arXiv research

Study shows normal distribution in divisor counts of random sections on complex manifolds.

Estimates holomorphic sections on Bohr-Sommerfeld Lagrangian submanifolds.

Study shows mass distribution of random holomorphic sections follows a central limit theorem.

We find the entropy's infinite-size behavior in complex manifold sections.

The study connects norms and filtrations on section rings of projective manifolds.

The study bounds dimensions and proves existence of holomorphic sections on Kähler Ricci shrinkers.

In this work we prove an universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular Hermitian holomorphic line bundles on a compact Kähler complex space $X$. Namely, under mild moment assumptions, we show that the asymptotic distribution of zeros of r…

The study proves a central limit theorem for Gaussian holomorphic sections on Kähler manifolds.

The paper equidistributes zeros of random polynomials and sections on manifolds.

Study Bergman kernels and zero distributions of random sections on Kähler manifolds.

Geometric quantization extended to big line bundles.

We obtain asymptotics of sequences of the holomorphic sections of the pluricanonical bundles on ball quotients associated to closed geodesics. A nonvanishing result follows.

Surveying random sections on Kähler manifolds, leading to metrics.

The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions a…

Geometric quantization results for Riemann surfaces with semi-positive line bundles.

Study asymptotics of extension and orthogonal Bergman kernels for high tensor powers of positive line bundles.

Study optimal holomorphic extensions for jets along submanifolds as tensor powers increase.

Odd-dimensional manifolds have contact maps of non-zero degree.

Study optimal holomorphic extensions on complex manifolds with transitivity property.

Extends Donaldson's techniques to symplectic orbifolds, proving existence of sections and computing cohomology.

The paper studies random systems of holomorphic sections on compact Kähler manifolds and proves equidistribution results.

We study some asymptotic properties of the sequences of symplectic Lefschetz pencils constructed by Donaldson. In particular we prove that the vanishing spheres of these pencils are, for large degree, conjugated under the action of the symplectomorphism group of the fiber. This implies the non-existence of homologicall…

We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. We give further Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line b…

We study the asymptotics of Fubini-Study currents and zeros of random holomorphic sections associated to a sequence of singular Hermitian line bundles on a compact normal Kaehler complex space.

New proof shows holomorphic sectional curvature fully determines curvature tensor.

By using asymptotic Morse inequalities we give a lower bound for the space of holomorphic sections of high tensor powers in a positive line bundle over a q-concave domain. The curvature of the positive bundle induces a hermitian metric on the manifold. The bound is given explicitely in terms of the volume of the domain…

Generalizes jet differential bounds and proves asymptotic Serre duality.

Let $\mathcal{O}(D)$ be an equivariant line bundle which is big and nef on a complex projective nonsingular toric variety $X$. Given a continuous toric metric $\|\cdot\|$ on $\mathcal{O}(D)$, we define the energy at equilibrium of $(X,φ_{\bar{D}})$ where $φ_{\bar{D}}$ is the weight of the metrized toric divisor $\bar{D…

The paper proves that Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.

Classifies Kähler metrics with constant holomorphic curvature.

The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.

Study extends complex sections on non-holomorphic objects on Kähler manifolds.

The paper quantizes Kähler manifolds using differential operators.

The paper proves properties of complex manifolds with nonnegative holomorphic sectional curvature.

Study Bergman and spectral kernels for non-compact complex manifolds.

We generalize a construction of Hitchin to prove that, given any compact Kähler manifold $M$ with positive holomorphic sectional curvature and any holomorphic vector bundle $E$ over $M$, the projectivized vector bundle ${\mathbb P}(E)$ admits a Kähler metric with positive holomorphic sectional curvature.

The analysis of holomorphic sections of high powers $L^N$ of holomorphic ample line bundles $L\to M$ over compact Kähler manifolds has been widely applied in complex geometry and mathematical physics. The Tian-Yau-Zelditch's asymptotic expansion of the Szegö kernel of a circle bundle plays an important role in Kähler-E…

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative c…

Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.

Directly proves Wu's theorem on negative curvature metrics.

Linear statistics of random zero sets are integrals of smooth differential forms over the zero set and as such are smooth analogues of the volume of the random zero set inside a fixed domain. We derive an asymptotic expansion for the variance of linear statistics of the zero divisors of random holomorphic sections of p…

Geometrically proves Zabrodin-Wiegmann conjecture for integer QH states.

The paper develops quantitative estimates for holomorphic sections over bounded domains.

The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the SU(2) character manifold of the peripheral torus. We compute the asymptotics of the torus knot states in terms of the Alexander polynomial, the Reidemeister torsion and the Chern-Simons invariant…

We study the space of holomorphic discs with boundary on a surface in a real 2-dimensional vector bundle over a compact 2-manifold. We prove that, if the ambient 4-manifold admits a fibre-preserving transitive holomorphic action, then a section with a single complex point has $C^{2,α}$-close sections such that any (non…

We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension $-\infty$. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although th…

We use Donaldson's approximately holomorphic techniques to build embeddings of a closed symplectic manifold with symplectic form of integer class in the grassmannians Gr(r,N). We assure that these embeddings are asymptotically holomorphic in a precise sense. We study first the particular case of embeddings in the proje…

The paper proves a Schwarz lemma for mappings between specific types of manifolds.