The Quillen metric, analytic torsion and tunneling for high powers of a   holomorphic line bundle
  Let L be a line bundle over a compact complex manifold X (possibly non-Kahler) and denote by h_{L} and h_{X} fixed Hermitian metrics on L and TX, respectively. We generalize the asymptotics for the induced Quillen metric on the determinant line associated to a higher tensor power of L to the non-Kahler setting. In the case when L is ample we also obtain the leading asymptotics for the Ray-Singer analytic torsion of a (possbly non-positively curved) metric on L, without assuming h_{X} is K\"ahler. The key point of the proofs is to relate the asymptotics of the torsions above to "tunneling", i.e. to the distribution of the exponentially small eigenvalues of the corresponding Dolbeault-Kodaira Laplacians. The proof thus avoids the use of the exact (i.e. non-asymptotic) deep results of Bismut-Gillet-Soul\'e for the Quillen metric, which are only known to hold under the assumption that h_{X} be Kahler. Accordingly the proofs are comparatively simple also in the Kahler case. A brief comparison with the tunneling effect for Witten Laplacians and large deviation principles for fermions is also made. 
Robert J. Berman
math.DG math-ph math.AG math.MP