}{{-}}\newcommand{\from}{\colon}\newcommand{\IRpos}{\mathbb{R}_{\ge 0}}\newcommand{\H}{\mathcal{H}}$For a ...
- 1,153
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