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George Marinescu

Publications and source records attributed to George Marinescu.

At least 19 recordsLinked to original sources
arXiv2026

Nakano-Griffiths inequality, holomorphic Morse inequalities, and extension theorems for $q$-concave domains

We establish a general Nakano-Griffiths inequality with boundary conditions and apply it to derive holomorphic Morse inequalities for domains satisfying analytic convexity assumptions. The proof of these inequalities relies on analyzing the spectral spaces of the Laplace operator with $\overline{\partial }$-Neumann boundary conditions. As an application, we obtain a criterion for Moishezon $1$-concave manifolds. We further apply the holomorphic Morse inequalities on Levi $q$-concave domains in conjunction with the Kohn-Rossi extension theorem to obtain the following result. Let $X$ be a compact complex manifold of dimension $n$ equipped with a holomorphic line bundle that is semi-positive everywhere and positive at least at one point, and let $D\subset X$ be a smooth $q$-concave domain ($1 \le q \le n-1$). We prove that every $\bar{\partial }_{b}$-closed $(0,\ell )$-form on $bD$ with values in a holomorphic vector bundle, admits a meromorphic extension to $D$ for all $q \le \ell \le n-1$.

math.CV↗
arXiv2026

Asymptotic expansion of induced Grassmannian Chern forms and distribution of random degeneracy sets

For the Grassmannian embeddings defined by the spaces $H^0(X,L^p\otimes E)$, where $L$ is a positive line bundle and $E$ is a holomorphic vector bundle over a compact complex manifold, we prove a complete asymptotic expansion of the induced Grassmannian Chern forms and compute the first coefficients explicitly. As an application of the first-order asymptotics and of the theory of meromorphic transforms by Dinh and Sibony, we prove that on a compact Kähler manifold, the normalized currents of integration over the loci where several random sections become linearly dependent converge almost surely to the corresponding power of the curvature form of the positive line bundle, with a quantitative estimate for the speed of convergence. Moreover, in the determinant case, we additionally present an alternative method based on the Wishart distribution, together with variance estimates.

math.CV↗
arXiv2026

Semi-classical heat kernel asymptotics on complex manifolds with boundary

Let $M$ be a relatively compact open subset of a complex manifold $M'$ with smooth boundary $X$ and let $L$ be a holomorphic line bundle over $M'$. Assuming that condition $Z(q)$ holds, we establish the semi-classical asymptotic behavior of $e^{-\frac{t}{k}\Box^{q}_k}$ near the boundary $X$ as $k\to\infty$, where $\Box^{q}_k$ is the $\bar{\partial}$-Neumann Laplacian acting on $(0,q)$-forms on $M$ with values in $L^k$. Our results extend the seminal work of Bismut to complex manifolds with boundary. As applications of our results, we provide a heat kernel-based proof of the holomorphic Morse inequalities for complex manifolds with boundary and derive a semi-classical Weyl law for the $\bar{\partial}$-Neumann Laplacian.

math.CV↗
arXiv2026

Berezin-Toeplitz Quantization of non-compact manifolds

We develop Berezin-Toeplitz quantization in a non-compact complex geometric setting. Let $(X,Θ)$ be a Hermitian manifold, $(L,h^L)$ a positive holomorphic line bundle, and $(E,h^E)$ a holomorphic Hermitian vector bundle. Assuming that the Kodaira Laplacian on $(0,1)$-forms with values in $L^p\!\otimes E$ has a spectral gap growing linearly in $p$, we prove that the Bergman projection onto the $L^2$-holomorphic space $H^0_{(2)}(X,L^p\!\otimes E)$ enjoys the usual off-diagonal decay and admits a full asymptotic expansion on compact subsets as $p\to\infty$. As a consequence, for every smooth symbol $f\in\mathcal{C}^\infty_{\mathrm{const}}(X,\operatorname{End}(E))$ (constant outside a compact set), the associated Toeplitz operators $T_{f,p}=P_p f P_p$ form a closed algebra and satisfy a complete composition expansion, yielding a star-product on $\mathcal C^\infty_{\mathrm{const}}(X,\operatorname{End}(E))$ and the expected semiclassical commutator formula. We also give intrinsic criteria characterizing Toeplitz families with compactly supported kernels. We then provide geometric conditions guaranteeing the spectral gap on large classes of non-compact manifolds, via fundamental $L^2$-estimates for $\bar\partial$ on complete Hermitian manifolds (including bounded-geometry complete Kähler manifolds, Kähler-Einstein manifolds, pseudoconvex/weakly $1$-complete, and quasi-projective manifolds). Finally, for compactly supported bounded symbols, we prove a Szegő-type theorem describing the eigenvalue distribution of the compact Toeplitz operators $T_{f,p}$ as $p\to\infty$.

math.DG↗
arXiv2026

Bergman kernels and Poincaré series

We show that the Bergman kernel of a finite-volume quotient of a Hermitian manifold $\widetilde{X}$ with bounded geometry by a discrete group $Γ$ of its isometries is the same as the averaging over $Γ$ of the Bergman kernel on $\widetilde{X}$. We then use these results when $\widetilde{X}$ is a Hermitian symmetric space to show that a large class of relative Poincaré series does not vanish. This extends the results of Borthwick-Paul-Uribe and Barron (formerly Foth) to the case of general locally symmetric spaces of finite volume.

math.DG↗
arXiv2025

Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections

Let $(X,ω)$ be a compact Kähler manifold, $(L,h^L)$ be a positive line bundle, and $(E,h^E)$ be a Hermitian holomorphic vector bundle of rank $r$ on $X$. We prove that the pullback by the Kodaira embedding associated to $L^p\otimes E$ of the $k$-th Chern class of the dual of the universal bundle over the Grassmannian converges as $p\to\infty$ to the $k$-th power of the Chern form $c_1(L,h^L)$, for $0\leq k\leq r$. If $c_1(L,h^L)=ω$ we also determine the second term in the semiclassical expansion, which involves $c_1(E,h^E)$. As a consequence we show that the limit distribution of zeros of random sequences of holomorphic sections of high powers $L^p\otimes E$ is $c_1(L,h^L)^r$. Furthermore, we compute the expectation of the currents of integration along degeneracy sets of random holomorphic sections.

math.CV↗
arXiv2024

Tian's theorem for Moishezon spaces

We prove that the Fubini-Study currents associated to a sequence of singular Hermitian holomorphic line bundles on a compact normal Moishezon space distribute asymptotically as the curvature currents of their metrics.

math.DG↗
arXiv2024

Toeplitz operators and zeros of square-integrable random holomorphic sections

We use the theory of abstract Wiener spaces to construct a probabilistic model for Berezin-Toeplitz quantization on a complete Hermitian complex manifold endowed with a positive line bundle. We associate to a function with compact support (a classical observable) a family of square-integrable Gaussian holomorphic sections. Our focus then is on the asymptotic distributions of their zeros in the semiclassical limit, in particular, we prove equidistribution results, large deviation estimates, and central limit theorems of the random zeros on the support of the given function. One of the key ingredients of our approach is the local asymptotic expansions of Berezin-Toeplitz kernels with non-smooth symbols.

math.CV↗
arXiv2024

Zeros of random holomorphic sections of big line bundles with continuous metrics

Let $X$ be a compact normal complex space, $L$ be a big holomorphic line bundle on $X$ and $h$ be a continuous Hermitian metric on $L$. We consider the spaces of holomorphic sections $H^0(X, L^{\otimes p})$ endowed with the inner product induced by $h^{\otimes p}$ and a volume form on $X$, and prove that the corresponding sequence of normalized Fubini-Study currents converge weakly to the curvature current $c_1(L,h_{\mathrm{eq}})$ of the equilibrium metric $h_{\mathrm{eq}}$ associated to $h$. We also show that the normalized currents of integration along the zero divisors of random sequences of holomorphic sections converge almost surely to $c_1(L,h_{\mathrm{eq}})$, for very general classes of probability measures on $H^0(X, L^{\otimes p})$.

math.CV↗
arXiv2024

Induced Fubini-Study metrics on strictly pseudoconvex CR manifolds and zeros of random CR functions

Let $X$ be a compact strictly pseudoconvex embeddable Cauchy-Riemann manifold and let $T_P$ be the Toeplitz operator on $X$ associated with a first-order pseudodifferential operator $P$. In our previous work we established the asymptotic expansion for $k$ large of the kernel of the operators $χ(k^{-1}T_P)$, where $χ$ is a smooth cut-off function supported in the positive real line. By using these asymptotics, we show in this paper that $X$ can be projectively embedded by maps with components of the form $χ(k^{-1}λ)f_λ$, where $λ$ is an eigenvalue of $T_P$ and $f_λ$ is a corresponding eigenfunction. We establish the asymptotics of the pull-back of the Fubini-Study metric by these maps and we obtain the distribution of the zero divisors of random Cauchy-Riemann functions. We then establish a version of the Lelong-Poincaré formula for domains with boundary and obtain the distribution of the zero divisors of random holomorphic functions on strictly pseudoconvex domains.

math.CV↗
arXiv2023

Semi-classical asymptotics of partial Bergman kernels on $\mathbb{R}$-symmetric complex manifolds with boundary

Let $M$ be a relatively compact connected open subset with smooth connected boundary of a complex manifold $M'$. Let $(L,h^L)\rightarrow M'$ be a positive line bundle over $M'$. Suppose that $M'$ admits a holomorphic $\mathbb{R}$-action which preserves the boundary of $M$ and lifts to $L$. We establish the asymptotic expansion of a partial Bergman kernel associated to a package of Fourier modes of high frequency with respect to the $\mathbb{R}$-action in the high powers of $L$. As an application, we establish an $\mathbb{R}$-equivariant analogue of Fefferman's and Bell-Ligocka's result about smooth extension up to the boundary of biholomorphic maps between weakly pseudoconvex domains in $\mathbb{C}^n$. Another application concerns the embedding of pseudoconcave manifolds.

math.CV↗
arXiv2023

Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds

Let $X$ be a compact strictly pseudoconvex embeddable CR manifold and let $T_P$ be the Toeplitz operator on $X$ associated with some first order pseudodifferential operator $P$. We consider $χ_k(T_P)$ the functional calculus of $T_P$ by any rescaled cut-off function $χ$ with compact support in the positive real line. In this work, we show that $χ_k(T_P)$ admits a full asymptotic expansion as $k\to+\infty$. As applications, we obtain several CR analogous of results concerning high power of line bundles in complex geometry but without any group action assumptions on the CR manifold. In particular, we establish a Kodaira type embedding theorem, Tian's convergence theorem and a perturbed spherical embedding theorem for strictly pseudoconvex CR manifolds.

math.CV↗
arXiv2023

Singular holomorphic Morse inequalities on non-compact manifolds

We study asymptotic estimates of the dimension of cohomology on possibly non-compact complex manifolds for line bundles endowed with Hermitian metrics with algebraic singularities. We give a unified approach to establishing singular holomorphic Morse inequalities for hyperconcave manifolds, pseudoconvex domains, $q$-convex manifolds and $q$-concave manifolds, and we generalize related estimates of Berndtsson. We also consider the case of metrics with more general than algebraic singularities.

math.CV↗
arXiv2023

Geometric quantization results for semi-positive line bundles on a Riemann surface

In earlier work the authors proved the Bergman kernel expansion for semipositive line bundles over a Riemann surface whose curvature vanishes to atmost finite order at each point. Here we explore the related results and consequences of the expansion in the semipositive case including: Tian's approximation theorem for induced Fubini-Study metrics, leading order asymptotics and composition for Toeplitz operators, asymptotics of zeroes for random sections and the asymptotics of holomorphic torsion.

math.DG↗
arXiv2023

Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface

We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the relation of the Bochner Laplacian on tensor powers with the sub-Riemannian (sR) Laplacian.

math.DG↗
arXiv2023

Restricted spaces of holomorphic sections vanishing along subvarieties

Let $X$ be a compact normal complex space of dimension $n$ and $L$ be a holomorphic line bundle on $X$. Suppose that $Σ=(Σ_1,\ldots,Σ_\ell)$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $τ=(τ_1,\ldots,τ_\ell)$ is an $\ell$-tuple of positive real numbers, and let $H^0_0(X,L^p)$ be the space of holomorphic sections of $L^p:=L^{\otimes p}$ that vanish to order at least $τ_jp$ along $Σ_j$, $1\leq j\leq\ell$. If $Y\subset X$ is an irreducible analytic subset of dimension $m$, we consider the space $H^0_0 (X|Y, L^p)$ of holomorphic sections of $L^p|_Y$ that extend to global holomorphic sections in $H^0_0(X,L^p)$. Assuming that the triplet $(L,Σ,τ)$ is big in the sense that $\dim H^0_0(X,L^p)\sim p^n$, we give a general condition on $Y$ to ensure that $\dim H^0_0(X|Y,L^p)\sim p^m$. When $L$ is endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces $H^0_0(X|Y,L^p)$ converge to a certain equilibrium current on $Y$. We apply this to the study of the equidistribution of zeros in $Y$ of random holomorphic sections in $H^0_0(X|Y,L^p)$ as $p\to\infty$.

math.CV↗
arXiv2023

Semi-classical spectral asymptotics of Toeplitz operators on strictly pseudodonvex domains

On a relatively compact strictly pseudoconvex domain with smooth boundary in a complex manifold of dimension $n$ we consider a Toeplitz operator $T_R$ with symbol a Reeb-like vector field $R$ near the boundary. We show that the kernel of a weighted spectral projection $χ(k^{-1}T_R)$, where $χ$ is a cut-off function with compact support in the positive real line, is a semi-classical Fourier integral operator with complex phase, hence admits a full asymptotic expansion as $k\to+\infty$. More precisely, the restriction to the diagonal $χ(k^{-1}T_R)(x,x)$ decays at the rate $O(k^{-\infty})$ in the interior and has an asymptotic expansion on the boundary with leading term of order $k^{n+1}$ expressed in terms of the Levi form and the pairing of the contact form with the vector field $R$.

math.CV↗