Publications
XXXXIX. Induced Fubini-Study metrics and zeros of random CR functions (jointly with Hendrik Herrmann, George Marinescu and Wei-Chuan Shen), arXiv:2401.09143
--Let $X$ be a compact strictly pseudoconvex embeddable Cauchy-Riemann manifold and let $T_P$ be the Toeplitz operator on $X$ associated with a first-order pseudodifferential operator $P$. In our previous work we established the asymptotic expansion for $k$ large of the kernel of the operators $\chi(k^{-1}T_P)$, where $\chi$ is a smooth cut-off function supported in the positive real line. By using these asymptotics, we show in this paper that $X$ can be projectively embedded by maps with components of the form $\chi(k^{-1}\lambda)f_\lambda$, where $\lambda$ is an eigenvalue of $T_P$ and $f_\lambda$ is a corresponding eigenfunction. We establish the asymptotics of the pull-back of the Fubini-Study metric by these maps and we obtain the distribution of the zero divisors of random Cauchy-Riemann functions. We then establish a version of the Lelong-Poincar\'e formula for domains with boundary and obtain the distribution of the zero divisors of random holomorphic functions on strictly pseudoconvex domains.
XXXXVIII. Semi-classical spectral asymptotics of Toeplitz operators on strictly pseudodonvex domains (jointly with George Marinescu), arXiv:2308.09820
--On a relatively compact strictly pseudoconvex domain with smooth boundary in a complex manifold of dimension $n$ we consider a Toeplitz operator $T_R$ with symbol a Reeb-like vector field R near the boundary. We show that the kernel of a weighted spectral projection $\chi(k^{-1}T_R)$, where $\chi$ is a cut-off function with compact support in the positive real line,is a semi-classical Fourier integral operator with complex phase, hence admits a full asymptotic expansion as $k\to+\infty$. More precisely, the restriction to the diagonal $\chi(k^{-1}T_R)(x,x)$ decays at the rate $O(k^{-\infty})$ in the interior and has an asymptotic expansion on the boundary with leading term of order $k^{n+1}$ expressed in terms of the Levi form and the pairing of the contact form with the vector field R.
XXXXVII. $G$-invariant Bergman kernel and geometric quantization on complex manifolds with boundary (jointly with Rung-Tzung-Huang, Xiaoshan Li and Guokuan Shao), arXiv:2305.00601, to appear in Math. Ann..
--Let $M$ be a complex manifold with boundary X, which admits a holomorphic Lie group $G$-action preserving $X$. We establish a full asymptotic expansion for the $G$-invariant Bergman kernel under certain assumptions. As an application, we get $G$-invariant version of Fefferman’s result about regularity of biholomorphic maps on strongly pseudoconvex domains of $\mathbb C^n$ . Moreover, we show that the Guillemin-Sternberg map on a complex manifold with boundary is Fredholm by developing reduction to boundary technique, which establish “quantization commutes with reduction” in this case.
XXXXVI. Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds (jointly with Hendrik Herrmann, George Marinescu and Wei-Chuan Shen), arXiv:2303.17319 ,
--Let $X$ be a compact strictly pseudoconvex embeddable CR manifold and let $T_P$ be the Toeplitz operator on $X$ associated with some first order pseudodifferential operator $P$. We consider $\chi_k(T_P)$ the functional calculus of $T_P$ by any rescaled cut-off function $\chi$ with compact support in the positive real line. In this work, we show that $\chi_k(T_P)$ admits a full asymptotic expansion as $k\to+\infty$. As applications, we obtain several CR analogous of results concerning high power of line bundles in complex geometry but without any group action assumptions on the CR manifold. In particular, we establish a Kodaira type embedding theorem , Tian's convergence theorem and a perturbed spherical embedding theorem for strictly pseudoconvex CR manifolds.
XXXXV. Embedding theorems for quantizable pseudo-K\"ahler manifolds (jointly with Andrea Galasso), arXiv:2209.10269,
--Given a compact quantizable pseudo-Kähler manifold $(M,\omega)$ of constant signature, there exists a Hermitian line bundle $(L,h)$ over $M$ with curvature $-2\pi i\omega$. We show that the asmptotic expansion of the Bergman kernels fpr $L^k$-valued $(0,q)$-forms implies more or less immediately a number of analogues of welll-known results, such as Kodaira embedding theorem and Tian's almost-isometry theorem.
XXXXIV. Semi-classical Bergman kernel asymptotics on complex manifolds with boundary (jointly with Xiaoshan Li and George Marinescu), 55 pages, arXiv:2208.12412,
--Let $M$ be a relatively compact connected open subset with smooth connected boundary of a complex manifold $M'$. Let $(L, h^L)\rightarrow M'$ be a positive line bundle over $M'$. Suppose that $M'$ admits a holomorphic $\mathbb R$-action which preserves the boundary of $M$ and the $\mathbb R$-action can be lifted to $L$. In this work, we establish an asymptotic expansion for the Bergman kernel of the $\overline\partial$-Neumann operator on $\overline M$ with respect to high powers of $L$.
XXXXIII. On the singularities of the Szeg\H{o} kernels on CR orbifolds (jointly with Andrea Galasso), 23 pages, arXiv:2208.03690,
--In this paper we study the microlocal properties of the Szeg\H{o} kernel of a given compact connected orientable CR orbifold whose Kohn Laplacian has closed range. This last assumption is satisfied if certain geometric conditions hold true, as in the smooth case. As applications, we give a pure analytic proof of Kodaira-Bailey theorem and explain how to generalize a CR version of quantization commutes with reduction to orbifolds.
XXXXII. Functional calculus and Quantization commutes with reduction for Toeplitz operators on CR manifolds (jointly with Andrea Galasso), 34 pages, arXiv:2112.11257,
--Given a CR manifold with non-degenerate Levi form, we show that the functional calculus for Toeplitz operators are complex Fourier integral operators of Szeg\H{o} type. As an application, we establish semi-classical Weyl law for Toeplitz operators. We then consider a CR manifold with a compact Lie group acion $G$ and we establish quantization commutes with reduction for Toeplitz operators. Moreover, we also obtain semi-classical Weyl law for $G$-invariant Toeplitz operators.
XXXXI. Toeplitz operators on CR manifolds and group action (jointly with Andrea Galasso), 34 pages, arXiv:2108.11061, J. Geom. Anal. 33 (2023), no. 1, 21.
--Let $(X,\,T^{1,0}X)$ be a connected orientable compact CR manifold of dimension $2n+1,\, n\geq 1$ with non-degenerate Levi curvature.
In this paper, we study the algebra of Toeplitz operators on $X$ and we establish star product for some class of symbols on $X$. In the second part of this paper, we consider a compact locally free Lie group $G$ acting on $X$ and we investigate the associated algebra of $G$-invariant Toeplitz operators.
XXXX. Heat kernel asymptotics for Kohn Laplacians on CR manifolds (jointly with Weixia Zhu), J. Funct. Anal. 284(2023), no. 2, Paper No. 109755.
--Let $X$ be an abstract orientable not necessarily compact
CR manifold of dimension $2n+1$, $n\geq1$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Suppose that condition $Y(q)$ holds at each point of $X$, we establish asymptotics of the heat kernel of Kohn Laplacian with values in $L^k$. As an application, we give a heat kernel proof of Morse inequalities on compact CR manifolds. When $X$ admits a transversal CR $\mathbb R$-action, we also
establish asymptotics of the $\mathbb R$-equivariant heat kernel of Kohn Laplacian with values in $L^k$. As an application, we get $\mathbb R$-equivariant Morse inequalities on compact CR manifolds with transversal CR $\mathbb R$-action.
XXXIX. On the second coefficient of the asymptotic expansion of Boutet de Monvel--Sj\"ostrand (jointly with Wei-Chuan Shen), Bulletin of the institute of Mathematics, Academia Sinica. 15 (2020) no. 4, 329--365.
-- In this paper, we calculate the second coefficient of the asymptotic expansion of Boutet de Monvel--Sj\"ostrand.
XXXVIII. Szeg\H{o} kernel asymptotics on some non-compact complete CR manifolds (jointly with George Marinescu and Huan Wang), 36 pages. arXiv:2012.11457, J. Geom. Anal. 32 (2022), no. 11, Paper No. 266, 53 pp.
-- We establish Szeg\H{o} kernel asymptotic expansions on non-compact strictly pseudoconvex complete CR manifolds with transversal CR R-action under certain natural geometric conditions.
XXXVII. On the coefficients of the equivariant Szeg\H{o} kernel asymptotic expansions (jointly with Rung-Tzung Huang and Guokuan Shao), 22 pages. arXiv:2009.10529, J. Geom. Anal. 32 (2022), no.1, Paper No. 31.
-- Let $(X, T^{1,0}X)$ be a compact connected orientable strongly pseudoconvex CR manifold of dimension $2n+1$, $n\geq1$. Assume that $X$ admits a connected compact Lie group $G$ action and a transversal CR $S^1$ action, we compute the coefficients of the first two lower order terms of the equivariant Szeg\H{o} kernel asymptotic expansions with respect to the $S^1$ action.
XXXVI. Bergman-Szeg\H{o} kernel asymptotics in weakly pseudoconvex finite type cases (jointly with Nikhil Savale), 44 pages. arXiv:2009.07159, J. Reine Angew. Math. 791 (2022), 173--223.
-- We construct a pointwise Boutet de Monvel-Sjöstrand parametrix for the Szeg\H{o} kernel of a weakly pseudoconvex three dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman's boundary asymptotics of the Bergman kernel to weakly pseudoconvex domains in $\mathbb{C}^{2}$, in agreement with D'Angelo's example. Finally our results generalize a three dimensional CR embedding theorem of Lempert .
XXXV. On the singularities of the Bergman projections for lower energy forms on complex manifolds with boundary (jointly with George Marinescu), 54 pages. arXiv:1911.10928, to appear in Analysis &PDE.
-- Let $M$ be a complex manifold of dimension $n$ with smooth boundary $X$.
Given $q\in\set{0,1,\ldots,n-1}$, let $\Box^{(q)}$ be the $\ddbar$-Neumann Laplacian for $(0,q)$ forms. We show that the spectral kernel of $\Box^{(q)}$ admits a full asymptotic expansion near the non-degenerate part of the boundary $X$ and the Bergman projection admits an asymptotic expansion under some local closed range condition. As applications, we establish Bergman kernel asymptotic expansions for some domains with weakly pseudoconvex boundary and $S^1$-equivariant Bergman kernel asymptotic expansions and embedding theorems
XXXIV. Geometric quantization on CR manifolds (jointly with Xiaonan Ma and George Marinescu), 47 pages. arXiv:1906.05627, to appear at Communications in Contemporary Mathematics.
-- Let $X$ be a compact connected orientable CR manifold of dimension greater than five with action of a connected compact Lie group $G$. Assuming that the Levu form of $X$ is positive definite near the inverse image $Y$ of $0$ by the momentum map and that the tangential Cauchy-Riemann operator has closed range on the reduction $Y/G$, we prove that there is a canonical Fredholm operator between the space of global $G$-invariant $L^2$ CR functions on $X$ and the space of global $L^2$ CR functions on the reduction $Y/G$.
XXXIII. $G$-equivariant embedding theorems for CR manifolds of high codimension (jointly with Kevin Fritsch and Hendrik Herrmann), 36 pages. arXiv:1810.09629, Michigan Math. J. 71 (2022), no. 4, 765--808.
-- Let $(X,T^{1,0}X)$ be a $(2n+1+d)$-dimensional compact CR manifold with codimension $d+1$, $d\geq1$, and let $G$ be a $d$-dimensional compact Lie group with CR action on $X$ and $T$ be a globally defined vector field on $X$ such that $\mathbb C TX=T^{1,0}X\oplus T^{0,1}X\oplus\mathbb C T\oplus\mathbb C\underline{\mathfrak{g}}$, where $\underline{\mathfrak{g}}$ is the space of vector fields on $X$ induced by the Lie algebra of $G$. In this work, we show that if $X$ is strongly pseudoconvex in the direction of $T$ and $n\geq 2$, then there exists a $G$-equivariant CR embedding of $X$ into $\mathbb C^N$, for some $N\in\mathbb N$. We also establish a CR orbifold version of Boutet de Monvel's embedding theorem.
XXXII. Solution of the tangential Kohn Laplacian on a class of non-compact CR manifolds (jointly with Po-Lam Yung), arXiv:1808.07212 , Calc. Var. Partial Differential Equations 58 (2019), no. 2, Art. 71, 62 pp.
--We solve Kohn Laplacian on a class of non-compact 3-dimensional strongly pseudoconvex CR manifolds via a certain conformal equivalence. The idea is to make use of a related $\boxb$ operator on a compact 3-dimensional strongly pseudoconvex CR manifold, which we solve using a pseudodifferential calculus. The way we solve Kohn Laplacian works whenever $\dbarb$ on the compact CR manifold has closed range in $L^2$; in particular, as in Beals-Greiner, it does not require the CR manifold to be the boundary of a strongly pseudoconvex domain in $\mathbb{C}^2$. Our result provides in turn a key step in the proof of a positive mass theorem in 3-dimensional CR geometry, by Cheng, Malchiodi and Yang , which they then applied to study the CR Yamabe problem in 3 dimensions.
XXXI. Torus equivariant Szeg\H{o} kernel asymptotics on strongly pseudoconvex CR manifolds (jointly with Hendrik Herrmann and Xiaoshan Li), 20 pages . arXiv:1808.01927, Acta Math. Vietnam. 45 (2020), no. 1, 113–135.
-- Let $(X, T^{1,0}X)$ be a compact strongly pseudoconvex CR manifold of dimension $2n+1$. Assume that $X$ admits a Torus action $T^d$. In this work, we study the behavior of the torus equivariant Szeg\H{o} kernels and prove that weighted torus equivariant Szeg\H{o} kernels admit asymptotic expansions.
XXX. $S^1$-equivariant Index theorems and Morse inequalities on complex manifolds with boundary (jointly with Rung-Tzung Huang, Xiaoshan Li and Guokuan Shao), 34 pages . arXiv:1711.05537, J. Funct. Anal. 279 (2020), no. 3, 108558, 51 pp.
-- Let $M$ be a complex manifold of dimension $n$ with smooth connected boundary $X$. Assume that $\ol M$ admits a holomorphic $S^1$-action preserving the boundary $X$ and the $S^1$-action is transversal and CR on $X$. We show that the $\ddbar$-Neumann Laplacian on $M$ is transversally elliptic and as a consequence, the $m$-th Fourier component of the $q$-th Dolbeault cohomology group $H^q_m(\ol M)$ is finite dimensional, for every $m\in\mathbb Z$ and every $q=0,1,\ldots,n$. This enables us to define $\sum^{n}_{j=0}(-1)^j{\rm dim\,}H^q_m(\ol M)$ the $m$-th Fourier component of the Euler characteristic on $M$ and to study large $m$-bahavior of $H^q_m(\ol M)$. In this paper, we establish an index formula for $\sum^{n}_{j=0}(-1)^j{\rm dim\,}H^q_m(\ol M)$ and Morse inequalities for $H^q_m(\ol M)$.
XXIX. Szeg\H{o} kernels and equivariant embedding theorems for CR manifolds (jointly with Hendrik Herrmann and Xiaoshan Li), 30 pages . arXiv:1710.04910, Math. Res. Lett. 29 (2022), no. 1, 193–246.
-- We consider a compact connected CR manifold with a transversal CR locally free $\mathbb R$-action endowed with a rigid positive CR line bundle. We prove that a certain Fourier-Szeg\H{o} kernel of the CR sections on the high tensor powers admits a full asymptotic expansion and we establish $\mathbb R$-equvariant Kodaira embedding theorem for CR manifolds. Using similar methods, we also establish an analytic proof of an $\mathbb R$-equivariant Boutete de Monvel embedding theorems for irregular Sasakian manifolds. As applications of our results, we obtain Torus equivariant Kodaira and Boutet de Monvel embedding theorems for CR manifolds and Torus equivariant Kodaira and embedding theorems for complex manifolds.
XXVIII. Equidistribution theorems on strongly pseudoconvex domains (jointly with Guokuan Shao), arXiv:1708.01094, Trans. Amer. Math. Soc. 372 (2019), no. 2,
-- This work consists of two parts. In the first part, we consider a compact connected strongly pseudoconvex CR manifold $X$ with a transversal CR $S^{1}$ action. We establish an equidistribution theorem on zeros of CR functions. The main techniques involve a uniform estimate of Szeg\H{o} kernel on $X$.
In the second part, we consider a general complex manifold $M$ with a strongly pseudoconvex boundary $X$. By using classical result of Boutet de Monvel-Sj\"ostrand about Bergman kernel asymptotics, we establish an equidistribution theorem on zeros of holomorphic functions on $\overline{M}$.
XXVII. An explicit formula for Szeg\H{o} kernels on the Heisenberg group (jointly with Hendrik Herrmann and Xiaoshan Li), arXiv:1706.09762, Acta Math. Sin. (Engl. Ser.) 34 (2018), no. 8, 1225-1247.
-- In this paper, we give an explicit formula for the Szeg\H{o} kernel for $(0,q)$ forms on the Heisenberg group $H_{n+1}$.
XXVI. $G$-invariant Szeg\"o kernel asymptotics and CR reduction (jointly with Rung-Tzung Huang), 60 pages . arXiv:1702.05012, Calculus of Variations and Partial Differential Equations 60 (2021), no. 1, Paper No. 47, 48 pp.
-- Let $(X, T^{1,0}X)$ be a compact connected orientable CR manifold of dimension $2n+1$ with non-degenerate Levi curvature. Assume that $X$ admits a connected compact Lie group action $G$. Under certain natural assumptions about the group action $G$, we show that the $G$-invariant Szeg\"o kernel for $(0,q)$ forms is a complex Fourier integral operator, smoothing away $\mu^{-1}(0)$ and there is a precise description of the singularity near $\mu^{-1}(0)$, where $\mu$ denotes the CR moment map. We apply our result to the case when $X$ admits a transversal CR $S^1$ action and deduce an asymptotic expansion for the $m$-th Fourier component of the $G$-invariant Szeg\"o kernel for $(0,q)$ forms as $m$ goes to infinity.
XXV. Szeg\"o kernel asymptotic expansion on strongly pseudoconvex CR manifolds with $S^1$ action (jointly with Hendrik Herrmann and Xiaoshan Li), arXiv:1610.04669, International Journal of Mathematics, Vol. 29, No. 9 (2018).
-- Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n \ge 1$ with a transversal CR $S^1$ action on $X$. We establish an asymptotic expansion for the $m$-th Fourier component of the Szeg\H{o} kernel function as $m\To\infty$, where the expansion involves a contribution in terms of a distance function from lower dimensional strata of the $S^1$ action. We also obtain explicit formulas for the first three coefficients of the expansion.
XXIV. On the stability of equivariant embedding of compact CR manifolds with circle action (jointly with Xiaoshan Li and George Marinescu), arXiv:1608.00893, Math. Z. 289 (2018), no. 1-2, 201--222.
-- We prove the stability of equivariant embedding of compact strictly pseudoconvex CR manifolds with transversal CR circle action under circle invariant perturbations of the CR structures.
XXIII. The asymptotics of analytic torsion on CR manifolds with $S^1$ action (jointly with Rung-Tzung Huang), arXiv:1605.07507, Commun. Contemp. Math. 21 (2019), no. 4, 1750094, 35 pp.
-- Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n \ge 1$ with a transversal CR $S^1$-action on $X$. We introduce the Fourier components of the Ray-Singer analytic torsion on $X$ with respect to the $S^1$-action. We establish an asymptotic formula for the Fourier components of the analytic torsion with respect to the $S^1$-action. This generalizes the asymptotic formula of Bismut and Vasserot on the holomorphic Ray-Singer torsion associated with high powers of a positive line bundle to strongly pseudoconvex CR manifolds with a transversal CR $S^1$-action.
XXII. Equivariant Kodaira embedding for CR manifolds with circle action (jointly with Xiaoshan Li and George Marinescu ), 38 pages . arXiv:1603.08872, Michigan Mathematical Journal 70 (2021), no. 1, 550--113.
-- We consider a compact CR manifold with a transversal CR locally free circle action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szeg\H{o} kernel of the CR sections in the high tensor powers admits a full asymptotic expansion. As a consequence, we establish an equivariant Kodaira embedding theorem.
XXI. Szegö kernel expansion and equivariant embedding of CR manifodls with circle action (jointly with Hendrik Herrmann and Xiaoshan Li ), arXiv:1512.03952 Ann. Glob. Anal. Geom. 52 (2017), no. 3, 313--340.
-- Let $X$ be a compact quasi-regular Sasakian manifold. In this paper, we establish the asymptotic expansion of Szeg\"o kernel of positive Fourier components and by using the asymptotics, we show that $X$ can be equivariant CR embedded into a Sasakian manifold in $\mathbb C^N$ with transversal CR \emph{simple} $S^1$ action and this embedding is compatible with the respective Reeb vector field.
XX. Extremal metrics for the $Q^\prime$-curvature in three dimensions (jointly with Jeffrey Caes and Paul Yang ), arXiv:1511.05248, C. R. Math. Acad. Sci. Paris, 354(2016-04), no. 4, 407-410.
-- This is an announcement of the main results of XIX.
XIX. Extremal metrics for the $Q^\prime$-curvature in three dimensions (jointly with Jeffrey Caes and Paul Yang ), J. European Mathematical Society, 21 (2019), no. 2, 585--626.
-- We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial steps are to show that the $P^\prime$-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green's function for $\sqrt{P^\prime}$.
XVIII. Heat kernel asymptotics and a local index theorem for CR manifolds with $S^1$ action (jointly with Jih-Hsin Cheng and I-Hsun Tsai ), arXiv:1511.00063, Mémoires de la Société Mathématique de France, No. 112, 2019, 152 pages.
-- Let $(X, T^{1,0}X)$ be a compact CR manifold of dimension $2n+1$ with a transversal CR locally free $S^1$ action and let $E$ be a rigid CR vector bundle over $X$. For every $m\in\mathbb Z$, let $H^j_{b,m}(X,E)$ be the $m$-th $S^1$ Fourier coefficient of the $j$-th $\ddbar_b$ Kohn-Rossi cohomology group with values in $E$. In this paper, we prove that the Euler characteristic $\sum\limits^n_{j=0}(-1)^j{\rm dim}H^j_{b,m}(X,E)$can be computed in terms of the tangential Chern character of $E$, the tangential Todd class of $T^{1,0}X$, and the Chern polynomial of the Levi curvature of $X$. As applications, we can produce many CR functions on a weakly pseudoconvex CR manifold with such an $S^1$ action and many CR sections on some class of CR manifolds. In some cases, we can reinterpret Kawasaki's Hirzebruch-Riemann-Roch formula for a complex orbifold with an orbifold holomorphic line bundle by an integral over a smooth CR manifold.
XVII. Morse inequalities for Fourier components of Kohn-Rossi cohomology of CR manifolds with $S^1$ action (jointly with Xiaoshan Li), arXiv:1506.06459, Math. Z. 284 (2016), no. 1-2, 441--468.
-- Let $X$ be a compact connected orientable CR manifold of dimension $2n-1, n\geq 2$ with a transversal CR $S^1$ action on $X$. We study the positive Fourier coefficients defined by Epstein. By studying the Szeg\"o kernel respect to the positive Fourier coefficients we establish the Morse inequalities on $X$. Using the Morse inequalities we have established on $X$ we prove that there are abundant CR functions on $X$ when $X$ is weakly pseudoconvex and strongly pseudoconvex at a point.
XVI. Szegö kernel asymptotics and Morse inequalities on CR manifolds with $S^1$ action (jointly with Xiaoshan Li), arXiv:1502.02365, Asian Journal of Mathematics, Vol. 22, No. 3 (2018), pp. 413--450.
-- Let X be a compact orientable CR manifold. We assume there is a transversal CR S^1 action on X. Let L^k be the k-th power of a rigid CR line bundle L over X. Without any assumption on the Levi form of X, we obtain a scaling upper bound for the partial Szegoe kernel on (0,q) forms with values in L^k. After integration, this gives the weak Morse inequalities. By a refined spectral analysis, we also obtain the strong Morse inequalities in CR setting. We apply the strong Morse inequalities to show that the Grauert-Riemenschneider criterion is also true in CR setting.
XV. Szegö kernel asymptotics and Kodaira embedding theorems of Levi-flat CR manifolds (jointly with G. Marinescu), arXiv:1502.01642, Math. Res. Lett. 24 (2017), no. 5, 1385--1451.
-- Let X be an orientable comp-act Levi-flat CR manifold and let L be a positive CR complex line bundle over X. We prove that certain microlocal conjugation of the associated Szego kernel admits an asymptotic expansion with respect to high power of L. As an application, we give a Szego kernel proof of the Kodaira type embedding theorem on Levi-flat CR manifodls due to Ohsawa and Sibony.
XIV. Berezin-Toeplitz quantization for lower energy forms(jointly with G. Marinescu), arXiv:1411.6654, Comm. Partial Differential Equations 42 (2017), no. 6, 895--942.
-- Let M be an arbitrary complex manifold and let L be a Hermitian holomorphic line bundle over M. We introduce the Berezin-Toeplitz quantization of the open set of M where the curvature on L is non-degenerate. The quantum spaces are the spectral spaces corresponding to [0, k−N ] (N > 1 fixed), of the Kodaira Laplace operator acting on forms with values in tensor powers L^k. We establish the asymptotic expansion of associated Toeplitz operators and their composition as k → ∞ and we define the corresponding star-product. If the Kodaira Laplace operator has a certain spectral gap this method yields quantization by means of harmonic forms. As applications, we obtain the Berezin-Toeplitz quantization for semi-positive and big line bundles.
XIII. Bergman kernel asymptotics and a pure analytic proof of Kodaira embedding theorem. to appear in Complex Analysis and Geometry, Springer Proc. Math. Stat., Available at arXiv:1411.5441.
-- In this paper, we survey recent results about the asymptotic expansion of Bergman kernel and we give a Bergman kernel proof of Kodaira embedding theorem.
XII. On the singularities of the Szegö projections on lower energy formss(jointly with G. Marinescu), arXiv:1407.6305, J. Differential Geom. 107 (2017), no. 1, 83--155.
-- Let X be an abstract not necessarily compact orientable CR manifold of dimension 2n-1, n>1. Let $\Box^{(q)}_b$ be the Gaffney extension of Kohn Laplacian for (0,q) forms. We show that the spectral function of $\Box^{(q)}_b$ admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if X is compact and the Levi form is non-degenerate of constant signature on X, then the spectrum of $\Box^{(q)}_b$ in $]0,\infty[$ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation of the associated Szegoe kernel admits an asymptotic expansion under a local closed range condition. As applications, we establish the Szegoe kernel asymptotic expansions on some weakly pseudoconvex CR manifolds and on CR manifolds with transversal CR $S^1$ actions. By using theses asymptotics, we establish some local embedding theorems on CR manifolds and we give an analytic proof of a theorem of Lempert asserting that a compact strictly pseudoconvex CR manifold of dimension three with a transversal CR $S^1$ action can be CR embedded into complex space.
XI. On CR Paneitz operators and CR pluriharmonic functions, arXiv:1405.0158. Math. Ann. 362 (2015), no. 3-4, 903--929.
Let X be a compact orientable embeddable strongly pseudoconvex CR manifold of dimensional three. Let P be the associate Paneitz operator and let \hat{\mathcal{P}} be the space of L^2 CR pluriparmonic functions. The operator P and the space \hat{\mathcal{P}} play important roles in CR embedding problems and CR conformal geometry. The operator P is a real, symmetric, fourth order non-hypoelliptic partial differential operator and \hat{\mathcal{P}} is an infinite dimensional subspace of L^2(X). In CR embedding problems and CR conformal geometry, it is crucial to be able to answer the following fumdamental analytic problems abou P and \hat{\mathcal{P}}: (I) Is P seld-adjoint? Does P has L^2 closed range? What is the spectrum of P? (II) If we have Pu=f, where f is in some Sobolev space H^s(X) , s\in\mathbb Z, and u is orthogonal to {\rm Ker\,}P. Can we have u\in H^{s'}(X) for some s'\in\mathbb Z?(III) It is well-known that \hat{\mathcal{P}}\subset{\rm Ker\,}p and if X has torsion zero then \hat{\mathcal{P}}={\rm Ker\,}P. It remains an important problem to determine the precise geometrical condition under which the kernel of P is exactly the CR pluriharmonic functions or eben a direct sum of a finite dimensional sunspace with CR pluriharmonic functions. (IV) Let \Pi be the orthogonal projection onto {\rm Ker\,}P and let $\tau$ be the orthogonal projection onto \hat{\mathcal{P}}. Let \Pi(x,y) and \tau(x,y) denote the distribution kernels of \Pi and \tau respectively. The {\rm P'\,} operator induced by Case and Yang recently plays a critical role in CR conformal geometry. To understand the operator {\rm P'\,}, it is crucial to be able to know the exactly forms of \Pi(x,y) and \tau(x,y). The purpose of this work is to completely answer these questions.
X. Szegö kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds, arXiv:1401.6647, Memoirs of the American Mathematical Society, 254 (2018), no. 1217.
Let X be an abstract not necessarily compact orientable CR manifold of dimension 2n-1, n>1, and let let Lk be the k-th tensor power of a CR complex line bundle L over X. Given a non-negative integer q, q<n, let \Box^{(q)}_{b,k} denote the Gaffney extension of Kohn Laplacian for (0,q) forms with values in Lk. For \lambda is greater than 0, let \Pi^{(q)}_{k,\leq\lambda}:=E([0,\lambda]), where E denotes the spectral measure of \Box^{(q)}_{b,k}. The purpose of this work is to completely study the asymptotic behavior of the the kernel of \Pi^{(q)}_{k,\leq\lambda}with respect to k. The difficulty of this problem comes from the presence of positive eigenvalues of the curvature of the line bundle and negative eigenvalues of the Levi form of the associated CR manifold X and hence the semi-classical characteristic manifold of the associated Kohn Laplacian is always degenerate at some point. This difficulty is also closely related to the fact that in the L2estimates for the tangential Cauchy-Riemann operator of Kohn-Hörmander there is a curvature term from the line bundle as well from the Levi form and, in general, it is very difficult to control the sign of the total curvature contribution. In this work, we introduce some kind of microlocal cut-off function F_k and we prove that \Pi^{(q)}_{k,\leq k^{-N_0}}F^*_k, F_k\Pi^{(q)}_{k,\leq k^{-N_0}}F^*_k, N_0 is greater than 1, admit asymptotic expansions on the non-degenerate part of the characteristic manifold of \Box^{(q)}_{b,k}. Moreover, we show that F_k\Pi^{(q)}_{k,\leq0}F^*_k admits a full asymptotic expansion if \Box^{(q)}_{b,k} has small spectral gap property with respect to F_k and \Pi^{(q)}_{k,\leq0} is k-negligible away the diagonal with respect to F_k. By using these asymptotics, we establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR S1 actions.
IX. Soving the Kohn Laplacian on asymptotically flat CR manifolds of dimension 3 (jointly with Po Lam Yung), arXiv:1303.6557. Advances in Mathematics, volume 281, pages 734-822.
The study of this paper was motivated by the positive p-mass theorem in CR geometry. In this work, we show that the weighted Kohn Laplacian on an asymptotically flat pseudohermitian 3-manifold has L2 closed range and the associated partial inverse and the Szegö projection enjoy some regularity properties near infinity. As an application, we prove the existence of some special functions in the kernel of the Kohn Laplacian on an asymptotically flat pseudohermitian 3-manifold of specific growth rate near infinity, which are used in the proof of the positive p-mass theorem in CR geometry for dimension 3.
VIII. The tangential Cauchy-Riemann complex on the Heisenberg group Via Conformal Invariance (jointly with Po Lam Yung), Bulletin of the institute of Mathematics, Academia Sinica(New Series) 8, (2013), no. 3, Page 359--375. arXiv:1303.6547, .
The Heisenberg group H1 is known to be conformally equivalent to the CR sphere S3 minus a point. We use this fact, together with the knowledge of the tangential Cauchy-Riemann operator on the compact CR manifold S3, to solve the corresponding operator on H1.
VII. The second coefficient of the asymptotic expansion of the weighted Bergman kernel on Cn, Bulletin of the Institute of Mathematics Academia Sinica, volume(11) 2016, Page 521-570.
In the work of S. K. Donaldson, it was shown that the precise formula of the second coefficient of the asymptotic expansion for Bergman kernel for high powers of positive line bundles plays an important role in the investigations about the relation between canonical metrics in Kähler geometry and stability in algebraic geometry. When q>0, the second coefficient of the asymptotic expansion for Bergman kernel will give us a new geometric invariant and it is interesting to know how to generalize the work of Donaldson and the concept of Mabuchi functional to the case when the curvature of L is non-degenerate. In this paper, we give for the first time a formula of the second coefficient of the expansion of the Bergman kernel for q>0 on Cn with respect to a high power of trivial line bundle over Cn endowed with non-degenerate curvature.
VI. Existence of CR sections for high power of semi-positive generalized Sasakian CR line bundles over generalized Sasakian CR manifolds, Ann. Glob. Anal. Geom. 47 (2015), no. 1, 13–62. arXiv:1204.4810
Let X be a compact CR manifold of dimension 2n-1, n>1. When the Levi form of X has negative eigenvalues, global embedding problems for X are a very important subject in CR geometry. In this case, the space of global CR functions could be finite dimensional (even trivial) and in general, X can not be globally CR embedded into complex space. Since many interesting examples live in the projective space, it is thus natural to consider a setting analogue to the Kodaira embeding theorem and ask if X can be embeded in the projective space by means of CR sections of a CR line bundle of positive curvature. Inspired by Kodaira, we introduced the idea of embedding CR manifolds by means of CR sections of k-th tensor powers Lk of a CR line bundle L over X. If the dimension of the space of CR sections of Lk is large, when k large, one should find many CR sections to embed X into projective space. In analogy to the Kodaira embedding theorem, it is natural to ask if X can be globally embedded into projective space when it carries a CR line bundle with positive curvature?
To understand this question, it is crucial to be able to know if L is big when L is positive? The difficulty of this problem comes from the presence of positive eigenvalues of the curvature of the line bundle and negative eigenvalues of the Levi form of X and it causes the associated Kohn Laplacian to have no semi-classical spectral gap. In this paper, by carefully studying semi-classical behaviour of microlocal Fourier transforms of the extreme functions for the spaces of lower energy forms of the associated Kohn Laplacian, we could prove that L is big if L is positive under rigidity conditions on L and the manifold X. The rigidity conditions we introduced in this paper can be seen as a generalization of the concept of Sasakian manifolds and Sasakian line bundles in CR geometry.
V. Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles(jointly with G. Marinescu), Communications in Analysis and Geometry, 22(2014), Page 1-108. arXiv:1112.5464
In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degenerate. As application we obtain the Bergman kernel asymptotics for adjoint semi-positive line bundles over complete Kaehler manifolds, on the set where the curvature is positive. We also prove the asymptotics for big line bundles endowed with singular Hermitian metrics with strictly positive curvature current. In this case the full asymptotics holds outside the singular locus of the metric.
IV. On the coefficients of the asymptotic expansion of the kernel of Berezin-Toeplitz quantization, Annals of Global Analysis and Geometry: Volume 42, Issue 2 (2012), Page 207-245. arXiv:1108.0498
We give new methods for computing the coefficients of the asymptotic expansions of the kernel of Berezin-Toeplitz quantization obtained recently by Ma-Marinescu, and of the composition of two Berezin-Toeplitz quantizations. Our main tool is the stationary phase formula of Melin-Sjöstrand.
III. Szegö kernel asymptotics and Morse inequalities on CR manifolds(jointly with G. Marinescu), Math. Z. 271 (2012), Page 509-553. arXiv:1005.5471
Let X be an abstract compact orientable CR manifold of dimension 2n-1, n>1, and let Lk be the k-th tensor power of a CR complex line bundle L over X. We assume that condition Y(q) holds at each point of X. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in Lk for large k. After integration, this gives weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities. We apply the strong Morse inequalities to the embedding of some convex-concave manifolds.
II. Projections in several complex variables, Mémoires de la Société Mathématique de France, 123 (2010), 131 pages. arXiv:0810.4083
This work consists two parts. In the first part, we completely study the heat equation method of Menikoff-Sjöstrand and apply it to the Kohn Laplacian defined on a compact orientable connected CR manifold. We then get the full asymptotic expansion of the Szegö projection for (0,q) forms when the Levi form is non-degenerate. This generalizes a result of Boutet de Monvel and Sjöstrand for (0,0) forms. Our main tool is Fourier integral operators with complex valued phase functions of Melin and Sjöstrand. In the second part, we obtain the full asymptotic expansion of the Bergman projection for (0,q) forms when the Levi form is non-degenerate. This also generalizes a result of Boutet de Monvel and Sjöstrand for (0,0) forms. We introduce a new operator analogous to the Kohn Laplacian defined on the boundary of a domain and we apply the heat equation method of Menikoff and Sjöstrand to this operator. We obtain a description of a new Szegö projection up to smoothing operators. Finally, by using the Poisson operator, we get our main result. This work is a part of my Phd thesis under the direction of Prof. Johannes Sjöstrand.
I. Projections in several complex variables, Phd thesis at Ecole Polytechnique, Palaiseau, Paris,, 2008. 254 pages. tel-00332787