arXiv research

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

The paper studies geodesic-Einstein metrics on line bundles over fibration.

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

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

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

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

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

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

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

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

We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, …

Let $X\rightarrow S$ be a smooth projective surjective morphism of relative dimension $n$, where $X$ and $S$ are integral schemes over $\mathbb C$. Let $L\rightarrow X$ be a relatively very ample line bundle. For every sufficiently large positive integer $m$, there is a canonical isomorphism of the Deligne pairing $\la…

Let $X\to B$ be a proper flat morphism between smooth quasi-projective varieties of relative dimension $n$, and $L\to X$ a line bundle which is ample on the fibers. We establish formulas for the first two terms in the Knudsen-Mumford expansion for $\det (π_* L^k)$ in terms of Deligne pairings of $L$ and the relative ca…

Defines geodesic-Einstein metrics and their relation to nonlinear stabilities.

The paper proves a CM line bundle is ample on K-moduli spaces.

We prove that the $L^2$ metric on the direct image of an adjoint positive line bundle by a locally trivial submersion between projective manifolds is Nakano positive, under the assumption that the typical fiber has zero first Betti number. As a consequence, we get that the symmetric powers of an ample vector bundle ten…

Characterizes projective submanifolds in high dimensions.

The paper establishes conditions for optimal sampling configurations on complex manifolds.

Study shows Kähler-Ricci flow singularity type is consistent over time.

Let X be a complex manifold fibered over the base S and let L be a relatively ample line bundle over X. We define relative Kahler-Ricci flows on the space of all Hermitian metrics on L with relatively positive curvature. Mainly three different settings are investigated: the case when the fibers are Calabi-Yau manifolds…

Let X be a smooth complex projective variety of dimension d. It is classical that ample line bundles on X satisfy many beautiful geometric, cohomological, and numerical properties that render their behavior particularly tractable. By contrast, examples due to Cutkosky and others have led to the common impression that t…

Estimates curvature for long-time continuity method solutions.

In this paper, we study the relations between the log term of the Szegö kernel of the unit circle bundle of the dual line bundle of an ample line bundle over a compact Kählermanifold. We proved a local rigidity theorem. The result is related to the classical Ramadanov Conjecture.

Study of monodromy and vanishing cycles for complete intersection curves.

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

Study stability thresholds of big line bundles, proving bounds and generalizing results.

For Fano homogeneous toric bundles, we obtain a formula of the greatest lower bound on Ricci curvature. We also give a criteria for the ampleness of a kind of line bundles over general homogeneous toric bundles.

Develops moduli theory for Calabi-Yau pairs, constructing a projective space.

Analyzes curvature and torsion on Teichmüller space for large k.

Study of line bundles on spherical varieties leads to Calabi-Yau metrics.

Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds of dimension $n$ and a relatively ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*Ω^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. We give an explicit formula for the curvature tensor of these direct images.…

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

The spectrum of the Laplace-Dolbeault operator for any line bundle with parallel curvature on a flat complex torus is computed. The Ray-Singer analytic torsion is then deduced, generalizing thus Bost's result for ample line bundles and Ray-Singer's ones for flat bundles, of which we a geometric interpretation is given.

Let $Δ\subset \mathbb{R}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_Δ$ and the very ample $(\mathbb{C}^\times)^n$-equivariant line bundle $L_Δ$ on $X_Δ$ associated with $Δ$. In the present paper, we give a necessary and sufficient …

Study on positivity of CM line bundles on moduli space of klt good minimal models with κ=1.

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

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

Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds and a relative ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*Ω^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. Using the explicit formula for the curvature tensor of these direct images, we prove that the d…

Quantizes semipositive line bundles on complex manifolds.

The study of Chern numbers on vector bundles uses combinatorial methods to establish bounds and ordering.

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

We compute the log canonical thresholds of non-negatively curved singular hermitian metrics on ample linearized line bundles on bi-equivariant group compactifications of complex reductive groups. To this end, we associate to any such metric a convex function whose asymptotic behavior determines the log canonical thresh…

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

Let $L$ be a holomorphic line bundle over a compact Kähler manifold $X$. Motivated by mirror symmetry, we study the deformed Hermitian-Yang-Mills equation on $L$, which is the line bundle analogue of the special Lagrangian equation in the case that $X$ is Calabi-Yau. We show that this equation is the Euler-Lagrange equ…

Sufficient condition for log-continuity of complex Monge-Ampère solutions.

Let $X$ be a smooth projective complex variety with an ample line bundle $L$, and let $D$ be a simple normal crossing divisor. We establish the Kobayashi-Hitchin correspondence between tame harmonic bundles on $X-D$ and $μ_L$-stable parabolic $λ$-flat bundles with trivial characteristic numbers on $(X,D)$. Especially, …

Study theta functions and adiabatic curvature on Abelian varieties.

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