## Continuity of a differential of a Banach-valued holomorphic map

Originally posted on MSE.

Let $$U$$ be an open set in $$\mathbb{C}^{n}$$ let $$F$$ be a Banach space (in my case even a dual Banach space), and let $$\varphi:U\to F$$ be a holomorphic map. I seem to be able to prove that the differential map $$D\varphi:U\times\mathbb{C}^{n}\to F$$ defined by $$D\varphi (z,v)= \lim\limits_{t\to 0}\frac{\varphi(z+tv)-\varphi(z)}{t}$$ is holomorphic.

Is there a reference for this assertion? (Or at least for continuity)

I tried to look into some sources on infinite-dimensional holomorphicity and could not find such a statement, but some of those sources are rather complicated, and so it is likely I missed it.

## Is chern classes of holomorphic vector bundles enough to generate Hodge cycles

Let $$X$$ ba a smooth projective variety of dimension $$n$$. Hodge Conjecture states that every Hodge cycle in $$Hdg^k(X,\mathbb{Q})$$ comes from a Chern class of codimension $$k$$ in $$CH^k(X,\mathbb{Q})$$. Now the $$k$$-th Chern class of holomorphic vector bundles generates a subgroup $$CH^k_{vec}(X,\mathbb{Q})$$. Is it possible that every Hodge cycle in $$Hdg^k(X,\mathbb{Q})$$ comes from $$CH^k_{vec}(X,\mathbb{Q})$$? Is there any counterexamples or results?

## Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\mathrm{deg} L)\omega$

I’m reading this paper and at page 67, he states that for any line bundle $$L$$ over a Rieman surface there is a connection $$A$$ whose curvature is $$F(A)=(\mathrm{deg}\mbox{ }L)\omega,$$ where $$\omega$$ is a positive form. Does anyone know how can I prove it without using algebraic geometry arguments?

## Cousin Problem – Hypersurface is defined by holomorphic function

how do I show that any hypersurface $$D \subset \mathbb{C}^n$$ is the defined by a global holomorphic function $$f: \mathbb{C}^n \rightarrow \mathbb{C}^n$$?

This is an exercise from Daniel Huybrechts’ “Complex Geometry: An Introduction” and there the exercise says one should use the Poincaré lemma to prove this statement.

## Holomorphic versus algebraic $\mathbb C^*$-actions

I believe that a holomorphic $$\mathbb C^*$$-action on a complex projective manifold is algebraic if and only if it has a fixed point. Where can I find a proof of this result? (this must be super classical).

## Can we find a holomorphic function $g$ on an open disk such that $\sum_{i \in \mathbb{N}} |(f-g)(a_i)|^2 < +\infty$?

Let $$f : \mathbb{C} \to \mathbb{C}$$ be a continuous function with $$f(0)=0$$.

Let $$\{a_i\}_{i\in \mathbb{N}}$$ be a set of scalars in $$\mathbb{C}$$ such that $$\exists C > 0 : \forall i\in \mathbb{N} : |a_i| \leq C$$ Can we always find a holomorphic function $$g$$ on $$B(0,C+1)$$ (the open disk of radius $$C+1$$) such that $$g(0)=0$$ and $$\sum_{i \in \mathbb{N}} |(f-g)(a_i)|^2 < +\infty$$ ?

## About maxima of injective holomorphic maps on $\mathbb{C}^n$

I am hoping the following is true. Mention of related ideas/topics are appreciated.

Suppose $$F:\mathbb{C}^n \to \mathbb{C}^n$$ is a injective holomorphic mapping such that $$F(0)=0$$ and $$dF(0) = I_n$$ where $$I_n$$ is the $$n \times n$$ identity matrix. Let $$\partial B$$ denote the boundary of the unit ball centered at the origin in $$\mathbb{C}^n$$. Let $$M = \sup_{x \in \partial B} ||F(x)||$$ where $$|| \cdot ||$$ is the usual Euclidean norm. Then $$\partial B \cap \{x: M = ||F(x)|| \}$$ is equal to one of three things: i) $$\{p\}$$ for some point $$p$$, ii) $$\{\alpha p : |\alpha|=1 \}$$ for some point $$p$$, iii) $$\partial B$$

## “Square root” of a holomorphic automorphism

Suppose $$F \in Aut(\mathbb{C}^n)$$. Does there exist a $$G \in Aut(\mathbb{C}^n)$$ s.t. $$G\circ G = F$$?

## Reference for “holomorphic contact geoemtry”

Just like holomorphic symplectic geometry is a complexification of real symplectic geometry, I am wondering is there any good survey paper or book talking about holomorphic version of real contact geometry?

## Milnor Number of real and imaginary parts of holomorphic germs?

By performing some computations using the Singular software, I’ve noticed the following pattern: if $$\mu$$ is the Milnor Number of a holomorphic germ $$f\in \mathcal{O}_n$$ at the origin, then the Milnor Number of its real and imaginary parts are equal and are $$\mu^2$$, that is, $$\mu_{(\mathcal{R}e(f),0)} = \mu_{f,0}^2$$

By Milnor number of the real part of $$f$$, I mean $$u=\mathcal{R}e(f)$$ as a germ of real analytic function of $$2n$$ real variables (the real and imaginary parts of each complex variable). If $$\mathcal{A_{2n}}$$ is the ring of germs of such real analytic functions, then $$\mu_{(\mathcal{R}e(f),0)}=\text{dim}_{\mathbb{R}} \dfrac{\mathcal{A}_{2n}}{\left<\frac{\partial u}{\partial x_1},…, \frac{\partial u}{\partial x_n}, \frac{\partial u}{\partial y_1},…, \frac{\partial u}{\partial y_n}\right> }$$

I would like to know if there’s any generalisation to this. I’ve tried using some direct sum properties on ideals but got nowhere. I suspect there might be some tensor products involved, but also got nowhere.