Microlocal Analysis and Complex Geometry
Final Report Abstract
The goal of this project was to study geometric objects (complex manifolds) using analytic tools. These include methods of microlocal analysis (Fourier integral operators) and semicalssical analysis (spectral analysis involving a parameter converging to zero). To get a good understanding of the holomorphic functions or sections on a complex manifold, one introduces the Bergman kernel, which is the integral kernel of the orthogonal projections on the space of square integrable holomorphic functions or sections. It can be understood as the local density of states of the space of global holomorphic sections. The Szego kernel is the analogue object on the boundary of a domain (one projects on the space of square integrable boundary values of holomorphic functions). The Kodaira embedding theorem asserts that a compact complex manifold X can be embedded in the projective space if and only if it admits a positive line bundle L. In this case there are a lot of sections of the high tensor powers Lk , which provide the embedding. Hence it is natural to study the Bergman kernel of high tensor powers Lk and take the limit as k → ∞. The semiclassical parameter here is = 1/k, which can be thought as Planck’s constant in the related physical problem of a particle moving on X in the magnetic field given by the curvature of Lk. A first result within the project is the asymptotics as k → ∞ of the density of states of the spectrum of the Kodaira Laplace operator on an arbitrary manifold, which may not posses holomorphic sections. Under some mild hypothesis this density of states approximates well the density of states of holomorphic sections (Bergman kernel). For example we obtain the asymptotic of the Bergman kernel on the set of positive curvature of a semipositive line bundle over a compact manifold. The asymptotics was known before only for bunldes which have everywhere positive curvature. The coefficients of this asymptotics are given in terms of geometric data of the manifold and line bundle, which turn out to be universal (in the sense that the formula is the same whenever the expansion exists). Furthermore, we determine the singularity of the Szego kernel on a CR manifold under mild hypotheses and give as application embedding theorems in the Euclidean space. We generalize in this way well-known results of Boutet de Monvel/Sjostrand about the Szego kernel on strictly pseudoconvex CR manifolds, as well as embedding theorems of Boutet de Monvel, Epstein and Lempert. We also consider the framework of semiclassical limit k → ∞ on a CR manifold endowed with a CR line bundle L. In this case we consider the asymptotics of the Szego kerk nel associated to the CR sections of Lk. The outcome are estimates for the dimension of the space of CR sections of Lk, called Morse inequalites. We can show that in important situations these CR sections give embeddings in the projective space, just as in the Kodaira embedding theorem. Our results bear further on the Berezin-Toeplitz quantization, which is a procedure to pass from smooth functions on a complex manifolds to operators on a Hilbert space of sections by composing the multiplication with the function with the projection on the space at hand. The operators obtained in this way are called Toeplitz operators. We determine again the asymptotic as k → ∞ of the Toeplitz operators associated to spectral spaces of the Kodaira Laplacian, and under mild hypotheses, we deduce the Berezin-Toeplitz quantization for the space of holomorphic sections (for semipositive or big line bundles). We also calculate the coefficients of the respective expansions, which again prove to be useful for geometric problems, such as the quantization of the Mabuchi energy. The results obtained in the project made the object of numerous talks at seminars and conferences at, among others, Academia Sinica Taipei, Chern Institute Tianjin, IAS Princeton, KIAS Seoul, Nagoya University, Simons Center for Geometry and Physics (Stony Brook), Tel Aviv University.
Publications
- On the coefficients of the asymptotic expansion of the kernel of Berezin-Toeplitz quantization, Ann. Global Anal. Geom. 42 (2012), no. 2, 207– 245
C.-Y. Hsiao
(See online at https://doi.org/10.1007/s10455-011-9309-6) - Szego kernel asymptotics and Morse inequalities on CR manifolds, Math. Z. 271 (2012), 509–553
C.-Y. Hsiao and G. Marinescu
(See online at https://doi.org/10.1007/s00209-011-0875-x) - Asymptotics of spectral function for lower energy forms and of Bergman kernel for semi-positive and big line bundles, Comm. Anal. Geom. 22 (2014), No. 1, 1–108
C.-Y. Hsiao and G. Marinescu
(See online at https://dx.doi.org/10.4310/CAG.2014.v22.n1.a1) - Berezin-Toeplitz quantization for lower energy forms, 38 pages, 2014
C.-Y. Hsiao and G. Marinescu
- On the singularities of the Szego projections on lower energy forms, 57 pages, 2014
C.-Y. Hsiao and G. Marinescu
- Exponential estimate for the asymptotics of Bergman kernels, Math. Ann. 362 (2015), no. 3-4, 1327–1347
X. Ma and G. Marinescu
(See online at https://doi.org/10.1007/s00208-014-1137-0) - Szego kernel asymptotics and Kodaira embedding theorems of Levi-flat CR manifolds, 36 pages, 2015
C.-Y. Hsiao and G. Marinescu
- Szego kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds, 116 pages, Memoirs of the AMS
C.-Y. Hsiao
