Fully faithful Fourier–Mukai conjecture for moduli spaces of vector bundles

Let $X$ be a connected smooth projective curve of genus $g\ge2$, let $r\ge2$ and let $d<r$ be coprime positive integers, let $L$ be a line bundle of degree $d$ on $X$, and let $\mathrm{M}(r,L)$ be the moduli space of stable rank-$r$ vector bundles with determinant $L$. Let $\mathcal{E}$ be the normalized Poincaré bundle on $X\times\mathrm{M}(r,L)$, and let $\Phi_{\mathcal{E}}:D^b\operatorname{Coh}(X)\to D^b\operatorname{Coh}(\mathrm{M}(r,L))$ be the Fourier–Mukai functor with kernel $\mathcal{E}$. Fully faithful Fourier–Mukai conjecture. The functor $\Phi_{\mathcal{E}}$ is fully faithful. Therefore, $D^b\operatorname{Coh}(X)$ is embedded into $D^b\operatorname{Coh}(\mathrm{M}(r,L))$. This was known for $r=2,d=1$ and for $d=1$ with $g\ge r+3$; the paper proves it for all coprime $(r,d)$ when $g\ge r+3$.