Turns out there really are theoretic bases for this phenomenon, and, although Podkletnov is apparently difficult to work with, his observations really should be taken seriously. For example:
"The London equations for superconductors in a gravitational field." D K Ross, J. Phys A: Math Gen. 16 (1983) 1331-1335."Gravitational shielding." Donald H. Eckhardt, Physical Review D v. 42 n. 6 (15 Sep 1990) 2144-2145.
"Effects of a gravitomagnetic field on pure superconductors." N. Li and D. G. Torr, Physical Review D v. 43 n. 2 (15 Jan 1991) 457-459.
"Interaction between gravity and moving superconductors." H. Peng, G. Lind, and Y.S. Chin, Univ. Alabama (CSPAR), General Relativity and Gravitation v. 23 (11), p. 1231-1250, 1991
"Absorption of gravitational waves" Corneliu D. Ciubotariu, Physics Letters A v. 158, 19 Aug 1991, 27-30.
"Gravitational effects on the magnetic attenuation of superconductors." N. Li and D. G. Torr, Physical Review B v. 46 n. 9 (1 Sept 1992) 5489-5495.
"Absence of a Gravitational Analog to the Meissner Effect." C. Ciubotariu and M. Agop, Technical University of Iaso (Romania), Department of Physics, General Relativity and Gravitation v.28 (4), p. 405-412, Apr 1996.
"Does a Superconductor Shield Gravity?" C.S. Unnikrishnan, Physica C v. 266 (1-2), p133-137, 20 Jul 1996.
"A Unified Theory Based on SO(5) Symmetry of Superconductivity and Antiferromagnetism." Shou-Cheng Zhang, Science, v. 275, 21 Feb 1997.
"Superconductor-mediated modification of gravity? AC motor experiments with bulk YBCO disks in rotating magnetic fields." David Noever, Ron Koczor, and Rick Roberson*, NASA MSFC ES76, Space Sciences Lab, Huntsville, AL 35812, *Tomorrow Tools, NASA MSFC, Huntsville, AL 35812. To be presented on Monday, July 13 in Cleveland at the 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Cleveland Convention Center Cleveland, OH July 12-15, 1998, Special Session of Breakthrough Propulsion Physics (Session 6-APC-1), Monday Morning, 9:00 AM Chaired by: M. MILLIS, NASA Lewis Research Center, Cleveland, OH, 10:30 AM AIAA-98-3139
Without getting too arcane, what are the theoretical bases for this in a nutshell?