Effective Local Finite Generation of Multiplier Ideal Sheaves

Let $\phi$ be a psh function on a bounded pseudoconvex open set \Omega \subset \C^n, and let ${\cal I}(\phi)$ be the associated multiplier ideal sheaf. Motivated by resolution of singularities issues, we establish an effective version of the coherence property of ${\cal I}(m\phi)$ as $m\to +\infty$. Namely, we estimate the order of growth in $m$ of the number of generators needed to engender ${\cal I}(m\phi)$ on a fixed compact subset, as well as the growth of the coefficients featuring in the decomposition of local sections as linear combinations over ${\cal O}_{\Omega}$ of finitely many generators. The main idea is to use Toeplitz concentration operators involving Bergman kernels associated with singular weights. Our approach relies on asymptotic integral estimates of singularly weighted Bergman kernels of independent interest. In the second part of the paper, we estimate the additivity defect of multiplier ideal sheaves already known to be subadditive by a result of Demailly, Ein, and Lazarsfeld. This implies that the decay rate of ${\cal I}(m\phi)$ is not far from being linear if the singularities of $\phi$ are reasonable.Comment: 27 page

Limits of Moishezon Manifolds under Holomorphic Deformations

Given a (smooth) complex analytic family of compact complex manifolds, we prove that the central fibre must be Moishezon if the other fibres are Moishezon. Using a "strongly Gauduchon metric" on the central fibre whose existence was proved in our previous work on limits of projective manifolds, we show that the irreducible components of the relative Barlet space of divisors contained in the fibres are proper over the base even under the weaker assumption that the $\partial\bar\partial$-lemma hold on all the fibres except, possibly, the central one. This implies that the algebraic dimension of the central fibre cannot be lower than that of the generic fibre. Since the latter is already maximal thanks to the Moishezon assumption, the central fibre must be of maximal algebraic dimension, hence Moishezon.Comment: 13 page

Degeneration at E2E_2 of Certain Spectral Sequences

We propose a Hodge theory for the spaces $E_2^{p,\,q}$ featuring at the second step either in the Fr\"olicher spectral sequence of an arbitrary compact complex manifold $X$ or in the spectral sequence associated with a pair $(N,\,F)$ of complementary regular holomorphic foliations on such a manifold. The main idea is to introduce a Laplace-type operator associated with a given Hermitian metric on $X$ whose kernel in every bidegree $(p,\,q)$ is isomorphic to $E_2^{p,\,q}$ in either of the two situations discussed. The surprising aspect is that this operator is not a differential operator since it involves a harmonic projection, although it depends on certain differential operators. We then use this Hodge isomorphism for $E_2^{p,\,q}$ to give sufficient conditions for the degeneration at $E_2$ of the spectral sequence considered in each of the two cases in terms of the existence of certain metrics on $X$. For example, in the Fr\"olicher case we prove degeneration at $E_2$ if there exists an SKT metric $\omega$ (i.e. such that $\partial\bar\partial\omega=0$) whose torsion is small compared to the spectral gap of the elliptic operator $\Delta' + \Delta"$ defined by $\omega$. In the foliated case, we obtain degeneration at $E_2$ under a hypothesis involving the Laplacians $\Delta'_N$ and $\Delta'_F$ associated with the splitting $\partial = \partial_N + \partial_F$ induced by the foliated structure.Comment: 40 page

Stability of Strongly Gauduchon Manifolds under Modifications

In our previous works on deformation limits of projective and Moishezon manifolds, we introduced and made crucial use of the notion of strongly Gauduchon metrics as a reinforcement of the earlier notion of Gauduchon metrics. Using direct and inverse images of closed positive currents of type $(1, \, 1)$ and regularisation, we now show that compact complex manifolds carrying strongly Gauduchon metrics are stable under modifications. This stability property, known to fail for compact K\"ahler manifolds, mirrors the modification stability of balanced manifolds proved by Alessandrini and Bassanelli.Comment: 8 page

A Simple Proof of a Theorem by Uhlenbeck and Yau

A subbundle of a Hermitian vector bundle $(E, h)$ can be metrically and differentiably defined by the orthogonal projection onto this subbundle. A weakly holomorphic subbundle of a Hermitian holomorphic bundle is, by definition, an orthogonal projection $\pi$ lying in the Sobolev space $L^2_1$ of $L^2$ sections with $L^2$ first order derivatives in the sense of distributions, which satisfies furthermore $(\mathrm{Id}-\pi)\circ D''\pi=0$. We give a new simple proof of the fact that a weakly holomorphic subbundle of $(E, h)$ defines a coherent subsheaf of ${\cal O}(E),$ that is a holomorphic subbundle of $E$ in the complement of an analytic set of codimension $\geq 2.$ This result was the crucial technical argument in Uhlenbeck's and Yau's proof of the Kobayashi-Hitchin correspondence on compact K\"ahler manifolds. We give here a much simpler proof based on current theory. The idea is to construct local meromorphic sections of $\mathrm{Im} \pi$ which locally span the fibers. We first make this construction on every one-dimensional submanifold of $X$ and subsequently extend it via a Hartogs-type theorem of Shiffman's.Comment: 19 page

Limits of Projective and ∂∂ˉ\partial\bar\partial-Manifolds under Holomorphic Deformations

We prove that if in a (smooth) holomorphic family of compact complex manifolds all the fibres, except one, are projective, then the remaining (limit) fibre must be Moishezon. In an earlier work, we proved this result under the extra assumption that the limit fibre carries a strongly Gauduchon metric. In the present paper, we remove the extra assumption by proving that if all the fibres, except one, are $\partial\bar\partial$-manifolds, then the limit fibre carries a strongly Gauduchon metric. The $\partial\bar\partial$-assumption on the generic fibre is much weaker than the projective, K\"ahler and even {\it class} ${\cal C}$ assumptions, but it implies the Hodge decomposition and symmetry, while being called the 'validity of the $\partial\bar\partial$-lemma' by many authors. Our method consists in starting off with an arbitrary smooth family $(\gamma_t)_{t\in\Delta}$ of Gauduchon metrics on the fibres $(X_t)_{t\in\Delta}$ and in correcting $\gamma_0$ in a finite number of steps to a strongly Gauduchon metric by repeated uses of the $\partial\bar\partial$-assumption on the generic fibre and of estimates of minimal $L^2$-norm solutions for $\partial$-, $\bar\partial$- and $d$-equations.Comment: To appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. The results in this paper correspond to those in the last section 4 of the 2009 version of this posting and the revised 2016 second versio

Deformation Openness and Closedness of Various Classes of Compact Complex Manifolds; Examples

We review the relations between compact complex manifolds carrying various types of Hermitian metrics (K\"ahler, balanced or {\it strongly Gauduchon}) and those satisfying the $\partial\bar\partial$-lemma or the degeneration at $E_1$ of the Fr\"olicher spectral sequence, as well as the behaviour of these properties under holomorphic deformations. The emphasis will be placed on the notion of {\it strongly Gauduchon} (sG) manifolds that we introduced recently in the study of deformation limits of projective and Moishezon manifolds. Various examples of sG and non-sG manifolds are exhibited while a range of constructions already known in the literature are reviewed and reinterpreted from this new standpoint.Comment: 48 page

L2L^2 Extension for Jets of Holomorphic Sections of a Hermitian Line Bundle

Let $(X, \omega)$ be a weakly pseudoconvex K\"ahler manifold, $Y \subset X$ a closed submanifold defined by some holomorphic section of a vector bundle over $X,$ and $L$ a Hermitian line bundle satisfying certain positivity conditions. We prove that for any integer $k\geq 0,$ any section of the jet sheaf $L\otimes {\cal O}_X/{\cal I}_Y^{k+1},$ which satisfies a certain $L^2$ condition, can be extended into a global holomorphic section of $L$ over $X$ whose $L^2$ growth on an arbitrary compact subset of $X$ is under control. In particular, if $Y$ is merely a point, this gives the existence of a global holomorphic function with an $L^2$ norm under control and with prescribed values for all its derivatives up to order $k$ at a point. This result generalizes the $L^2$ extension theorems of Ohsawa-Takegoshi and of Manivel to the case of jets of sections of a line bundle. A technical difficulty is to achieve uniformity in the constant appearing in the final estimate. In this respect, we make use of the exponential map and of a Rauch-type comparison theorem for complete Riemannian manifolds

Transcendental K\"ahler Cohomology Classes

Associated with a smooth, $d$-closed $(1, 1)$-form $\alpha$ of possibly non-rational De Rham cohomology class on a compact complex manifold $X$ is a sequence of asymptotically holomorphic complex line bundles $L_k$ on $X$ equipped with $(0, 1)$-connections $\bar\partial_k$ for which $\bar\partial_k^2\neq 0$. Their study was begun in the thesis of L. Laeng. We propose in this non-integrable context a substitute for H\"ormander's familiar $L^2$-estimates of the $\bar\partial$-equation of the integrable case that is based on analysing the spectra of the Laplace-Beltrami operators $\Delta_k"$ associated with $\bar\partial_k$. Global approximately holomorphic peak sections of $L_k$ are constructed as a counterpart to Tian's holomorphic peak sections of the integral-class case. Two applications are then obtained when $\alpha$ is strictly positive : a Kodaira-type approximately holomorphic projective embedding theorem and a Tian-type almost-isometry theorem for compact K\"ahler, possibly non-projective, manifolds. Unlike in similar results in the literature for symplectic forms of integral classes, the peculiarity of $\alpha$ lies in its transcendental class. This approach will be hopefully continued in future work by relaxing the positivity assumption on $\alpha$.Comment: 49 page

Adiabatic Limit and Deformations of Complex Structures

Based on our recent adaptation of the adiabatic limit construction to the case of complex structures, we prove the fact, that we first dealt with in 2009 and 2010, that the deformation limiting manifold of any holomorphic family of Moishezon manifolds is Moishezon. Two new ingredients, hopefully of independent interest, are introduced. The first one canonically associates with every compact complex manifold $X$, in every degree $k$, a holomorphic vector bundle over $\mathbb{C}$ of rank equal to the $k$-th Betti number of $X$. This vector bundle shows that the degenerating page of the Fr\"olicher spectral sequence of $X$ is the holomorphic limit, as $h\in\mathbb{C}^\star$ tends to $0$, of the $d_h$-cohomology of $X$, where $d_h=h\partial + \bar\partial$. A relative version of this vector bundle is canonically associated with every holomorphic family of compact complex manifolds. The second new ingredient is a relaxation of the notion of strongly Gauduchon (sG) metric that we introduced in 2009. For a given positive integer $r$, a Gauduchon metric $\gamma$ on an $n$-dimensional compact complex manifold $X$ is said to be $E_r$-sG if $\partial\gamma^{n-1}$ represents the zero cohomology class on the $r$-th page of the Fr\"olicher spectral sequence of $X$. Strongly Gauduchon metrics coincide with $E_1$-sG metrics.Comment: 41 pages; very minor rewordings in the opening lines of the abstract and the introductio