arXiv research

The study constructs universal invariants for non-Archimedean metrics on projective varieties.

Study shows Calabi-Yau metrics converge to a specific form under certain conditions.

Solves non-Archimedean Calabi-Yau equation on complex log pairs.

The paper describes a stratification of a compactified Hurwitz space using combinatorial trees.

Defines volume and Monge-Ampère energy on polarized affine varieties.

Paper generalizes K-stability to non-algebraic spaces for Kähler metrics.

We study the convergence of volume forms on a degenerating holomorphic family of log-Calabi-Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let $(X,B)$ be a holomorphic family of sub log canonical, log-Calabi-Yau complex varieties parameterized by the punctured un…

We compactify the classical moduli variety of compact Riemann surfaces by attaching moduli of (metrized) graphs as boundary. The compactifications do not admit the structure of varieties and patch together to form a big connected moduli space in which $\sqcup_{g} M_{g}$ is open dense. The metrized graphs, which are oft…

We construct universal geometric spaces over the real spectrum compactification $Ξ^{\mathrm{RSp}}$ of the character variety $Ξ$ of a finitely generated group $Γ$ in $\mathrm{SL}_n$, providing geometric interpretations of boundary points. For an algebraic set $Y(\mathbb{R})$ on which $\mathrm{SL}_n(\mathbb{R})$ acts by …

Non-archimedean SYZ fibration constructed for Calabi-Yau hypersurfaces.

This paper constructs a non-Archimedean Teichmüller space using tropical geometry.

Defines plurisubharmonic metrics on hybrid spaces and proves their canonical extensions.

Construct special Lagrangian fibrations on abelian varieties using retraction techniques.

The Bergman measure converges to the Zhang measure on a hybrid space.

Synthetic approach to pluripotential theory measures finite energy.

The paper proves a Basmajian identity for non-Archimedean local fields.

In this paper, we study the dynamics of degenerating sequences of rational maps on Riemann sphere $\hat{\mathbb{C}}$ using $\mathbb{R}$-trees. Given a sequence of degenerating rational maps, we give two constructions for limiting dynamics on $\mathbb{R}$-trees: one geometric and one algebraic. The geometric constructio…

Study Monge-Ampère equations on Calabi-Yau hypersurfaces, proving unique solutions and implications for special Lagrangian fibrations.

Study of non-archimedean μ-entropy and its connection to K-stability.

Let $X$ be a compact Kähler manifold with vanishing Riemann curvature. We prove that there exists a manifold $X'$, deformation equivalent to $X$, which is not an analytification of any projective variety, if and only if $H^0(X, Ω^2) \neq 0$. Using this, we recover a recent theorem of Catanese and Demleitner, which stat…

New homotopy theory reveals the structure of stable curves.

We define the notion of a smooth pseudo-Riemannian algebraic variety $(X,g)$ over a field $k$ of characteristic $0$, which is an algebraic analogue of the notion of Riemannian manifold and we study, from a model-theoretic perspective, the algebraic differential equation describing the geodesics on $(X,g)$. When $k$ is …

Mathematically proves SYZ conjecture for conifold transition.

Lecture notes on using non-Archimedean geometry for complex variety degenerations.

Suppose $(X_n)$ is a sequence of positive-dimensional smooth projective complete intersections over $\mathbb{F}_q$ with dimensions bounded from above and with characteristic zero lifts $(\tilde{X}_n)$ to smooth projective geometrically connected varieties. Suppose each complex variety $\tilde{X}^{an}_n$ has (underlying…

Signed seminorms linked to real tropical spaces and matroids.

Researchers resolve a SYZ conjecture for A_n singularities using quantum-corrected T-duality.