It might not be as expensive as it sounds. First of all a shadow can be cast using very very light-weight material, maybe even get the mass from objects already in space. Another benefit is that it can be controlled much more carefully than other proposals that have been proposed by serious publications that involve dispersing stuff into the atmosphere itself.
> First of all a shadow can be cast using very very light-weight material
Not in space. Unless the material is an absolutely perfect reflector across the entire EM spectrum, which simply isn't possible, any structure would melt pretty much instantly. You'd have to make radiators that are as big or bigger than the shade structure in order to radiate that heat away and you'd have to make the entire structure much bigger than the shade part because otherwise you'd just be radiating the absorbed heat back at the Earth.
> perfect reflector across the entire EM spectrum, which simply isn't possible
You don't need it to be a perfect reflector across the entire spectrum, 200 to 2000 nm is enough.
And according to "Solar Absorptance and Thermal Emittance of Some Common Spacecraft Thermal-Control Coatings", e.g. Kapton film reflects 92% of the entire solar spectrum.
Sorry I meant be completely transparent or reflective. If even < 1% is absorbed in any way, the heat will steadily build up and eventually start to melt the material, which changes its properties usually causing it to absorb more and more heat (causing a runaway heating effect). You can either radiate that heat back towards the sun (not going to work) or towards the earth which defeats the purpose. This is one of the most basic design constraints in any spacecraft simply because there is no way to convect heat away like we do within our atmosphere.
Since black body radiators are very inefficient, you'd have to build a massive umbrella with an oversized "heatsink" that dumps way more heat to the sides than it does towards the earth or absorbs energy from the sun. I don't even want to imagine how massive of an engineering challenge that would be.
Again, that would require perfect reflection for that 85% of spectrum and perfect transparency and/or a perfect radiator for the other 15%. If that kind of material is possible, we're capable of producing it, and the x% is capable of feeding all of the photosynthetic organisms on Earth it would be totally fine.
However, there is zero evidence that such a perfect material can even exist let alone be produced and maintained at a scale capable of shielding the earth. Any imperfections will absorb heat and steadily destroy the shielding material in runaway overheating.
Look into the research and tests on solar sails. It's not a difficult problem. Most of the light is reflected, and the rest is not enough to overheat it. Casting a shadow does not require 100% of photons be blocked.
Solar sails are designed to move away from a large source of photons which decreases their energy input exponentially as they move. They are also attached to a spacecraft with its own heat dumping mechanism which means most of the energy can conduct away from the sails into the rest of the more massive thermal system.
》 The thermal mass of the rest doesn't matter, only how much it can radiate over the long term. And an enormous film makes a good radiator.
Unless you've got a perfect material that can be accelerated by radiation pressure and radiate all heat away, then the rest of the spacecraft mass very much matters because that's where 99+% of the heat will migrate to. Whether a "film" is a good raditor or not (AND a good solar sail) depends entirely on the type of material so I don't know where you're getting your assumptions from.
》 Certain designs can't go very close to the sun, some can. Many designs would work just fine for a solar shade.
Do you have a link to such a design? I would be greatly interested in learning about it as I have not heard of a single practical design that can do what you're claiming.