In one of my exercises we should finde a connection to a hyperplane bundle corresponding to a divisor of the form $\sum_ia_i\gamma_i=0$ with $\gamma_i$ are the homogenious coordinates of $\mathbb{P}^n$.
The local coordinates of a patch $U_j$ with $z_j\neq0$ are $\beta^i_{(j)}=\frac{\gamma_i}{\gamma_j}$. So localy the divisor looks like $a_j+\sum_{i:i\neq j}a_i\beta^i_{(j)}=0$.
Next I learned that I can get transition functions wich are holomorphic by dividing two of this divisors (meromorphic functions) with intersecting patches wich would look like this: $$ g_{jk}=\frac{a_j+\sum_{i:i\neq j}a_i\beta^i_{(j)}}{a_k+\sum_{m:m\neq k}a_m\beta^m_{(k)}} $$ After some transformation I get: $$ \frac{a_j+\sum_{i:i\neq j}a_i\frac{\gamma_i}{\gamma_j}}{a_k+\sum_{i:i\neq k}a_i\frac{\gamma_i}{\gamma_k}}=\frac{\gamma_k}{\gamma_j}\frac{\sum_ia_i\gamma_i}{\sum_ma_m\gamma_m}=\frac{\gamma_k}{\gamma_j}=g_{jk}=\beta^k_{(j)} $$ So my transition function would only be a local coordinate. Is this true so far?
Next I should guess a connection and should show that this connection is compatible with my transition function. I think that my connection should look like this: $$ \theta_j=\partial ln(1+\sum_{i:i\neq j}|\beta^i_{(j)}|^2) $$
To show that this connection is compatible with my transition functions I can use this transformation formula (we derived it earlier): $$\theta_k=g_{kj}\theta_jg^{-1}_{kj}+dg_{kj}g^{-1}_{kj} $$ Because my transition function are only scalar valued I get: $$\theta_k=\theta_j+dg_{kj}g^{-1}_{kj} $$ And this would look like: $$\theta_k=\partial ln(1+\sum_{i:i\neq j}|\beta^i_{(j)}|^2)+(\partial+\overline{\partial})ln(\beta^j_{(k)}) $$ The $\overline{\partial}ln(\beta^j_{(k)})$ is zero because this are holomorphic coordinates. So in my function I would be able to combine the ln to get this expression: $$ \theta_k=\partial ln((1+\sum_{i:i\neq j}|\beta^i_{(j)}|^2)*\beta^j_{(k)}) $$ And this looks kind of wrong. This would change the patch for the holomorphic part of $|\beta^i_{(j)}|^2$ but not the antiholomorphic one. Did I guessed a connection which is not possible or did I make an error much earlier? Edit: I think I found a solution. It should be possible to add $$\partial ln(\overline{\beta}^j_{(k)})=0$$ to my transformation of the connection and therefor I can change $$ |\beta^i_{(j)}|^2*\beta^j_{(k)}*\overline{\beta}^j_{(k)}=|\beta^i_{(k)}|^2$$ Is my Idea correct?
Question: "Is my Idea correct?"
Answer: If $L:=\mathcal{O}(d)$ is a line bundle on complex projective space $X$, there is an exact sequence
$$0 \rightarrow \Omega^1_X \otimes L \rightarrow J^1(L) \rightarrow L \rightarrow 0$$ and an extension class
$$a(L) \in Ext^1(L,\Omega^1_X\otimes L ) \cong H^1(X, \Omega^1_X)$$
and $a(L)=0$ iff $L$ has a holomorphic (or algebraic) connection
$$\nabla:L \rightarrow \Omega^1_X \otimes L.$$
Few linebundles (I believe none except for the trivial one) on projective space have a holomorphic (or algebraic) connection and it seems like this is what you are trying to construct in your post. In you case you could at first try to calculate the class $a(L)$ for your line bundle and check if this class is zero. The above sequence is the "Atiyah sequence" and the class is the "Atiyah class". The group $H^1(X, \Omega^1_X)$ can be calculated using Cech-cohomology (as in Hartshorne).