I've just managed to compress my LVs, extend my VG to my new SSD, move the LVs to the SSD and remove the old disk from the VG. All is well and good

(mind you, I did kill my /home partition but it's sorted now...)
My next step was to shrink the PV (I think that's what I want to shrink???) so that I can add another partition to the SSD (to reduce compile times). See below for my current setup. I have a 35G PV with LVs totalling around 33G but the "partition" of the LVM is still 58G.
I have a feeling I'm missing something simple, but I have absolutely nfi... Any info/guidance would be appreciated
Code:
#pvdisplay
--- Physical volume ---
PV Name /dev/sda2
VG Name vg_desktop
PV Size 35.00 GiB / not usable 31.81 MiB
Allocatable yes
PE Size 32.00 MiB
Total PE 1119
Free PE 68
Allocated PE 1051
PV UUID ht42dd-eXaK-JnBo-B2bW-D8dJ-iU5C-5sb6df
#lvdisplay
--- Logical volume ---
LV Name /dev/vg_desktop/lv_swap
VG Name vg_desktop
LV UUID Ok3rSJ-YJSl-1Zad-zmTh-btf7-4j3l-vmFr3U
LV Write Access read/write
LV Status available
# open 2
LV Size 7.84 GiB
Current LE 251
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Block device 253:0
--- Logical volume ---
LV Name /dev/vg_desktop/lv_root
VG Name vg_desktop
LV UUID 5gZdio-dr9i-frhV-oo2s-x2Xk-0HAY-K5sdln
LV Write Access read/write
LV Status available
# open 1
LV Size 15.00 GiB
Current LE 480
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Block device 253:1
--- Logical volume ---
LV Name /dev/vg_desktop/lv_home
VG Name vg_desktop
LV UUID rewqnr-BIdy-kdz2-A3l4-Ci63-sIYT-0A21cs
LV Write Access read/write
LV Status available
# open 1
LV Size 10.00 GiB
Current LE 320
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Block device 253:2
#fdisk -s /dev/sda2
58100969