Holomorphic Poisson structures on $C P^{n-1}$ and homogeneous Poisson structures on $C^n$

Is it correct that any holomorphic Poisson structure on $ C P^{n-1}$ can be lifted to a homogeneous Poisson structure on $ C^n$ ? By homogeneous I mean a quadratic Poisson structure of the form $ \{z_i,z_j\}=q_{ij}^{kl}z_kz_l$ where coefficients $ q_{ij}^{kl}$ are constants.

I suspect that this is correct but do not know any reference nor any idea of possible proof. Could you please help me with these?

Reference/Known results on the singular behaviour of the fibres of a holomorphic map between compact Kähler manifolds

I have been interested in the following situation of late: Let $ X$ and $ Y$ be compact Kähler manifolds with $ \dim_{\mathbb{C}}(Y) < \dim_{\mathbb{C}}(X)$ and let $ f : X \to Y$ be a surjective holomorphic map with connected fibres. Let $ S = \{ s_1, …, s_k \}$ denote the critical values of $ f$ , which is a subvariety of $ Y$ .

I cannot find a detailed account of how bad the singular behaviour of the fibres of $ f$ can be. For example, do the fibres contain $ (-1)$ curves (i.e., curves with self-intersection number $ -1$ ) or $ (-2)$ curves?

If anyone can provide references where I can get a better understanding of this, that would be tremendously appreciated.

Compilation of representations of holomorphic functions

Holomorphic functions are my muse. As my muse, I love drawing them different ways. Allow me to frame this as though an artist talking about his muse.

A holomorphic function $ f$ on the unit disk $ \mathbb{D}$ is completely determined (and not only determined, represented) by $ \{f^{(j)}(0)\}_{j=0}^\infty$ .

$ $ f(z) = \sum_{j=0}^\infty f^{(j)}(0) \frac{z^j}{j!}$ $

Similarly a holomorphic function $ f$ on $ \mathbb{D}$ is completely determined (and not only determined, represented) by its values on any contour $ \mathcal{C} \subset \mathbb{D}$ .

$ $ f(z) = \int_{\mathcal{C}} \frac{f(\zeta)}{\zeta – z}\,d\zeta$ $

Adding more constraints, and restricting my muse to certain poses, you can find better and more nuanced art:

If $ f$ is entire, and $ |f(z)| < C e^{\tau |z|^\rho}$ everywhere, for arbitrary constants $ C,\rho,\tau \in \mathbb{R}^+$ , then $ f$ is completely determined (and not only determined, $ nearly$ represented) by its zeroes $ \{a_j\}_{j=1}^\infty$ .

$ $ f(z) = e^{p(z)}\prod_{j=1}^\infty (1 – \frac{z}{a_j})e^{-\frac{z}{a_j} – \frac{z^2}{2a_j^2} -…-\frac{z^n}{na_j^n}}$ $

where $ n$ is the closest greatest integer to $ \rho$ and $ p$ is a polynomial of at most degree $ n$ .

My muse also has rare representations, that bring out specificity and still beauty. Thanks to Ramanujan’s careful deliberations,

A holomorphic function $ f$ on $ \mathbb{C}_{\Re(z)>0}$ , such that $ |f(z)|< Ce^{\rho |\Re(z)| + \tau |\Im(z)|}$ , for arbitrary constants $ C, \rho, \tau \in \mathbb{R}^+$ with $ \tau < \pi/2$ , then $ f(z)$ is completely determined (and not only determined, represented) by $ f \big{|}_{\mathbb{N}}$ .

$ $ f(z)\Gamma(1-z) = \int_0^\infty \vartheta(-x)x^{-z}\,dx$ $

where $ \vartheta(x) = \sum_{j=0}^\infty f(j+1) \frac{x^j}{j!}$ , $ \Gamma$ is the Gamma function, and $ 0 < \Re(z) < 1$ . This can be extended to the expression

$ $ f(z)\Gamma(1-z) = \sum_{j=0}^\infty f(j+1)\frac{(-1)^j}{j!(j+1-z)} + \int_1^\infty \vartheta(-x)x^{-z}\,dx$ $

which works for all $ \mathbb{C}_{\Re(z) > 0}$ .

What other instances do holomorphic functions (on any domain subject to whatever constraints) admit a unique representation theorem based on a sliver of information about the function. Slightly different than identity theorems, as these determine, but rather examples that also represent.

If need be this can be Community wiki.

Thank you and Happy New Year,

Richard Diagram

Constructing a homotopy of nonzero holomorphic functions using local homotopies

I’ll denote by $ \mathbb{C}^*$ the punctured complex plane $ \mathbb{C} \setminus \{0\}$ . Say that I’ve got some open cover $ \{V_j\}_{j \in J}$ of the closed unit interval $ [0,1]$ , and continuous functions $ w_j: V_j \times \mathbb{C}^* \to \mathbb{C}^*$ such that, for any $ t \in V_j$ ,

(a) $ w_j(t, \cdot)$ is a holomorphic function $ \mathbb{C}^* \to \mathbb{C}^*$ and

(b) $ w_j(t, \cdot)$ has a holomorphic antiderivative, say $ F: \mathbb{C}^* \to \mathbb{C}$ .

I want to find a function $ u: [0,1] \times \mathbb{C}^* \to \mathbb{C}^*$ that such that, for fixed $ t \in [0,1]$ , $ u(t, \cdot)$ is still holomorphic and still has a holomorphic antiderivative.

My first thought was to use partitions of unity with respect to this cover, say $ \{p_j\}_{j \in J}$ and to set $ u(t, \cdot) = \sum_{j} p_j(t)w_j(t, \cdot)$ . Of course, such a $ u$ does still have an antiderivative for each $ t$ but now we can’t be sure whether the image of $ u$ still lies in $ \mathbb{C}^*$ .

I get that this is a bit vague, but I’d appreciate any good advice and/or ideas…can someone think of another way of constructing $ u$ . Or perhaps I can pursue the partitions of unity idea, provided $ w_j$ and/or the open cover have some additional features, and if so, what kinds of features would enable this? I’d appreciate any ideas.