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In this article, we solve the strong openness conjecture on the multiplier ideal sheaf associated to any plurisubharmonic function, which was posed by Demailly.

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Given $q\in\set{0,1,\ldots,n-1}$, let $\Box^{(q)}_{b,k}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$ forms with values in $L^k$. For $\lambda\geq0$, let $\Pi^{(q)}_{k,\leq\lambda}:=E((-\infty,\lambda])$, where $E$ denotes the spectral measure of $\Box^{(q)}_{b,k}$. In this work, we prove that $\Pi^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $F_k\Pi^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $N_0\geq1$, admit asymptotic expansions with respect to $k$ on the non-degenerate part of the characteristic manifold of $\Box^{(q)}_{b,k}$, where $F_k$ is some kind of microlocal cut-off function. Moreover, we show that $F_k\Pi^{(q)}_{k,\leq0}F^*_k$ admits a full asymptotic expansion with respect to $k$ 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 $S^1$ action.

Let $(X,T^{1,0}X)$ be a compact orientable embeddable three dimensional strongly pseudoconvex CR manifold and let ${\rm P\,}$ be the associated CR Paneitz operator. In this paper, we show that (I) ${\rm P\,}$ is self-adjoint and ${\rm P\,}$ has $L^2$ closed range. Let $N$ and $\Pi$ be the associated partial inverse and the orthogonal projection onto ${\rm Ker\,}{\rm P\,}$ respectively, then $N$ and $\Pi$ enjoy some regularity properties. (II) Let $\hat{\mathcal{P}}$ and $\hat{\mathcal{P}_0}$ be the space of $L^2$ CR pluriharmonic functions and the space of real part of $L^2$ global CR functions respectively. Let $S$ be the associated Szeg\"o projection and let $\tau$, $\tau_0$ be the orthogonal projections onto $\hat{\mathcal{P}}$ and $\hat{\mathcal{P}_0}$ respectively. Then, $\Pi=S+\ol S+F_0$, $\tau=S+\ol S+F_1$, $\tau_0=S+\ol S+F_2$, where $F_0, F_1, F_2$ are smoothing operators on $X$. In particular, $\Pi$, $\tau$ and $\tau_0$ are Fourier integral operators with complex phases and $\hat{\mathcal{P}}^\perp\bigcap{\rm Ker\,}{\rm P\,}$, $\hat{\mathcal{P}_0}^\perp\bigcap\hat{\mathcal{P}}$, $\hat{\mathcal{P}_0}^\perp\bigcap{\rm Ker\,}{\rm P\,}$ are all finite dimensional subspaces of $C^\infty(X)$ (it is well-known that $\hat{\mathcal{P}_0}\subset\hat{\mathcal{P}}\subset{\rm Ker\,}{\rm P\,}$). (III) ${\rm Spec\,}{\rm P\,}$ is a discrete subset of $\Real$ and for every $\lambda\in{\rm Spec\,}{\rm P\,}$, $\lambda\neq0$, $\lambda$ is an eigenvalue of ${\rm P\,}$ and the associated eigenspace $H_\lambda({\rm P\,})$ is a finite dimensional subspace of $C^\infty(X)$.

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 K\"ahler 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.