arXiv research

Corrected proof for complex Monge-Ampère equations

Let (X,D) be a klt pair. Assuming either K_X+D big or -(K_X+D) ample, and that the coefficients of D are greater than 1/2, we show that the Kähler-Einstein metric attached to (X,D) -whenever it exists- has cone singularities along D on the log-smooth locus of the pair intersected with the ample locus of K_X+D (in the n…

The paper proves boundedness of envelopes in complex manifolds.

Study on Bergman kernels near analytic hypersurfaces with exponential decay.

In this paper we prove that the Kähler-Einstein metrics for a toroidal canonical degeneration family of Kähler manifolds with ample canonical bundles Gromov-Hausdorff converge to the complete Kähler-Einstein metric on the smooth part of the central fiber when the base locus of the degeneration family is empty. We also …

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

Let $(X,ω)$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $\MAH(X,ω)$ of the complex Monge-Am…

Conic Kähler-Einstein metrics glued from fibers of a map are positive.

Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere has at most 3 non-free orbits. Using a certain one-to-one correspondence, Kollár formulated the algebraic version of the Montgomery-Yang problem: every projective surface $S$ with the second Betti number $b_2…

In this paper, we study the boundary behavior of the negatively curved Kähler-Einstein metric attached to a log canonical pair $(X,D)$ such that $K_X+D$ is ample. In the case where $X$ is smooth and $D$ has simple normal crossings support (but possibly negative coefficients), we provide a very precise estimate on the p…

New system solves curvature for ample vector bundles, proving Griffiths conjecture.

Let $X$ be a smooth variety and let $L$ be an ample line bundle on $X$. If $π^{alg}_{1}(X)$ is large, we show that the Seshadri constant $ε(p^{*}L)$ can be made arbitrarily large by passing to a finite étale cover $p:X'\rightarrow X$. This result answers affirmatively a conjecture of J.-M. Hwang. Moreover, we prove an …

We study degenerate complex Monge-Ampère equations of the form $(ω+dd^c \varphi)^n = e^{t \varphi} μ$ where $ω$ is a big semi-positive form on a compact Kähler manifold $X$ of dimension $n$, $t \in \R^+$, and $μ=fω^n$ is a positive measure with density $f\in L^p(X,ω^n)$, $p>1$. We prove the existence and unicity of bou…

Solves Demailly's system for direct sums of ample line bundles on Riemann surfaces.

Study local curvature of Kähler-Ricci flow on semi-ample manifolds.

Proves polynomial injectivity of Fubini-Study map for ample line bundles.

The paper studies volumes of direct images for high tensor powers of ample bundles.

Explains differences and similarities of strictly nef and ample vector bundles.

Study on Kähler-Ricci flow's infinite-time singularities.

Monodromy and vanishing cycles computed for ample linear systems on simply connected surfaces.

We prove that the CM line bundle is ample on the proper moduli space which parametrizes KSBA stable varieties.

Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric h on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k th tensor power of L. In this paper various convergence results are obtained for the corr…

The paper studies positivity properties of cotangent bundles in complex hyperbolic manifolds with cusps.

Shows CM line bundles are ample on K-stable varieties.

Constructs currents and heights on K3 surfaces.

It is conjectured that the moduli b-divisor of the Kawamata-Kodaira canonical bundle formula associated to a klt-trivial fibration $(X,B)\to Z$ is semi-ample. In this paper, we show the semi-ampleness of an arbitrarily small perturbation of the moduli b-divisor by a fixed appropriate divisor which roughly speaking come…

Continuous metrics on ample bundles lie in infinite-dimensional cones.

Proves projectivity and ampleness of a Kähler manifold using complex Monge-Ampère equation.

Compact Hermitian manifolds with quasi-negative curvature have ample canonical line bundles.

Study on curvature of direct image bundle for higher powers of relatively ample line bundles.

In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if $E$ is an ample vector bundle over a compact Kähler manifold $X$, $S^kE\…

This paper generalizes the cut locus concept for almost distance functions.

Proves complex Finsler bundles with positive curvature are ample and biholomorphic to projective space.

AMPL is a new software pipeline for drug discovery models.

Study the geometry of lightlike loci on mixed type surfaces in Lorentz-Minkowski 3-space.

Existence proven for special metrics on certain complex manifolds.

Study non-existence of complex ball quotients in Torelli locus.

In this study, we investigate the locus of the centers of the Meusnier spheres. Just as focal curve is the locus of the centers of the osculating spheres, we investigate the geometrical interpretation on the locus of the centers of the Meusnier spheres. We proved that if the curve is a principal line, the locus of the …

The paper solves a 25-year-old problem about maximal growth distributions on manifolds.

We study the singular locus of solutions to Hamilton-Jacobi equations with a Hamiltonian independent of $u$. In a previous paper, we proved that the singular locus is what we call a balanced split locus. In this paper, we find and classify all balanced split sets, identifying the cases where the only balanced split loc…

The paper describes the CR umbilical locus of a real ellipsoid in complex space.

New 2-spheres of revolution with simple cut locus structures.

Laplacian of distance function shows negative infinity at cut locus points.

Study on Blaschke locus with covariance metric properties.

Let $X$ be a canonically polarized variety, i.e. a complex projective variety such that its canonical class $K_{X}$ defines an ample $\Q-$line bundle, and satisfying the conditions $G_1$ and $S_2$. Our main result says that $X$ admits a Kähler-Einstein metric iff $X$ has semi-log canonical singularities i.e. iff $X$ is…

Study describes deformation of topological locus for Hopf link.

The paper studies geometric loci and their invariants in complex dynamics.

Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. …