Questions tagged [analytic-geometry]
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115
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Smooth analogue of Cartan's Theorem B
Cartan's Theorem B can be stated as follows:
Let $X$ be a space let $\mathcal{F}$ be a sheaf on $X$.
Consider the following three conditions:
$X$ is "simple";
$\mathcal{F}$ is "nice&...
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Centroid of $\Omega$ and $\partial\Omega$ concides then $\Omega$ must be a ball
Hi I just happened to have a small question. If we have
$$\frac{\int_\Omega x}{|\Omega|}=\frac{\int_{\partial\Omega} x}{|\partial\Omega|}$$
for a simply connected set $\Omega$ with analytic boundary. ...
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Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions
I asked this question on MSE here.
Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, ...
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Are the higher direct images of pluricanonical bundles torsion-free?
Suppose $f: X\rightarrow S$ is a projective smooth morphism to a smooth variety $S$. Let $m\geq 2$ be a natural number.
It is known that the first higher direct image sheaf $R^1f_*\mathcal{O}_X(mK_X)$ ...
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Is $\mathbf{C}_p(X)$ self-dual?
Let $X$ be a set. Consider $\mathbf{Q}_p$ and $\mathbf{Z}_p$ as the $p$-adic numbers and $p$-adic integers, respectively. For any finite subset $F \subseteq X$, one can construct the topological ...
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Deform a divisor from a fiber in a fibration
Suppose $X\rightarrow Z$ is a projective smooth morphism. Let $0\in Z$ be a closed point, $X_0$ the corresponding fiber. Suppose $H^1(X_0,\mathcal{O})=H^2(X_0,\mathcal{O})=0$, then a line bundle $L$ ...
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The intersection number $C\cdot D=\deg(D_{/C})$
Let $S$ be an algebraic complex surface, and $D=[(U_\alpha,f_{\alpha})]$ is a Cartier divisor over $S$, and let $\cal{O}_S(D)$ be the sheaf associated to $D$. And let $C$ be a complex compact curve in ...
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Is there a notion of a complex/analytic diffeological space?
I have a bit of a general question. This seems like something you can do, but I can't seem to find much reference for this.. Perhaps something like this already exists in a different guise. But, is ...
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Pullback of coherent sheaves on Stein manifolds
Let $i:X\to Y$ be a closed embedding of Stein spaces, $G$ be a coherent $O_Y$-module. Set $I=\ker(i^*:O(Y)\to O(X))$. Then $I$ is an ideal of the ring $O(Y)$. Is that true that $\Gamma(X,i^*G)=G(Y)/...
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Differentiable functions on analytic varieties
Let $\iota\colon X\to \Omega\subseteq \mathbb{C}^n$ be a complex analytic variety $X$ in an open subset $\Omega$ of $\mathbb{C}^n$. If $N$ is a smooth manifold and $h\colon M\to X$ is a continuous map,...
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Detecting topology change of tubular neighbourhoods via smoothness of volume function
Let $M$ be an embedded closed manifold in $\mathbb R^n$, define $M_r=\{x\in\mathbb R^n:d(x,M)<r\}$.
Define $r\in\mathcal S_M$ iff $M_r\subset M_{r+\epsilon}$ is not a homotopy equivalence for all ...
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What lies between algebraic geometry and analytic geometry?
Algebraic geometry and analytic geometry are closely related (witness GAGA). But the latter still seems much "bigger" than the former. I'd like to be able to get from algebraic geometry to ...
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Contractible real analytic varieties
If a real analytic variety $V$ in $\mathbb{R}^n$ is both bounded and contractible, is it true that $V$ must be a single point?
Here a real analytic variety is the set of zeros of a real analytic ...
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Proving Geometric Inequality Using Equation Discriminant
I met this question before:
An acute $\triangle ABC$ (you can imagine $BC$ below) has a point $D$ on side $AC$. The line parallel to BC through $D$ meets $AB$ at $E$, and the parallel line $BD$ ...
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What information does the topology of nonarchimedean Berkovich analytic spaces encode?
Given a finite type scheme $X$ over $\Bbb{C}$ we can associate to it an analytic space $X^\text{an}$. There are then comparison theorems comparing invariants of the topological space $X^\text{an}$ ...
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