  Fluctuating Cooper pairs formed by spatially separated electrons and holes
are precursors of their equilibrium condensation. Their presence strongly
impacts transport phenomena and interlayer tunneling in double-layer systems
above the transition temperature. Here, we consider a hybrid graphene/quantum
well double-layer system and focus on the dynamics of fluctuating Cooper pairs
formed by conventional electrons and Dirac holes. We demonstrate that the
chiral nature of Dirac holes is manifested in the presence of two (almost)
degenerate competing pairing channels, which are intertwined by effective
pseudospin-orbit interactions. We argue that the spectrum of the
Ginzburg-Landau Hamiltonian describing the energetics of fluctuating Cooper
pairs is geometrically nontrivial and can be characterized by the half-integer
topological Chern number. We derive a kinetic equation for fluctuating Cooper
pairs and demonstrate that their nontrivial geometries generate two anomalous
velocities of distinct geometrical origins. These velocities are intricately
connected with the Berry curvature and the quantum metric for the
Ginzburg-Landau Hamiltonian, respectively. The resulting anomalous
contributions to conductivity are singular at the transition temperature, and
we discuss possible setups for their experimental observation.
