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Dan Popovici

Publications and source records attributed to Dan Popovici.

At least 19 recordsLinked to original sources
arXiv2026

$m$-Positive Stability of Holomorphic Vector Bundles and Moduli Spaces

We first propose a notion of $m$-positivity for higher-rank vector bundles, a variant of which reduces to the classical Griffiths positivity when $m=1$. Based on this, we go on to propose a generalisation of the classical Mumford-Takemoto theory of stability by means of a smooth function that we associate with every proper coherent subsheaf ${\cal F}$ of a given holomorphic vector bundle $E$. This places the emphasis on the holomorphic structure and the Hermitian fibre metric of $E$, rather than on numerical invariants of the smooth structure of $E$, making our stability conditions into relative pointwise $m$-positivity properties of $E$ with respect to its proper coherent subsheaves ${\cal F}$. We establish links with Hermite-Einstein geometry, prove that Hermite-Einstein bundles are uniformly semi-stable, study the resulting moduli spaces, and compare the new notions with the classical Mumford-Takemoto (semi-)stability notions.

math.DG↗
arXiv2025

Generalised Hermite-Einstein Fibre Metrics and Slope Stability for Holomorphic Vector Bundles

Let $X$ be a compact complex manifold of dimension $n$ and let $m$ be a positive integer with $m\leq n$. Assume that $X$ admits a K\"ahler metric $\omega$ and a weakly positive, $\partial\bar\partial$-closed, smooth $(n-m,\,n-m)$-form $\Omega$. We introduce the notions of $(\omega,\,\Omega)$-Hermite-Einstein holomorphic vector bundles and $(\omega,\,\Omega)$(-semi)-stable coherent sheaves on $X$ by generalising the classical definitions depending only on $\omega$. We then prove that the $(\omega,\,\Omega)$-Hermite-Einstein condition implies the $(\omega,\,\Omega)$-semi-stability of a holomorphic vector bundle and its splitting into $(\omega,\,\Omega)$-stable subbundles. This extends a classical result by Kobayashi and L\"ubke to our generalised setting. In the appendix, we propose notions of both strongly and weakly (strictly) positive forms and currents and discuss their various properties.

math.AG↗
arXiv2025

Holomorphic $p$-Contact and $s$-Symplectic Line Bundles

We generalise the notions of scalar-valued holomorphic $p$-contact and $s$-symplectic structures introduced recently on compact complex manifolds by the second-named author jointly with H. Kasuya and L. Ugarte to their analogues with values in a holomorphic line bundle. We then study the resulting holomorphic $p$-contact and $s$-symplectic manifolds which, unlike their scalar counterparts that are never K\"ahler, can even be projective. In particular, we investigate the (lack of) positivity properties of the canonical bundle of these manifolds when it is given a possibly singular Hermitian fibre metric. One of the tools used is a very recent regularisation result for $m$-psh functions obtained jointly by S. Dinew and the second-named author.

math.DG↗
arXiv2025

Properties of Holomorphic $p$-Contact Manifolds

We continue the study of compact holomorphic $p$-contact manifolds $X$ that we introduced recently by expanding the discussion to include non-K\"ahler hyperbolicity issues and a differential calculus based on what we call the Lie derivative with respect to a $(0,\,q)$-form with values in the holomorphic tangent bundle of $X$. We also propose the notion of $p$-contact deformations for which we prove a Bogomolov-Tian-Todorov-type unobstructedness theorem to order two. This kind of small deformations of the complex structure is related to the essential horizontal deformations that we introduced in our previous work and forms part of a wider on-going project aimed at developing a non-K\"ahler mirror symmetry theory that was first tested on the Iwasawa manifold and subsequently on Calabi-Yau page-$1$-$\partial\bar\partial$-manifolds.

math.DG↗
arXiv2025

$m$-Pseudo-effectivity and a Monge-Amp\`ere-Type Equation for Forms of Positive Degree

Given an $n$-dimensional compact K\"ahler manifold, we continue our study of $m$-positivity in two ways. We first propose generalisations of the notions of pseudo-effective and big Bott-Chern cohomology classes of bidegree $(1,\,1)$ by relaxing the standard positivity hypotheses to their $m$-counterparts after we have proved a Lamari-type duality lemma in bidegree $(m,\,m)$. Independently, we propose a Monge-Amp\`ere-type non-linear pde whose distinctive feature is that its solutions, if any, are forms of positive degree rather than functions. We prove a form of uniqueness for the solutions and, under the assumption that a solution exists, we give a geometric application involving the $m$-bigness notion introduced in the first part.

math.DG↗
arXiv2025

$m$-Positivity and Regularisation

Starting from the notion of $m$-plurisubharmonic function introduced recently by Dieu and studied, in particular, by Harvey and Lawson, we consider $m$-(semi-)positive $(1,\,1)$-currents and Hermitian holomorphic line bundles on complex Hermitian manifolds and prove two kinds of results: vanishing theorems and $L^2$-estimates for the $\bar\partial$-equation in the context of $C^\infty$ $m$-positive Hermitian fibre metrics; global and local regularisation theorems for $m$-semi-positive $(1,\,1)$-currents whose proofs involve the use of viscosity subsolutions for a certain Monge-Amp\`ere-type equation and the associated Dirichlet problem.

math.DG↗
arXiv2025

Accelerating scientific discovery with Co-Scientist

Scientific discovery is driven by scientists generating novel hypotheses for complex problems that undergo rigorous experimental validation. To augment this process, we introduce Co-Scientist, a multi-agent AI system built on Gemini for structured scientific thinking and hypothesis generation. Co-Scientist aims to help scientists discover new original knowledge. Conditioned on their research objectives and prior scientific evidence, it formulates demonstrably novel research hypotheses for experimental verification. The system's design involves agents continuously generating, critiquing and refining hypotheses accelerated by scaling test-time compute. Key contributions include: (1) a multi-agent architecture with an asynchronous task execution framework for flexible compute scaling; (2) a tournament evolution process for self-improving hypotheses generation. Automated evaluations show continued benefits of test-time compute scaling, improving hypothesis quality over time. While general purpose, we focus the validation in three biomedical applications: drug repurposing, novel target discovery, and explaining mechanisms of anti-microbial resistance. Specifically, Co-Scientist helped identify new drug repurposing candidates and synergistic combination therapies for acute myeloid leukemia, which were validated through in vitro experiments. These real-world validations demonstrate the potential of Co-Scientist to accelerate scientific discovery and usher in an era of AI empowered scientists.

cs.AI↗
arXiv2025

Higher-Degree Holomorphic Contact Structures

We introduce the classes of holomorphic $p$-contact manifolds and holomorphic $s$-symplectic manifolds that generalise the classical holomorphic contact and holomorphic symplectic structures. After observing their basic properties and exhibiting a wide range of examples, we give two types of general conceptual results involving the former class of manifolds: structure theorems and unobstructedness theorems. The latter type generalises to our context the classical Bogomolov-Tian-Todorov theorem for a type of small deformations of complex structures that generalise the small essential deformations previously introduced for the Iwasawa manifold and for Calabi-Yau page-$1$-$\partial\bar\partial$-manifolds.

math.DG↗
arXiv2024

A Twisted Adiabatic Limit Approach to Vanishing Theorems for Complex Line Bundles

Given an $n$-dimensional compact complex Hermitian manifold $X$, a $C^\infty$ complex line bundle $L$ equipped with a connection $D$ whose $(0,\,1)$-component $D''$ squares to zero and a real-valued function $\eta$ on $X$, we prove that the $D''$-cohomology group of $L$ of any bidegree $(p,\,q)$ such that either $(p>q \hspace{1ex}\mbox{and}\hspace{1ex} p+q\geq n+1)$ or $(p<q \hspace{1ex}\mbox{and}\hspace{1ex} p+q\leq n-1)$ vanishes when two extra hypotheses are made. The first hypothesis requires a certain real-valued, not necessarily closed, $(1,\,1)$-form depending on $p,\,q$, on the curvature of $D$ and on a $(1,\,1)$-form induced by $\eta$ to be positive definite. The second hypothesis requires the norm of $\partial\eta$ to be small relative to $|\eta|$. This theorem, for which we also give a number of variants, is proved by generalising our very recent twisted adiabatic limit construction for complex structures to connections on complex line bundles. This twisting of $D$ induces first-order differential operators acting on the $L$-valued forms, for which we obtain commutation relations involving their formal adjoints, and two twisted Laplacians for which we obtain a comparison formula reminiscent of the classical Bochner-Kodaira-Nakano identity. The main features of our results are that $X$ need not be K\"ahler, $L$ need not be holomorphic and the types of $C^\infty$ functions that $X$ supports play a key role in our hypotheses, thus capturing some of their links with the geometry of manifolds.

math.DG↗
arXiv2024

Twisted Adiabatic Limit for Complex Structures

Given a complex manifold $X$ and a smooth positive function $\eta$ thereon, we perturb the standard differential operator $d=\partial + \bar\partial$ acting on differential forms to a first-order differential operator $D_\eta$ whose principal part is $\eta\partial + \bar\partial$. The role of the zero-th order part is to force the integrability property $D_\eta^2=0$ that leads to a cohomology isomorphic to the de Rham cohomology of $X$, while the components of types $(0,\,1)$ and $(1,\,0)$ of $D_\eta$ induce cohomologies isomorphic to the Dolbeault and conjugate-Dolbeault cohomologies. We compute Bochner-Kodaira-Nakano-type formulae for the Laplacians induced by these operators and a given Hermitian metric on $X$. The computations throw up curvature-like operators of order one that can be made (semi-)positive under appropriate assumptions on the function $\eta$. As applications, we obtain vanishing results for certain harmonic spaces on complete, non-compact, manifolds and for the Dolbeault cohomology of compact complex manifolds that carry certain types of functions $\eta$. This study continues and generalises the one of the operators $d_h=h\partial + \bar\partial$ that we introduced and investigated recently for a positive constant $h$ that was then let to converge to $0$ and, more generally, for constants $h\in\C$. The operators $d_h$ had, in turn, been adapted to complex structures from the well-known adiabatic limit construction for Riemannian foliations. Allowing now for possibly non-constant functions $\eta$ creates positivity in the curvature-like operator that stands one in good stead for various kinds of applications.

math.DG↗
arXiv2023

A Moment Map for the Space of Maps to a Balanced Manifold

Given a complex balanced manifold $X$ and a compact complex manifold $S$ equipped with a positive volume form $dV>0$ and satisfying an extra condition such that $\mbox{dim}\,S\geq\mbox{dim}\,X -1$, we construct a moment map for the action of the Lie group of biholomorphisms of $S$ that preserve $dV$ onto the space of holomorphic maps $f:S\longrightarrow X$ that satisfy a certain condition with respect to the Bott-Chern cohomology class of the balanced metric of $X$. The purpose is twofold: to study such maps as a possible addition to some very recent hyperbolicity notions involving holomorphic maps with a certain type of growth from some $\C^p$, rather than $S$, to $X$; and to lay the groundwork for a possible future construction of balanced quotients as an analogue of the classical symplectic quotients.

math.DG↗
arXiv2023

A Non-Integrable Ohsawa-Takegoshi-Type $L^2$ Extension Theorem

Given a complete Kähler manifold $(X,\,ω)$ with finite second Betti number, a smooth complex hypersurface $Y\subset X$ and a smooth real $d$-closed $(1,\,1)$-form $α$ on $X$ with arbitrary, possibly non-rational, De Rham cohomology class $\{α\}$ satisfying a certain assumption, we obtain extensions to $X$, with control of their $L^2$-norms, of smooth sections of the canonical bundle of $Y$ twisted by the restriction to $Y$ of any $C^\infty$ complex line bundle $L_k$ in a sequence of asymptotically holomorphic line bundles whose first Chern classes approximate the positive integer multiples $k\{α\}$ of the original class. Besides a known non-integrable $(0,\,1)$-connection $\bar\partial_k$ on $L_k$, the proof uses two twisted Laplace-type elliptic differential operators that are introduced and investigated, leading to Bochner-Kodaira-Nakano-type (in-)equalities, a spectral gap result and an a priori $L^2$-estimate. The main difference from the classical Ohsawa-Takegoshi extension theorem is that the objects need not be holomorphic, but only asymptotically holomorphic as $k\to\infty$. The possibility that $\bar\partial_k$ does not square to $0$ accounts for its lack of commutation with the Laplacian $Δ''_k$ it induces. We hope this study is a possible first step in a future attack on Siu's conjecture predicting the invariance of the plurigenera in Kähler families of compact complex manifolds.

math.CV↗
arXiv2023

Pluriclosed Star Split Hermitian Metrics

We introduce a class of Hermitian metrics, that we call pluriclosed star split, generalising both the astheno-Kähler metrics of Jost and Yau and the $(n-2)$-Gauduchon metrics of Fu-Wang-Wu on complex manifolds. They have links with Gauduchon and balanced metrics through the properties of a smooth function associated with any Hermitian metric. After pointing out several examples, we generalise the property to pairs of Hermitian metrics and to triples consisting of a holomorphic map between two complex manifolds and two Hermitian metrics, one on each of these manifolds. Applications include an attack on the Fino-Vezzoni conjecture predicting that any compact complex manifold admitting both SKT and balanced metrics must be Kähler, that we answer affirmatively under extra assumptions. We also introduce and study a Laplace-like differential operator of order two acting on the smooth $(1,\,1)$-forms of a Hermitian manifold. We prove it to be elliptic and we point out its links with the pluriclosed star split metrics and pairs defined in the first part of the paper.

math.DG↗
arXiv2023

A Variational Approach to SKT and Balanced Metrics

We investigate compact complex manifolds endowed with SKT or balanced metrics. In each case we define a new functional whose critical points are proved to be precisely the Kähler metrics, if any, on the manifold. As general manifolds of either type need not admit Kähler metrics, this provides an approach to new obstructions to Kählerianity within these two families of metrics.

math.DG↗
arXiv2023

Partially Hyperbolic Compact Complex Manifolds

We propose and investigate two types, the latter with two variants, of notions of partial hyperbolicity accounting for several classes of compact complex manifolds behaving hyperbolically in certain directions, defined by a vector subbundle of the holomorphic tangent bundle, but not necessarily in the other directions. A key role is played by certain entire holomorphic maps, possibly from a higher-dimensional space, into the given manifold $X$. The dimension of the origin $\C^p$ of these maps is allowed to be arbitrary, unlike both the classical $1$-dimensional case of entire curves and the $1$-codimensional case introduced in previous work of the second-named author with S. Marouani. The higher-dimensional generality necessitates the imposition of certain growth conditions, very different from those in Nevanlinna theory and those in works by de Th\'elin, Burns and Sibony on Ahlfors currents, on the entire holomorphic maps $f:\C^p\longrightarrow X$. The way to finding these growth conditions is revealed by certain special, possibly non-K\"ahler, Hermitian metrics in the spirit of Gromov's K\"ahler hyperbolicity theory but in a higher-dimensional context. We then study several classes of examples, prove implications among our partial hyperbolicity notions, give a sufficient criterion for the existence of an Ahlfors current and a sufficient criterion for partial hyperbolicity in terms of the signs of two curvature-like objects introduced recently by the second-named author.

math.DG↗
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↗