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

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        42 views

        Global interior estimate complex Monge-Ampere equation

        Suppose u is a smooth strictly plurisubharmonic function satisfying $(i\partial \bar{\partial} u)^n =f dV_{Euc}$ on the unit ball centered at the origin, where $f>0$ is a smooth function. Let $C$ ...
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        1answer
        51 views

        Riesz measure of a smooth subharmonic function on a ball

        Let $B(x_{0},r)$ be the ball of center $ x_{0} $ and radius $r>0$ in $ \mathbb{R}^{k} $ ($ k\geq2 $), and $u$ a subharmonic function on an open neighborhood of the closure of $B(x_{0},r)$. Let $\mu$...
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        3answers
        521 views

        How to find a conformal map of the unit disk on a given simply-connected domain

        By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of ...
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        0answers
        141 views
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        Integral with product of two infinite sums

        I am looking for references and results on integrals with product of two infinite sums: $$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$ Above integral is ...
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        0answers
        53 views

        Is a domain of a holomorphic flow pseudoconvex?

        Let $Z$ be a holomorphic vector field on $\mathbb{C}^n$. I would like to know whether (it seems that it is) the domain $D_\phi \subset \mathbb{C} \times \mathbb{C}^n$ of a maximal flow $\phi: D_\phi \...
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        votes
        1answer
        99 views

        A question about distribution of fractional part of $2^k\alpha$

        Let $\{x\}$ be the fractional part of $x$, i.e. $\{x\}=x-[x]$, where $[x]$ is the biggest integer $\leq x$. The question might be well known but I don't know where to look for: Assume $\alpha$ is an ...
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        1answer
        225 views

        Does $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converge for $\sigma > \frac{1}{2}$?

        Looking at @Lucia's answer to this question it appears $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converges for $\sigma > \frac{1}{2}$. Can someone point me to a proof or provide proof for this? If I ...
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        1answer
        238 views

        Cohomology of neighborhood of $\mathbb{C}\mathbb{P}^1$ in $\mathbb{C}\mathbb{P}^n$

        Let $\mathbb{C}\mathbb{P}^1$ be embedded linearly to $\mathbb{C}\mathbb{P}^n$ with $n>1$. (Such an embedding is given in coordinates by $[x:y]\mapsto [x:y:0:\dots: 0]$.) Is it true that for any ...
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        Borel-Ecalle re-summation and resurgence: Criteria and Results

        This is about the theory of Borel-$\acute{\textrm{E}}$calle re-summation and resurgence, see Refs below. This states that the perturbative series (say of the vacuum expectation value of an operator $\...
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        0answers
        115 views

        Cohomology of a tubular neiborhood of submanifold vs cohomology of the formal neighborhood

        Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $\hat Z$ be the formal neighborhood of $Z$. Let $U$ be an open neighborhood of $Z$. Is ...
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        1answer
        416 views

        Are there enough meromorphic functions on a compact analytic manifold?

        Let $X$ be a compact complex analytic manifold, $D\subset X$ an irreducible smooth divisor, given as zeroes of a global meromorphic function $f\in {\mathfrak M} (X)$. Are there enough other ...
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        0answers
        83 views

        Is there a probabilistic proof/interpretation of Mergelyan Theorem

        I came across Mergelyan's Theorem:- Let K be a compact subset of the complex plane C such that C?K is connected. Then, every continuous function $f : K \to C$, such that the restriction f to int(K) ...
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        0answers
        226 views

        Cohomology of complex manifold vs cohomology of its complex submanifold

        Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that $$H^i(Z, A|_Z)=0 \mbox{ for any } ...
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        1answer
        275 views

        A variant of Cauchy-type functional equation conjecture

        Let $f:\mathbb{C}\to \mathbb{C}$ be a complex function such that $$|f(x-y)|=|f(x)-f(y)|,\qquad x,y\in\mathbb{C}.$$ Is it true that $$f(x+y)=f(x)+f(y),\qquad x,y\in\mathbb{C}?$$ The answer is ...
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        2answers
        116 views

        Lelong numbers and integrability of psh functions

        Let $\varphi$ be a plurisubharmonic function in the unit ball $B_1\subset \mathbb{C}^n$ with $\varphi\le 0$. Suppose that the Lelong number $\nu(\varphi,0)<k$ for some $k>0$. Does it follow that ...

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