arXiv subjects

Dan Popovici

Publications and source records attributed to Dan Popovici.

48 records · Page 3Linked to original sources

arXiv2015

Aeppli Cohomology Classes Associated with Gauduchon Metrics on Compact Complex Manifolds

We propose the study of a Monge-Ampère-type equation in bidegree $(n-1,\,n-1)$ rather than $(1,\,1)$ on a compact complex manifold $X$ of dimension $n$ for which we prove uniqueness of the solution subject to positivity and normalisation restrictions. Existence will hopefully be dealt with in future work. The aim is to construct a special Gauduchon metric uniquely associated with any Aeppli cohomology class of bidegree $(n-1,\,n-1)$ lying in the Gauduchon cone of $X$ that we hereby introduce as a subset of the real Aeppli cohomology group of type $(n-1,\,n-1)$ and whose first properties we study. Two directions for applications of this new equation are envisaged: to moduli spaces of Calabi-Yau $\partial\bar\partial$-manifolds and to a further study of the deformation properties of the Gauduchon cone beyond those given in this paper.

math.DG↗

arXiv2015

Sufficient Bigness Criterion for Differences of Two Nef Classes

We prove the qualitative part of Demailly's conjecture on transcendental Morse inequalities for differences of two nef classes satisfying a numerical relative positivity condition on an arbitrary compact Kähler (and even more general) manifold. The result improves on an earlier one by J. Xiao whose constant $4n$ featuring in the hypothesis is now replaced by the optimal and natural $n$. Our method follows arguments by Chiose as subsequently used by Xiao up to the point where we introduce a new way of handling the estimates in a certain Monge-Ampère equation. This result is needed to extend to the Kähler case and to transcendental classes the Boucksom-Demailly-Paun-Peternell cone duality theorem if one is to follow these authors' method and was conjectured by them.

math.DG↗

arXiv2014

On the sum between a closable operator $T$ and a $T$-bounded operator

We provide several perturbation theorems regarding closable operators on a real or complex Hilbert space. In particular we extend some classical results due to Hess--Kato, Kato--Rellich and Wüst. Our approach involves ranges of matrix operators of the form $\left(\begin{array}{cc}\!\! I & A\!\!\\ \!\! -B& I\!\!\end{array}\right)$.

math.FA↗

arXiv2014

On operators which are adjoint to each other

Given two linear operators $S$ and $T$ acting between Hilbert spaces $\mathscr{H}$ and $\mathscr{K}$, respectively $\mathscr{K}$ and $\mathscr{H}$ which satisfy the relation \begin{equation*} \langle Sh, k\rangle=\langle h, Tk\rangle, \quad h\in\dom S, \ k\in\dom T, \end{equation*} i.e., according to the classical terminology of M.H. Stone, which are adjoint to each other, we provide necessary and sufficient conditions in order to ensure the equality between the closure of $S$ and the adjoint of $T.$ A central role in our approach is played by the range of the operator matrix $M_{S, T}=\begin{pmatrix} 1_{\dom S} & -T S & 1_{\dom T} \end{pmatrix}.$ We obtain, as consequences, several results characterizing skewadjointness, selfadjointness and essential selfadjointness. We improve, in particular, the celebrated selfadjointness criterion of J. von Neumann.

math.FA↗

arXiv2012

Transcendental Kähler Cohomology Classes

Associated with a smooth, $d$-closed $(1, 1)$-form $α$ 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 Hörmander'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 $Δ_k"$ 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 $α$ is strictly positive : a Kodaira-type approximately holomorphic projective embedding theorem and a Tian-type almost-isometry theorem for compact Kähler, possibly non-projective, manifolds. Unlike in similar results in the literature for symplectic forms of integral classes, the peculiarity of $α$ lies in its transcendental class. This approach will be hopefully continued in future work by relaxing the positivity assumption on $α$.

math.AG↗

arXiv2011

Factorizations of Linear Relations

Given two linear relations $A$ and $B$ we characterize the existence of a linear relation (operator) $C$ such that $A\subseteq BC$, respectively $A\subseteq CB.$ These factorizations extend and improve well-known results by R.G. Douglas and Z. Sebestyén.

math.FA↗

arXiv2011

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 (Kähler, balanced or {\it strongly Gauduchon}) and those satisfying the $\partial\bar\partial$-lemma or the degeneration at $E_1$ of the Frölicher 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.

math.AG↗

arXiv2010

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 Kähler manifolds, mirrors the modification stability of balanced manifolds proved by Alessandrini and Bassanelli.

math.CV↗

arXiv2010

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.

math.AG↗

arXiv2006

Effective Local Finite Generation of Multiplier Ideal Sheaves

Let $ϕ$ be a psh function on a bounded pseudoconvex open set $Ω\subset \C^n$, and let ${\cal I}(ϕ)$ be the associated multiplier ideal sheaf. Motivated by resolution of singularities issues, we establish an effective version of the coherence property of ${\cal I}(mϕ)$ as $m\to +\infty$. Namely, we estimate the order of growth in $m$ of the number of generators needed to engender ${\cal I}(mϕ)$ on a fixed compact subset, as well as the growth of the coefficients featuring in the decomposition of local sections as linear combinations over ${\cal O}_Ω$ 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 ${\cal I}(mϕ)$ is not far from being linear if the singularities of $ϕ$ are reasonable.

math.CV↗

arXiv2004

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

Let $(X, ω)$ be a weakly pseudoconvex Kähler 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.

math.CV↗

arXiv2003

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 $π$ 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 $(\mathrm{Id}-π)\circ D''π=0$. 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 Kähler manifolds. We give here a much simpler proof based on current theory. The idea is to construct local meromorphic sections of $\mathrm{Im} π$ 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.

math.CV↗