arXiv research

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

$\mathcal{H}-$holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic $1-$form as perturbation term. In this paper we study the asymptotics of $\mathcal{H}-$holomorphic curves defined on a sequence of degenerating cylinders.

Formula derived for torsion of modified Dirac operator.

Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.

Study of torsion forms for positive line bundles.

Formula found for energy slope in complex geometry.

Estimates holomorphic sections on Bohr-Sommerfeld Lagrangian submanifolds.

Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.

The purpose of this paper is first to give an asymptotic formula for the holomorphic analytic torsion forms of a fibration associated with increasing powers of a given line bundle. Secondly, we generalize this formula, thanks to the theory of Toeplitz operators, in the case where the powers of the line bundle is replac…

Investigates $J$-equation on holomorphic vector bundles over Kähler manifolds.

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

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

Master thesis proves Bergman kernel asymptotics for positive line bundles.

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…

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

Holomorphic quantum modular forms linked to knot volumes.

The paper proves heat kernel asymptotics for high power line bundles on complex manifolds.

We study the family of holomorphic maps from the polydisk to the disk which restrict to the identity on the diagonal. In particular, we analyze the asymptotics of the orbit of such a map under the conjugation action of a unipotent subgroup of $\text{PSL}_2(\mathbb{R})$. We discuss an application our results to the stud…

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

Study optimal holomorphic extensions on complex manifolds with transitivity property.

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

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

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

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…

We initiate the study of the asymptotic topology of groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers (these are called here as holomorphically convex groups). We prove the $H_1$-semistability conjecture of Geoghegan for holomorphically…

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

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

Motivated by gauge theory under special holonomy, we present techniques to produce holomorphic bundles over certain noncompact $3-$folds, called building blocks, satisfying a stability condition `at infinity'. Such bundles are known to parametrise solutions of the Yang-Mills equation over the $\rm G_2-$manifolds obtain…

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

Study limits of Kähler submanifolds and prove no holomorphic isometries.

For a polarized algebraic manifold $(X,L)$, let $T$ be an algebraic torus in the group of all holomorphic automorphisms of $X$. Then strong relative K-stability will be shown to imply asymptotic relative Chow-stability. In particular, by taking $T$ to be trivial, we see that asymptotic Chow-stability follows from stron…

The article develops deformation theory for ACyl associative submanifolds in ACyl G2-manifolds.

We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line C, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to TxP1.…

Geometric quantization extended to big line bundles.

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

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

The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincaré disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a bounded symmetric domain $Ω$ must necessarily be asymptotically totally geodesic. A…

We study the complex geometry of generalized Kepler manifolds, defined in Jordan theoretic terms, introduce Hilbert spaces of holomorphic functions defined by radial measures, and find the complete asymptotic expansion of the corresponding reproducing kernels for Kähler potentials, both in the flat and bounded setting.

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

Unified proofs of weak holomorphic Morse inequalities using Bergman kernel functions.

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

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…

The paper studies the asymptotic behavior of HCMA equations on ALE Kahler manifolds.

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

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…

The paper quantizes Kähler manifolds using differential operators.

We explore the relationship between contact forms on $\mathbb S^3$ defined by Finsler metrics on $\mathbb S^2$ and the theory developed by H. Hofer, K. Wysocki and E. Zehnder in \cite{HWZ,HWZ1}. We show that a Finsler metric on $\mathbb S^2$ with curvature $K\geq 1$ and with all geodesic loops of length $>π$ is dynamic…

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…