## Do complex schemes locally deformation retract onto closed subschemes in the analytic topology?

Let X be a scheme of finite type over C and let Z↪X be a closed subscheme. Consider the associated closed inclusion Zan↪Xan between their analytifications (regarded as topological spaces). Is this a strong neighborhood deformation retract? By this I mean, can I find for every neighborhood U of a point z in Zan another … Read more

## Can one integrate around a branch-cut?

How meaningful is it to try to integrate around the branch-cut of a function? For example lets say I have the function log(z2+a2) for a>0 and I choose my branch-cuts to be starting at ±ia and moving up and down the y−axis respectively. Now I am trying to integrate around a small circle around such … Read more

## classification of homogenous complex manifolds

Suppose X is a complex manifold (doesn’t assume it’s Kahler), and it’s holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ? Answer AttributionSource : Link , Question Author : user42804 , Answer Author : Community

## Automorphism groups of elliptic bundles

This is a question in complex geometry, but even for algebraic varieties I don’t know the answer: Let S be a smooth compact Kähler surface (for example a smooth complex projective surface) that is an elliptic bundle, i.e there exists a morphism with connected fibres φ:S→C onto a smooth projective curve C such that all … Read more

## Complex L^1 spaces; reference request

I have been doing a fair amount of research into a complex analytic modified version of the Mellin transform. I have hit a few roadblocks, and am hoping there may already be literature on the subject. And it is not the transform itself that is my interest but rather specific functions under the transform. I … Read more

## Linearizing an operator

This question is more about a curious identity I have come across, than to do with explicit research. The question is somewhat advanced so I’m posting it here rather than on math stackexchange. It comes across as a sort of juvenile identity, but the work put in to show the result is rather tenuous. For … Read more

## Topological invariance of the relative multiplicities

Let f:(Cn,0)→(C,0) be a reduced complex analytic function. We write f=fm+fm+1+⋯+fk+⋯ where each fk is a homogeneous polynomial of degree k and fm≠0. We have the decomposition of fm in irreducible polynomials fm=hk11⋯hkrr. Thus, we define the multiplicity of V(f) along of V(hi) for kV(f)(V(hi)):=ki. The numbers kV(f)(V(h1)),…,kV(f)(V(hr)) are called, as well, the multiplicities relatives … Read more

## Analytic maps φ:Cn→Cn\varphi: \mathbb C^n\to \mathbb C^n with degenerate differentials

Let Bn⊂Cn be a unit ball with center p . Let φ:Bn→Cn be a complex analytic map such that dφ has rank at most n−1 at p. I would like to know if there exist two distinct paths in Bn starting at p that are mapped to the same path in Cn. More formally, I … Read more

## Action of the monodromy on the cycle made of the real points

Let $f : \Bbb C^n \to \Bbb C$ be a polynomial function with real coefficients. Let $X_t = f^{-1}(t)$ denote the fiber above some $t \in \Bbb C$. Let assume that the set of real points of $X_t$, for $t > 0$ small enough, is compact and has dimension $n-1$. It defines a $(n-1)$-cycle, say … Read more

## Expansion of Jacobi theta function at pp-torsion

I am aware of the formula Θ(z,q)=zexp(−2∑k≥1z2k(2k)!E2k(q)) for the Jacobi theta function at the origin z=0. The definition I am using for the theta function is Θ(z,q)=(ez/2−e−z/2)∏m≥1(1−ezqm)(1−e−zqm)(1−qm)2. Is there a similar formula for the expansion at a p-torsion point z=2πrip? EDIT: It isn’t hard to show that the qn coefficients in the expansion of lnΘ(z,q) … Read more