Intereting Posts

How to compute $\mathbb{Z}_n^*$, the unit group of the integers modulo $n$?
How prove this inequality $(a^2+bc^4)(b^2+ca^4)(c^2+ab^4) \leq 64$
Every Hilbert space has an orthonomal basis – using Zorn's Lemma
Distance between two points on the Clifford torus
If a subsequence of a Cauchy sequence converges, then the whole sequence converges.
Endomorphisms of $V$ and the dual space
Entire function, Liouville and zeroes
Conjectures that have been disproved with extremely large counterexamples?
What comes after $\cos(\tfrac{2\pi}{7})^{1/3}+\cos(\tfrac{4\pi}{7})^{1/3}+\cos(\tfrac{6\pi}{7})^{1/3}$?
Is $x^{1-\frac{1}{n}}+ (1-x)^{1-\frac{1}{n}}$ always irrational if $x$ is rational?
Modular Inverses
Borel-Cantelli Lemma “Corollary” in Royden and Fitzpatrick
If $G$ is non-abelian group of order 6, it is isomorphic to $S_3$
Unique Decomposition of Primes in Sums Of Higher Powers than $2$
Proof that the irrational numbers are uncountable

As far as I am aware, partitions of unity for smooth manifolds require the use of smooth functions with compact support (e.g. bump functions). However, for a complex manifold, the transition maps have to be not only smooth, but holomorphic. And by the Identity Theorem any holomorphic function with compact support is identically zero.

**Question:** How does one surmount this obstacle when working with complex manifolds?

Naively it seems to me that because of this the only possible complex manifold to define would be the open unit disk in $\mathbb{C}^n$, i.e. a manifold with only one chart, since Wikipedia says that the atlas of a complex manifold consists of charts to the open unit disk in $\mathbb{C}^n$.

- Meaning of holomorphic Euler characteristics?
- Show that a complex expression is smaller than one
- complement of zero set of holomorphic function is connected
- Normal bundle of a section of a $\mathbb{P}^1$-bundle
- Exercise from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris
- Geometric interpretation of the determinant of a complex matrix

- pullback of density
- Relations between curvature and area of simple closed plane curves.
- Computing gradient in cylindrical polar coordinates using metric?
- shortest distance between two points on $S^2$
- What exactly is a Kähler Manifold?
- Problems that differential geometry solves
- Why isn't there a contravariant derivative? (Or why are all derivatives covariant?)
- Notation and hierarchy of cartesian spaces, euclidean spaces, riemannian spaces and manifolds
- Surface with non-zero mean curvature means orientable
- Examples of 2-dimensional foliations of a 4-sphere.

Note that complex manifolds are in particular smooth manifold (holomorphic $\Rightarrow$ smooth). Thus one can define partition or unity as in the smooth case. We cannot have something like “holomorphic” partition of unity, of course.

- Counting two ways, $\sum \binom{n}{k} \binom{m}{n-k} = \binom{n+m}{n}$
- Convergence of the Eisenstein series
- Maximize area of a rectangle
- Intersection of ellipse and hyperbola at a right angle
- Finding out the area of a triangle if the coordinates of the three vertices are given
- how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$
- Probability that two people see each other at the coffee shop
- Equivalence of the two cosine definitions
- Proving that irreducibility of a matrix implies strong connectedness of the graph
- Column or row of a matrix?
- How to tell if a directed graph is acyclic from the adjacency matrix
- Extract real and imaginary parts of $\operatorname{Li}_2\left(i\left(2\pm\sqrt3\right)\right)$
- intersection of non zero prime ideals of polynomial ring R over integral domain R is zero
- Simplifying polynomials
- In a Hausdorff space the intersection of a chain of compact connected subspaces is compact and connected