arXiv subjects

Siarhei Finski

Publications and source records attributed to Siarhei Finski.

At least 19 recordsLinked to original sources

arXiv2026

Metric entropy of the space of holomorphic functions

By Montel's theorem, the continuous functions on a compact subset of a complex manifold that admit uniformly bounded holomorphic extensions to the manifold form a compact set in the uniform topology. We render this compactness quantitative by determining the asymptotics of the associated metric entropy, thereby giving a new solution to a problem of Kolmogorov for domains in $\mathbb{C}^n$ and extending the solution to domains in arbitrary Stein manifolds.

math.CV↗

arXiv2026

Kodaira-Iitaka dimension and multiplicity: an analytic perspective

We express the Kodaira-Iitaka dimension and the multiplicity of graded linear series in terms of the intersection theory of the plurisubharmonic envelope associated with the linear series, and obtain two refined versions of these formulas at the pointwise and at the metric levels. At the pointwise level, we focus on the weak convergence of the partial Bergman kernel associated with the linear series and a Bernstein-Markov measure. At the metric level, we compute the asymptotic ratio of the volumes of unit balls defined by the sup-norms on the linear series. Based on our findings, we introduce a non-pluripolar version of the numerical Kodaira-Iitaka dimension for a line bundle, show that this invariant dominates the classical Kodaira-Iitaka dimension and is, in turn, bounded above by the numerical versions proposed so far.

math.CV↗

arXiv2025

Geometric quantization on big line bundles

We extend several geometric quantization results to the setting of big line bundles. More precisely, we prove the asymptotic isometry property for the map that associates to a metric on a big line bundle the corresponding sup-norms on the spaces of holomorphic sections of its tensor powers. Building on this, we show that submultiplicative norms on section rings of big line bundles are asymptotically equivalent to sup-norms. As an application, we show that any bounded submultiplicative filtration on the section ring of a big line bundle naturally gives rise to a Mabuchi geodesic ray, and the speed of this ray encodes the statistical invariants of the filtration.

math.CV↗

arXiv2025

On the Localization of the Bergman Kernel and applications to Toeplitz theory

For a compact complex manifold endowed with a big line bundle and a Radon measure, we study the localization phenomena of the associated Bergman (or Christoffel-Darboux) kernel. For Bernstein-Markov measures, this results in the determination of the limiting off-diagonal Bergman measure, thereby confirming a conjecture of Zelditch. We then turn to applications in the theory of Toeplitz operators, showing in particular that they form an algebra under composition. Building on this, we then show that for Bernstein-Markov measures, the spectrum of Toeplitz operators equidistributes.

math.CV↗

arXiv2025

Bernstein-Markov measures and Toeplitz theory

We prove that Toeplitz operators associated with a Bernstein-Markov measure on a compact complex manifold endowed with a big line bundle form an algebra under composition. As an application, we derive a Szeg\H{o}-type spectral equidistribution result for this class of operators. A key component of our approach is the off-diagonal asymptotic analysis of the Bergman kernel, also known as the Christoffel-Darboux kernel.

math.CV↗

arXiv2025

Small eigenvalues of Toeplitz operators, Lebesgue envelopes and Mabuchi geometry

We study small eigenvalues of Toeplitz operators on polarized complex projective manifolds. For Toeplitz operators whose symbols are supported on proper subsets, we prove the existence of eigenvalues that decay exponentially with respect to the semiclassical parameter. We moreover, establish a connection between the logarithmic distribution of these eigenvalues and the Mabuchi geodesic between the fixed polarization and the Lebesgue envelope associated with the polarization and the non-zero set of the symbol. As an application of our approach, we also obtain analogous results for Toeplitz matrices.

math.CV↗

arXiv2024

Toeplitz operators, submultiplicative filtrations and weighted Bergman kernels

We demonstrate that the weight operator associated with a submultiplicative filtration on the section ring of a polarized complex projective manifold is a Toeplitz operator. We further analyze the asymptotics of the associated weighted Bergman kernel, presenting the local refinement of earlier results on the convergence of jumping measures for submultiplicative filtrations towards the pushforward measure defined by the corresponding geodesic ray.

math.CV↗

arXiv2024

About Wess-Zumino-Witten equation and Harder-Narasimhan potentials

For a polarized family of complex projective manifolds, we identify the algebraic obstructions that govern the existence of approximate solutions to the Wess-Zumino-Witten equation. When this is specialized to the fibration associated with a projectivization of a vector bundle, we recover a version of Kobayashi-Hitchin correspondence. More broadly, we demonstrate that a certain auxiliary Monge-Amp\`ere type equation, generalizing the Wess-Zumino-Witten equation by taking into account the weighted Bergman kernel associated with the Harder-Narasimhan filtrations of direct image sheaves, admits approximate solutions over any polarized family. These approximate solutions are shown to be the closest counterparts to true solutions of the Wess-Zumino-Witten equation whenever the latter do not exist, as they minimize the associated Yang-Mills functional. As an application, in a fibered setting, we prove an asymptotic converse to the Andreotti-Grauert theorem conjectured by Demailly.

math.DG↗

arXiv2024

On Harder-Narasimhan slopes of direct images

For a flat family of complex projective varieties, we study the asymptotic distribution of the Harder-Narasimhan slopes of the direct images on the base of high tensor powers of a line bundle on the total space. We establish a theorem of Mehta-Ramanathan type, showing that this asymptotic distribution can be recovered from its analogues associated with the base changes of the family over general curves. As an application, we show that the multiplicity of the line bundle on the family can be estimated in terms of the Harder-Narasimhan slopes of the associated direct images and the multiplicity of the general fiber.

math.AG↗

arXiv2024

Lower bounds on fibered Yang-Mills functionals: generic nefness and semistability of direct images

The main goal of this paper is to generalize a part of the relationship between mean curvature and Harder-Narasimhan filtrations of holomorphic vector bundles to arbitrary polarized fibrations. More precisely, for a polarized family of complex projective manifolds, we establish lower bounds on a fibered version of Yang-Mills functionals in terms of the Harder-Narasimhan slopes of direct image sheaves associated with high tensor powers of the polarization. We discuss the optimality of these lower bounds and, as an application, provide an analytic characterisation of a fibered version of generic nefness. As another application, we refine the existent obstructions for finding metrics with constant horizontal mean curvature. The study of the semiclassical limit of Hermitian Yang-Mills functionals lies at the heart of our approach.

math.DG↗

arXiv2023

Semiclassical Ohsawa-Takegoshi extension theorem and asymptotics of the orthogonal Bergman kernel

We study the asymptotics of Ohsawa-Takegoshi extension operator and orthogonal Bergman projector associated with high tensor powers of a positive line bundle. More precisely, for a fixed complex submanifold in a complex manifold, we consider the operator which associates to a given holomorphic section of a positive line bundle over the submanifold the holomorphic extension of it to the ambient manifold with the minimal $L^2$-norm. When the tensor power of the line bundle tends to infinity, we obtain an explicit asymptotic expansion of this operator. This is done by proving an exponential estimate for the associated Schwartz kernel and showing that this Schwartz kernel admits a full asymptotic expansion. We prove similar results for the projection onto holomorphic sections orthogonal to those which vanish along the submanifold.

math.DG↗

arXiv2023

Geometry at the infinity of the space of positive metrics: test configurations, geodesic rays and chordal distances

From the work of Phong and Sturm in 2007, for a polarised projective manifold and an ample test configuration, one can associate the geodesic ray of plurisubharmonic metrics on the polarising line bundle using the solution of the Monge-Amp\`ere equation on an equivariant resolution of singularities of the test configuration. We prove that the Mabuchi chordal distance between the geodesic rays associated with two ample test configurations coincides with the spectral distance between the associated filtrations on the section ring. This gives an algebraic description of the boundary at the infinity of the space of positive metrics, viewed - as it is usually done for spaces of negative curvature - through geodesic rays.

math.DG↗

arXiv2022

Submultiplicative norms and filtrations on section rings

We show that submultiplicative norms on section rings of polarised projective manifolds are asymptotically equivalent to sup-norms associated with metrics on the polarisation. We then discuss some applications to the spectral theory of submultiplicative filtrations, the asymptotic study of the Narasimhan-Simha pseudonorms, and holomorphic extension theorem. As an unexpected byproduct, we show that injective and projective tensor norms on symmetric algebras of finite dimensional complex normed vector spaces are asymptotically equivalent.

math.DG↗

arXiv2022

On the metric structure of section ring

The main goal of this article is to study for a projective manifold and an ample line bundle over it the relation between metric and algebraic structures on the associated section ring. More precisely, we prove that once the kernel is factored out, the multiplication operator of the section ring becomes an approximate isometry (up to normalization) with respect to the $L^2$-norms and the induced Hermitian tensor product norm. We also show that the analogous result holds for the $L^1$ and $L^{\infty}$-norms if instead of the Hermitian tensor product norm, we consider the projective and injective tensor norms induced by $L^1$ and $L^{\infty}$-norms respectively. Then we show that $L^2$-norms associated with continuous plurisubharmonic metrics are actually characterized by the multiplicativity properties of this type. Using this, we refine the theorem of Phong-Sturm about quantization of Mabuchi geodesics from the weaker level of Fubini-Study convergence to the stronger level of norm equivalences.

math.DG↗

arXiv2022

The asymptotics of the optimal holomorphic extensions of holomorphic jets along submanifolds

We study the asymptotics of the $L^2$-optimal holomorphic extensions of holomorphic jets associated with high tensor powers of a positive line bundle along submanifolds. More precisely, for a fixed complex submanifold in a complex manifold, we consider the operator which for a given holomorphic jet along the submanifold of a positive line bundle associates the $L^2$-optimal holomorphic extension of it to the ambient manifold. When the tensor power of the line bundle tends to infinity, we give an explicit asymptotic formula for this extension operator. This is done by a careful study of the Schwartz kernels of the extension operator and related Bergman projectors. It extends our previous results, done for holomorphic sections instead of jets. As an application, we prove the asymptotic isometry between two natural norms on the space of holomorphic jets: one induced from the ambient manifold and another from the submanifold.

math.DG↗

arXiv2022

Complex embeddings, Toeplitz operators and transitivity of optimal holomorphic extensions

In a setting of a complex manifold with a fixed positive line bundle and a submanifold, we consider the optimal Ohsawa-Takegoshi extension operator, sending a holomorphic section of the line bundle on the submanifold to the holomorphic extension of it on the ambient manifold with the minimal $L^2$-norm. We show that for a tower of submanifolds in the semiclassical setting, i.e. when we consider a large tensor power of the line bundle, the extension operators satisfy transitivity property modulo some small defect, which can be expressed through Toeplitz type operators. We calculate the first significant term in the asymptotic expansion of this "transitivity defect". As a byproduct, we deduce the composition rules for Toeplitz type operators, the extension and restriction operators, and calculate the second term in the asymptotic expansion of the optimal constant in the semi-classical version of Ohsawa-Takegoshi extension theorem.

math.DG↗

arXiv2020

On Monge-Amp\`ere volumes of direct images

This paper is devoted to the study of the asymptotics of Monge-Amp\`ere volumes of direct images associated with high tensor powers of an ample line bundle. We study the leading term of this asymptotics and provide a classification of bundles saturating the topological bound of Demailly. In the special case of high symmetric powers of ample vector bundles, this provides a characterization of vector bundles admitting projectively flat Hermitian structures.

math.DG↗

arXiv2020

On characteristic forms of positive vector bundles, mixed discriminants and pushforward identities

We prove that Schur polynomials in Chern forms of Nakano and dual Nakano positive vector bundles are positive as differential forms. Moreover, modulo a statement about the positivity of a "double mixed discriminant" of linear operators on matrices, which preserve the cone of positive definite matrices, we establish that Schur polynomials in Chern forms of Griffiths positive vector bundles are weakly-positive as differential forms. This provides differential-geometric versions of Fulton-Lazarsfeld inequalities for ample vector bundles.An interpretation of positivity conditions for vector bundles through operator theory is in the core of our approach. Another important step in our proof is to establish a certain pushforward identity for characteristic forms, refining the determinantal formula of Kempf-Laksov for homolorphic vector bundles on the level of differential forms. In the same vein, we establish a local version of Jacobi-Trudi identity.Then we study the inverse problem and show that already for vector bundles over complex surfaces, one cannot characterize Griffiths positivity (and even ampleness) through the positivity of Schur polynomials, even if one takes into consideration all quotients of a vector bundle.

math.AG↗