# Tagged Questions

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

3k views

### A paper to the question, if the six dimensional sphere is a complex manifold

Hi, for a few days a paper was published on arxiv.org with the title "The six dimensional sphere is a complex manifold": http://arxiv.org/PS_cache/math/pdf/0505/0505634v3.pdf Because I am not able to ...
457 views

### “Three great cocycles” in Complex Analysis as cohomology generators

In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$ and the Schwarzian ...
423 views

### Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them. Define the set $\mathcal{E}$ of ...
974 views

### Is analytic capacity inner regular?

For a compact set $K$ in the complex plane, define the analytic capacity of $K$ by $$\gamma(K) := \sup |f'(\infty)|$$ where the supremum is taken over all functions $f$ holomorphic and bounded by $1$ ...
431 views

### Polynomials with roots in convex position

Let $\mathcal P_n$ denote the set of all monic polynomials of degree $n$ with real or complex coefficients such that $P\in\mathcal P_n$ if for all $k\in\lbrace 0,1,\dots,n-2\rbrace$ the $n-k$ roots of ...
757 views

902 views

Let $T=\mathbb C^n/\Lambda$ a complex torus. It is completely elementary to prove that the de Rham cohomology of $T$ in degree $q$ is isomorphic to the $q$-th exterior power of the dual of $\mathbb C^... 0answers 162 views ### How useful is knowing every torsionfree$\mathcal O(D)$module is flat? One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that$\mathcal O(\Bbb C)$, more generally$\mathcal O(D)$for some region$D$is such that every ... 0answers 246 views ### Cesaro summation of a particular Dirichlet series associated with$\zeta(s)$If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{... 0answers 637 views ### Convexity of Jacobi's theta function with zero argument This question may be elementary, I have asked it on math.stackexchange.com but have not received any answer yet. Note that I am not an expert on theta/elliptic functions. Define Jacobi's theta ... 0answers 1k views ### Meaning of Cauchy integral theorem - the (co)homology viewpoint I'm not sure what follows is not just a complicated way to deduce a blatant triviality, or if is even correct. Let's try. In the elementary theory of analytic functions of 1 complex variable, one ... 0answers 410 views ### Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks! Definition ... 0answers 443 views ### Adeles of Holomorphic Functions In number theory, an adele is a restricted product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular ... 0answers 130 views ### Smooth quotients of algebraic spaces that are varieties away from codimension \ge 2 subset This is a question about when a smooth complex algebraic space that is very close to being an algebraic variety is actually an algebraic variety. General question: Let X be a smooth separated ... 0answers 323 views ### rings of modular functions on the upper half plane Let \Gamma_1\le SL_2(\mathbb{Z}) be a noncongruence subgroup of finite index. Let \Gamma_2\le SL_2(\mathbb{Z}) be another subgroup of finite index. Let M_0(\Gamma_i) denote the ring of modular ... 0answers 159 views ### Functional equation or analytic continuation of certain approximations to \zeta^z(s)? Let z be a complex number and \omega(n) denote the number of distinct prime factors of the natural number n. I am considering the arithmetic functions |\mu(n)|z^{\omega(n)} and their ... 0answers 103 views ### What does this number tell me about a convex lattice polygon? EDIT: I realized I'd tricked myself by working with a too special case of f, the question is now updated (boundary lattice points replaced vertices). Suppose I have a convex lattice polygon P, ... 0answers 240 views ### Are the Chern numbers of a hyperbolic-type compact complex manifold bounded in terms of the Euler number? Let X be an n-dimensional compact Kahler manifold with negative first Chern class. Are its Chern numbers \prod_{i=1}^{n-1} c_i^{k_i}, over k _i \geq 0 with \sum ik_i = n, bounded in terms ... 0answers 362 views ### C^\infty function f:{\bf C}\mapsto {\bf C} such that f(z)\in\overline{{\bf Q}(z)} for all z\in {\bf C} Suppose that f:{\bf C}\mapsto {\bf C} is a C^\infty function such that f(z)\in\overline{{\bf Q}(z)} for all z\in {\bf C}, ie f(z) is algebraic over the field {\bf Q}(z) generated by z ... 0answers 163 views ### Biholomorphic neighborhoods of the boundary of Stein domains Let (X_1,J_1) and (X_2,J_2) be Stein domains with the same contact boundary (Y,\xi). Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ... 0answers 77 views ### Extremal length of graphs in surfaces Given a surface \Sigma with conformal structure \omega, the extremal length of a homotopy class \gamma of curves in \Sigma is defined to be$$ \sup_{g \in \omega} \frac{\ell_g(\gamma)^2}{A_g(\... 0answers 183 views ### Complex structures on$\Bbb R^4$Calabi & Eckmann proved that$S^{2p+1} \times S^{2q+1}$admits an integrable complex structure fibred by holomorphic tori, and this implies that$\Bbb R^{2p+2q+2}$, obtained by removing a point in ... 0answers 191 views ### Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups Let$M$be a (Hausdorff) smooth compact manifold and$G$a Lie group acting smoothly on$M$. If$G$is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding$M\to {\...
Suppose I have (roughly speaking) a multivalued meromorphic function $f(z)$ on all of $\mathbb{C}$ that is single-valued and holomorphic on the open unit disc and has some branch points of finite ...