LC Oscillator in Capacitive Divider Mode
Your oscillator's (inverting) amplifier is actually a two-port device, whose input signal is amplified with respect to ground. And it provides output power which sinks and sources currents with respect to ground. Like this:







simulate this circuit &ndash; Schematic created using CircuitLab
 The amplifier output port delivers power to the C1-L1-C2 network's input (at C1). Similarly, the amplifier input port accepts power from the C1-L1-C2 network.
The network C1-L1-C2 is not just a resonator - it is of the low-pass filter form. In an oscillator circuit, this network is underdamped, so that it has a "resonant" peak. Near this resonant peak, it provides nearly 180 degree phase shift between its input and output. The inverting amplifier provides a little more than 180 degrees to satisfy the condition for oscillation.
It may be possible for this circuit to use a resonator with very little capacitance to ground (for C1 and for C2) and still oscillate. In a breadboard, stray capacitances combined with amplifier input and output capacitances are often enough to start feeble oscillations with a crystal resonator, so that no added capacitors (C1 &amp; C2) are needed. C1 and C2 are often of similar value, especially where the inverting amplifier has similar input and output impedance.