I don't understand the question - all complex projective algebraic varieties are Kahler, via the pullback of the Fubini-Study form. Could you please clarify?

...is to interpret $F$ as $\operatorname{Cone}(F|_U\oplus F|_V\rightarrow F|_{U\cap V})[1],$ and similarly for $G$. Then the data described above gives you a morphism between the two morphisms inside the cone. However, because you don't have functoriality of cones, you can't upgrade this to a morphism $F\rightarrow G$ without upgrading to some other (e.g. DG) setting.

Here is my possibly incomplete understanding for why functorial cones (one big advantage of DG categories) are necessary for categorical descent. Assume you have an open cover of a variety $X$ by open sets $U,V$. To do categorical descent, you need to be able to reconstruct a morphism $f:F\rightarrow G$ from the data of the morphisms $f|_U$, $f|_V$, and a homotopy between their restrictions to $U\cap V.$ (In the derived world, you cannot keep track of this extra datum of a homotopy, which leads to non-functoriality of cones.) The natural way to do this...

@JonPridham Does Ayoub's recent proof of the conservativity conjecture in characteristic 0 suffice for this? It seems to me like it should but perhaps I am missing some subtlety...

I'm not sure what properties you want your projection to have, but I doubt there will be such a projection w/ any reasonable properties. For example, taking $G$ to be $SL_2(\mathbb{C})$ and $P=B$, you don't have any projection $G\rightarrow T$ that sends $T$ to itself (here I'm identifying $L$ and $T$ with diagonal matrices.) To see this, note that the map $\pi_1(T)\rightarrow\pi_1(G)$ is not injective.

@fedja, Timothy: Thanks for pointing that out. I agree that this is not as simple as I originally thought; let me think a bit about how to fix this...

Here is one way to see this: There is a map from the universal enveloping algebra to the algebra of differential operators landing in right-invariant operators. Your question seems to be why this map is an isomorphism; for this, it suffices to check this statement after taking associated gradeds w/r/t the PBW&order filtrations. The associated graded of $\mathcal{D}_G$ is isomorphic to $\mathcal{O}_G\otimes\operatorname{Sym}^{\bullet}\mathfrak{g},$ with $G$ acting purely on the first factor.

For KZ equations, you could try the book "Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations", Etingof-Frenkel-Kirillov. If you happen to be comfortable with physics terminology, the conformal field theory book of di Francesco-Mathieu-Senechal is the best reference I know. Unfortunately, although the majority of that book is math, it is written in entirely in physical terminology and will probably be impenetrable if you are not comfortable with that language...

@nfdc23: You are of course right. I'm not sure what I was thinking when I wrote that...

Take $X$ a smooth characteristic $0$ variety for simplicity (you can probably get away with much less.) $L$ defines a map from $Y$ to the Picard scheme of $X$, and your $Y_1$ is the fiber of the identity. So if you for instance, take $Y$ to be $\operatorname{Pic}_0(X)$ and $L$ to be the square of the tautological line bundle, then this map is the multiplication by $2$ map and has a non-reduced kernel.