Extensions of holomorphic line bundles

Let $ X$ be a complex manifold (e.g., a complex torus), $ Z$ an analytic subset of $ X$ with codimension $ \ge 2$ . Let $ U$ be the complement of $ Z$ in $ X$ . I wonder if any holomorphic line bundle on $ U$ can be extended to a holomorphic line bundle on $ X$ . I would greatly appreciate a reference (if such an extension always exists. The case of codimension $ \ge 3$ follows from results of R. Harvey, Amer. J. Math. 96 (1974), 498–504.).

Any holomorphic vector bundle over a compact Riemann surface can be defined by only one transition function?

It is known that any holomorphic bundle of any rank over a noncompact Riemann surface is trivial. A proof can be found in Forster’s “Lectures on Riemann surfaces”, section 30.

Let $ E$ be a holomorphic vector bundle over a compact Riemann surface $ X$ with gauge group $ G$ . A consequence of the above theorem is the restriction $ E|_{X-\{p\}}$ for any point $ p\in X$ is a trivial bundle. Thus $ E$ can be recovered by specifying the transition function $ g: D\cap (X-\{p\}) \rightarrow G$ where $ D$ is a small disk containing $ p$ .

Is this correct? If not, could you give a counter-example? I am mainly interested in learning about the moduli space of holomorphic bundles over $ X$ in a concrete way, e.g. using transition functions.

Branch of $m$th root of a holomorphic function

Let $ f$ be a holomorphic function in the open subset $ G$ of $ C$ . Let the point $ z_0$ of $ G$ be a zero of $ f$ of order $ m$ . I want to prove that there is a branch of $ f^{1/m}$ in some open disk centered at $ z_0$ .

I tried to prove this way but I am not sure about my solution:

Since $ z_0$ is a zero of $ f$ of order $ m$ , there exist an analytic function $ g$ s.t. $ f(z)=(z-z_0)^mg(z)$ ; $ g(z_0)\neq0$ . If there is a branch $ h$ of log$ f$ , then $ e^{h/m}$ is a branch of $ f^{1/m}$ . My problem is how to prove such $ h$ exists becuase $ f(z_0)=0$ .

I appreciate any help to solve this problem.

The holomorphic map from a compact smooth curve $C$ to $\mathbb{C}P^1$ when $H^0(C,\mathcal{O}(p))=2$

Let $ C$ be a compact smooth complex curve with $ H^0(C,\mathcal{O}(p))=2$ .

I feel confused with the following words:

Denote by $ a$ and $ b$ two non-collinear sections in $ H^0(C,\mathcal{O}(p))$ . Then one can consider the ratio $ f=a/b$ as a holomorphic map from $ C$ to $ \mathbb{C}P^1$ .

$ \bullet$ If both sections vanish at the same point $ q$ then $ f$ does not take the value $ 0$ so is constant. This contradicts the non-collinearity of $ a$ and $ b$ .

$ \bullet$ If $ a$ but not $ b$ vanishes at $ q$ then looking at the fiber of $ 0\in \mathbb{C}P^1$ the degree of the map $ f$ is $ 1$ .

I don’t understand “If both sections vanish at the same point $ q$ then $ f$ does not take the value $ 0$ so is constant.” And does non-collinear means $ a,b$ are not linear?

I also wonder how to deduce the degree is $ 1$ from $ a(q)=0,b(q)\not=0$ ?

At last, how is that related to $ p$ ?

Is there a way to categorise the valleys of a holomorphic function (potentially of $\geq 2$ variables) (multidimensional steepest descent)

More specifically, I am particularly interested in the question: given some $ f:\mathbb{C}^n \to \mathbb{C}$ , can we categorise $ \mathbb{C}^n$ by which valley the steepest descent curves of a point (that is, the curve that reduces $ Re(f)$ the fastest i.e following $ -\nabla Re(f) $ ) end up in?

For an entire function $ f$ , the input space $ \mathbb{C}$ is split into hills and valleys about the saddle points of $ Re(f)$ by the maximum modulus theorem. Visually it is quite obvious whether or not two points (say $ z_1,z_2$ ) are in the same valley. Is there a way to formalise this idea? In addition to this, is there some way to categorise $ \mathbb{C}$ around this quality? Ideally I would like to find some test that extends to functions $ f:\mathbb{C}^n \rightarrow \mathbb{C}$ for $ n\geq 2 $ .

I had thought one idea would be to take the steepest descent curves of starting from $ z_1,z_2$ . Then if $ arg(z_1),arg(z_2) \rightarrow \theta$ we can say the two points must be in the same valley. I think this would work pointwise; can we use it or some different idea to categorise $ \mathbb{C}$ based on which valley the point belongs to?

I hope this would then extend to functions of 2 complex variables by writing $ z,w=(z_1,z_2),(w_1,w_2) \in \mathbb{C}$ and checking that the arguments of $ z_1,w_1$ and $ z_2,w_2$ tend to the same limit. Does this sound sensible? Is there a better way to do it?

The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve

Working over the complex numbers, consider a function $ F\left(x,y\right)$ and a curve $ C$ defined by $ F\left(x,y\right)=0$ .

I know that to construct the Jacobian variety associated to $ C$ , one integrates a basis of global holomorphic differential forms over the contours of the curve’s homology group. I’m looking for information that is oriented toward actually computing things for given concrete examples; everything I’ve seen so far, however, has been uselessly abstract or non-specific. Note: I’m new to this—I’m an analyst who knows next to nothing about algebra and even less about differential geometry or topology.

In my quest for a sensible answer, I turned to a H.F. Baker’s wonderful (though densely written) text from the start of the 20th century. Just reading through the first few pages makes it abundantly clear that there is a general procedure for constructing a basis of holomorphic differential forms for a given curve. Ted Shifrin’s comment on this math-stack-exchange problem only makes me more certain than ever that the answers I seek are out there, somewhere.

Broadly speaking, my goals are as follows. In all of these, my aim is to be able to use the answers to these questions to compute various specific examples, either by hand, or with the assistance of a computer algebra system. So, I’m looking for formulae, explanations and/or step-by-step procedures/algorithms, and/or pertinent reference/reading material.

(1) In the case where $ F$ is a polynomial, what is/are the procedure(s) for determining a basis of holomorphic differential 1-forms over $ F$ ? If the procedure varies depending on certain properties of $ F$ (say, if $ F$ is an affine curve, or a projective curve, or of a certain form, or some detail like that), what are those variations?

(2) In the case where $ F$ is a polynomial of $ x$ -degree $ d_{x}$ , $ y$ -degree $ d_{y}$ , and $ C$ is a curve of genus $ g$ , I know that the basis of holomorphic differential 1-forms for $ C$ will be of dimension $ g$ . In the case, say, where $ C$ is an elliptic curve, with:

$ $ F\left(x,y\right)=4x^{3}-g_{2}x-g_{3}-y^{2}$ $

the classical Jacobi Inversion Problem arises from considering a function $ \wp\left(z\right)$ which parameterizes $ C$ , in the sense that $ F\left(\wp\left(z\right),\wp^{\prime}\left(z\right)\right)$ is identically zero. Using the equation: $ $ F\left(\wp\left(z\right),\wp^{\prime}\left(z\right)\right)=0$ $ we can write: $ $ \wp^{-1}\left(z\right)=\int_{z_{0}}^{z}\frac{ds}{4s^{3}-g_{2}s-g_{3}}$ $ and know that the multivaluedness of the integral then reflects the structure of the Jacobian variety associated to $ C$ .

That being said, in the case where $ C$ is of genus $ g\geq2$ , and where we can write $ F\left(x,y\right)=0$ as: $ $ y=\textrm{algebraic function of }x$ $ nothing stops us from performing the exact same computation as for the case with an elliptic curve. Of course, this computation must be wrong; my question is: where and how does it go wrong? How would the parameterizing function thus obtained relate to the “true” parameterizing function—the multivariable Abelian function associated to $ C$ ? Moreover, how—if at all—can this computation be modified to produce the correct parameterizing function (the Abelian function)?

(3) My hope is that by understanding both (1) and (2), I’ll be in a position to see what happens when these classical techniques are applied to non-algebraic plane curves defined but with $ F$ now being an analytic function (incorporating exponentials, and other transcendental functions, in addition to polynomials). Of particular interest to me are the transcendental curves associated to exponential diophantine equations such as: $ $ a^{x}-b^{y}=c$ $ $ $ y^{n}=b^{x}-a$ $

That being said, I wonder: has this already been done? If so, links and references would be much appreciated.

Even if it has, though, I would still like to know the answers to my previous questions, even if it’s merely for my personal edification alone.

Thanks in advance!

Find the largest subset of $\mathbb{C}$ where $g(z)$ is holomorphic?

I’m struggling a lot with this question:

—- Fix $ a \in \mathbb{C}$ where $ g(z) = \frac{1}{z^2-a^2}$ . Find the largest subset of $ \mathbb{C}$ where g(z) is holomorphic. Illustrate this with a sketch. —- I understand that a holomorphic function is a function is differentiable in the complex plane, I believe that the only time g(z) would not be holomorphic is when the function is invalid e.g $ \frac{1}{0}$ , however this leaves me with just assuming the only points g(z) is not holomorphic is where $ z^2 = a^2$ however this then leaves just the subset $ \mathbb{C}$ not including positive or negative a, which doesn’t seem right. Any help help would be fab.

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