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 PoincarĂ© lemma to prove this statement.

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$

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.