arXiv subjects

Dan Popovici

Publications and source records attributed to Dan Popovici.

At least 37 records · Page 2Linked to original sources

arXiv2022

Higher-Page Hodge Theory of Compact Complex Manifolds

On a compact $\partial\bar\partial$-manifold $X$, one has the Hodge decomposition: the de Rham cohomology groups split into subspaces of pure-type classes as $H_{dR}^k (X)=\oplus_{p+q=k}H^{p,\,q}(X)$, where the $H^{p,\,q}(X)$ are canonically isomorphic to the Dolbeault cohomology groups $H_{\bar\partial}^{p,\,q}(X)$. For an arbitrary nonnegative integer $r$, we introduce the class of page-$r$-$\partial\bar\partial$-manifolds by requiring the analogue of the Hodge decomposition to hold on a compact complex manifold $X$ when the usual Dolbeault cohomology groups $H^{p,\,q}_{\bar\partial}(X)$ are replaced by the spaces $E_{r+1}^{p,\,q}(X)$ featuring on the $(r+1)$-st page of the Frölicher spectral sequence of $X$. The class of page-$r$-$\partial\bar\partial$-manifolds coincides with the usual class of $\partial\bar\partial$-manifolds when $r=0$ but may increase as $r$ increases. We give two kinds of applications. On the one hand, we give a purely numerical characterisation of the page-$r$-$\partial\bar\partial$-property in terms of dimensions of various cohomology vector spaces. On the other hand, we obtain several classes of examples, including all complex parallelisable nilmanifolds and certain families of solvmanifolds and abelian nilmanifolds. Further, there are general results about the behaviour of this new class under standard constructions like blow-ups and deformations.

math.AG↗

arXiv2022

Balanced Hyperbolic and Divisorially Hyperbolic Compact Complex Manifolds

We introduce two notions of hyperbolicity for not necessarily Kähler $n$-dimensional compact complex manifolds $X$. The first, called {\it balanced hyperbolicity}, generalises Gromov's Kähler hyperbolicity by means of Gauduchon's balanced metrics. The second, called {\it divisorial hyperbolicity}, generalises the Brody hyperbolicity by ruling out the existence of non-degenerate holomorphic maps from $\C^{n-1}$ to $X$ that have what we term a subexponential growth. Our main result in the first part of the paper asserts that every balanced hyperbolic $X$ is also divisorially hyperbolic. We provide a certain number of examples and counter-examples and discuss various properties of these manifolds. In the second part of the paper, we introduce the notions of {\it divisorially Kähler} and {\it divisorially nef} real De Rham cohomology classes of degree $2$ and study their properties. They also apply to $C^\infty$, not necessarily holomorphic, complex line bundles and are expected to be implied in certain cases by the hyperbolicity properties introduced in the first part of the work. While motivated by the observation of hyperbolicity properties of certain non-Kähler manifolds, all these four new notions seem to have a role to play even in the Kähler and the projective settings.

math.CV↗

arXiv2022

Some Properties of Balanced Hyperbolic Compact Complex Manifolds

We prove several vanishing theorems for the cohomology of balanced hyperbolic manifolds that we introduced in our previous work and for the $L^2$ harmonic spaces on the universal cover of these manifolds. Other results include a Hard Lefschetz-type theorem for certain compact complex balanced manifolds and the non-existence of certain $L^1$ currents on the universal covering space of a balanced hyperbolic manifold.

math.CV↗

arXiv2021

A Generalised Volume Invariant for Aeppli Cohomology Classes of Hermitian-Symplectic Metrics

We investigate the class of compact complex Hermitian-symplectic manifolds $X$. For each Hermitian-symplectic metric $ω$ on $X$, we introduce a functional acting on the metrics in the Aeppli cohomology class of $ω$ and prove that its critical points (if any) must be Kähler when $X$ is $3$-dimensional. We go on to exhibit these critical points as maximisers of the volume of the metric in its Aeppli class and propose a Monge-Ampère-type equation to study their existence. Our functional is further utilised to define a numerical invariant for any Aeppli cohomology class of Hermitian-symplectic metrics that generalises the volume of a Kähler class. We obtain two cohomological interpretations of this invariant. Meanwhile, we construct an invariant in the form of an $E_2$-cohomology class, that we call the $E_2$-torsion class, associated with every Aeppli class of Hermitian-symplectic metrics and show that its vanishing is a necessary condition for the existence of a Kähler metric in the given Hermitian-symplectic Aeppli class.

math.DG↗

arXiv2021

Deformations of Higher-Page Analogues of $\partial\bar\partial$-Manifolds

We extend the notion of essential deformations from the case of the Iwasawa manifold, for which they were introduced recently by the first-named author, to the general case of page-$1$-$\partial\bar\partial$-manifolds that were jointly introduced very recently by all three authors. We go on to obtain an analogue of the unobstructedness theorem of Bogomolov, Tian and Todorov for page-$1$-$\partial\bar\partial$-manifolds. As applications of this discussion, we study the small deformations of certain Nakamura solvmanifolds and reinterpret the cases of the Iwasawa manifold and its $5$-dimensional analogue from this standpoint.

math.AG↗

arXiv2021

Higher-Page Bott-Chern and Aeppli Cohomologies and Applications

For every positive integer $r$, we introduce two new cohomologies, that we call $E_r$-Bott-Chern and $E_r$-Aeppli, on compact complex manifolds. When $r=1$, they coincide with the usual Bott-Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when $r\geq 2$. They provide analogues in the Bott-Chern-Aeppli context of the $E_r$-cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page-$(r-1)$-$\partial\bar\partial$-manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott-Chern and Aeppli cohomologies and for the spaces featuring in the Frölicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.

math.AG↗

arXiv2020

Limits of Projective and $\partial\bar\partial$-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, Kähler 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 $(γ_t)_{t\inΔ}$ of Gauduchon metrics on the fibres $(X_t)_{t\inΔ}$ and in correcting $γ_0$ 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.

math.AG↗

arXiv2020

Operator equalities and Characterizations of Orthogonality in Pre-Hilbert $C^*$-Modules

In the first part of the paper, we use states on $C^*$-algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^*$-module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^*$-module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^*$-module under the assumption that their inner product has negative real part. We introduce the concept of Pythagoras orthogonality and discuss its properties. We describe this notion for Hilbert space operators in terms of the parallelogram law and some limit conditions. We present several examples in order to illustrate the relationship between the Birkhoff--James, Roberts, and Pythagoras orthogonalities, and the usual orthogonality in the framework of Hilbert $C^*$-modules.

math.FA↗

arXiv2019

Non-Kähler Mirror Symmetry of the Iwasawa Manifold

We propose a new approach to the Mirror Symmetry Conjecture in a form suitable to possibly non-Kähler compact complex manifolds whose canonical bundle is trivial. We apply our methods by proving that the Iwasawa manifold $X$, a well-known non-Kähler compact complex manifold of dimension $3$, is its own mirror dual to the extent that its Gauduchon cone, replacing the classical Kähler cone that is empty in this case, corresponds to what we call the local universal family of essential deformations of $X$. These are obtained by removing from the Kuranishi family the two "superfluous" dimensions of complex parallelisable deformations that have a similar geometry to that of the Iwasawa manifold. The remaining four dimensions are shown to have a clear geometric meaning including in terms of the degeneration at $E_2$ of the Frölicher spectral sequence. On the local moduli space of "essential" complex structures, we obtain a canonical Hodge decomposition of weight $3$ and a variation of Hodge structures, construct coordinates and Yukawa couplings while implicitly proving a local Torelli theorem. On the metric side of the mirror, we construct a variation of Hodge structures parametrised by a subset of the complexified Gauduchon cone of the Iwasawa manifold using the sGG property of all the small deformations of this manifold proved in earlier joint work of the author with L. Ugarte. Finally, we define a mirror map linking the two variations of Hodge structures and we highlight its properties.

math.AG↗

arXiv2018

Holomorphic Deformations of Balanced Calabi-Yau $\partial\bar\partial$-Manifolds

Given a compact complex $n$-fold $X$ satisfying the $\partial\bar\partial$-lemma and supposed to have a trivial canonical bundle $K_X$ and to admit a balanced (=semi-Kähler) Hermitian metric $ω$, we introduce the concept of deformations of $X$ that are {\bf co-polarised} by the balanced class $[ω^{n-1}]\in H^{n-1,\,n-1}(X,\,\C)\subset H^{2n-2}(X,\,\C)$ and show that the resulting theory of balanced co-polarised deformations is a natural extension of the classical theory of Kähler polarised deformations in the context of Calabi-Yau or even holomorphic symplectic compact complex manifolds. The concept of Weil-Petersson metric still makes sense in this strictly more general, possibly non-Kähler context, while the Local Torelli Theorem still holds.

math.AG↗

arXiv2018

Positivity Cones under Deformations of Complex Structures

We investigate connections between the sGG property of compact complex manifolds, defined in earlier work by the second author and L. Ugarte by the requirement that every Gauduchon metric be strongly Gauduchon, and a possible degeneration of the Frölicher spectral sequence. In the first approach that we propose, we prove a partial degeneration at $E_2$ and we introduce a positivity cone in the $E_2$-cohomology of bidegree $(n-2,\,n)$ of the manifold that we then prove to behave lower semicontinuously under deformations of the complex structure. In the second approach that we propose, we introduce an analogue of the $\partial\bar\partial$-lemma property of compact complex manifolds for any real non-zero constant $h$ using the partial twisting $d_h$, introduced recently by the second author, of the standard Poincaré differential $d$. We then show, among other things, that this $h$-$\partial\bar\partial$-property is deformation open.

math.AG↗

arXiv2017

Compact Complex Manifolds with Small Gauduchon Cone

This paper is intended as the first step of a programme aiming to prove in the long run the long-conjectured closedness under holomorphic deformations of compact complex manifolds that are bimeromorphically equivalent to compact Kähler manifolds, known as Fujiki {\it class} ${\cal C}$ manifolds. Our main idea is to explore the link between the {\it class} ${\cal C}$ property and the closed positive currents of bidegree $(1,\,1)$ that the manifold supports, a fact leading to the study of semi-continuity properties under deformations of the complex structure of the dual cones of cohomology classes of such currents and of Gauduchon metrics. Our main finding is a new class of compact complex, possibly non-Kähler, manifolds defined by the condition that every Gauduchon metric be strongly Gauduchon (sG), or equivalently that the Gauduchon cone be small in a certain sense. We term them sGG manifolds and find numerical characterisations of them in terms of certain relations between various cohomology theories (De Rham, Dolbeault, Bott-Chern, Aeppli). We also produce several concrete examples of nilmanifolds demonstrating the differences between the sGG class and well-established classes of complex manifolds. We conclude that sGG manifolds enjoy good stability properties under deformations and modifications.

math.DG↗

arXiv2017

Volume and Self-Intersection of Differences of Two Nef Classes

Let $\{α\}$ and $\{β\}$ be nef cohomology classes of bidegree $(1,\,1)$ on a compact $n$-dimensional Kähler manifold $X$ such that the difference of intersection numbers $\{α\}^n - n\,\{α\}^{n-1}.\,\{β\}$ is positive. We solve in a number of special but rather inclusive cases the quantitative part of Demailly's Transcendental Morse Inequalities Conjecture for this context predicting the lower bound $\{α\}^n-n\,\{α\}^{n-1}.\,\{β\}$ for the volume of the difference class $\{α-β\}$. We completely solved the qualitative part in an earlier work. We also give general lower bounds for the volume of $\{α-β\}$ and show that the self-intersection number $\{α-β\}^n$ is always bounded below by $\{α\}^n-n\,\{α\}^{n-1}.\,\{β\}$. We also describe and estimate the relative psef and nef thresholds of $\{α\}$ with respect to $\{β\}$ and relate them to the volume of $\{α-β\}$. Finally, broadening the scope beyond the Kähler realm, we propose a conjecture relating the balanced and the Gauduchon cones of $\partial\bar\partial$-manifolds which, if proved to hold true, would imply the existence of a balanced metric on any $\partial\bar\partial$-manifold.

math.CV↗

arXiv2017

Adiabatic Limit and the Frölicher Spectral Sequence

Motivated by our conjecture of an earlier work predicting the degeneration at the second page of the Frölicher spectral sequence of any compact complex manifold supporting an SKT metric $ω$ (i.e. such that $\partial\bar\partialω=0$), we prove degeneration at $E_2$ whenever the manifold admits a Hermitian metric whose torsion operator $τ$ and its adjoint vanish on $Δ''$-harmonic forms of positive degrees up to $\mbox{dim}_\C X$. Besides the pseudo-differential Laplacian inducing a Hodge theory for $E_2$ that we constructed in earlier work and Demailly's Bochner-Kodaira-Nakano formula for Hermitian metrics, a key ingredient is a general formula for the dimensions of the vector spaces featuring in the Frölicher spectral sequence in terms of the asymptotics, as a positive constant $h$ decreases to zero, of the small eigenvalues of a rescaled Laplacian $Δ_h$, introduced here in the present form, that we adapt to the context of a complex structure from the well-known construction of the adiabatic limit and from the analogous result for Riemannian foliations of Álvarez López and Kordyukov.

math.CV↗

arXiv2017

The Albanese Map of sGG Manifolds and Self-Duality of the Iwasawa Manifold

We prove that the three-dimensional Iwasawa manifold $X$, viewed as a locally holomorphically trivial fibration by elliptic curves over its two-dimensional Albanese torus, is self-dual in the sense that the base torus identifies canonically with its dual torus under a sesquilinear duality, the Jacobian torus of $X$, while the fibre identifies with itself. To this end, we derive elements of Hodge theory for arbitrary sGG manifolds, introduced in earlier joint work of the author with L. Ugarte, to construct in an explicit way the Albanese torus and map of any sGG manifold. These definitions coincide with the classical ones in the special Kähler and $\partial\bar\partial$ cases. The generalisation to the larger sGG class is made necessary by the Iwasawa manifold being an sGG, non-$\partial\bar\partial$, manifold. The main result of this paper can be seen as a complement from a different perspective to the author's very recent work where a non-Kähler mirror symmetry of the Iwasawa manifold was emphasised. We also hope that it will suggest yet another approach to non-Kähler mirror symmetry for different classes of manifolds.

math.AG↗

arXiv2017

Regularisation of Currents with Mass Control and Singular Morse Inequalities

Let $X$ be a compact complex, not necessarily Kähler, manifold of dimension $n$. We characterise the volume of any holomorphic line bundle $L\to X$ as the supremum of the Monge-Ampère masses $\int_X T_{ac}^n$ over all closed positive currents $T$ in the first Chern class of $L$, where $T_{ac}$ is the absolutely continuous part in the Lebesgue decomposition. This result, new in the non-Kähler context, can be seen as holomorphic Morse inequalities for the cohomology of high tensor powers of line bundles endowed with arbitrarily singular Hermitian metrics. It gives, in particular, a new bigness criterion for line bundles in terms of existence of singular Hermitian metrics satisfying positivity conditions. The proof is based on the construction of a new regularisation for closed $(1, 1)$-currents with a control of the Monge-Ampère masses of the approximating sequence. To this end, we prove a potential-theoretic result in one complex variable and study the growth of multiplier ideal sheaves associated with increasingly singular metrics.

math.CV↗

arXiv2016

Degeneration at $E_2$ of Certain Spectral Sequences

We propose a Hodge theory for the spaces $E_2^{p,\,q}$ featuring at the second step either in the Frölicher 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 Frölicher case we prove degeneration at $E_2$ if there exists an SKT metric $ω$ (i.e. such that $\partial\bar\partialω=0$) whose torsion is small compared to the spectral gap of the elliptic operator $Δ' + Δ"$ defined by $ω$. In the foliated case, we obtain degeneration at $E_2$ under a hypothesis involving the Laplacians $Δ'_N$ and $Δ'_F$ associated with the splitting $\partial = \partial_N + \partial_F$ induced by the foliated structure.

math.DG↗

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↗