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Tran-Trung Nghiem

Publications and source records attributed to Tran-Trung Nghiem.

7 recordsLinked to original sources
arXiv2026

Examples of complete Calabi--Yau metrics on affine smoothings of irregular toric Calabi--Yau cones

We present infinitely many new examples of affine Calabi--Yau manifolds of Euclidean volume growth and quadratic curvature decay, whose tangent cones at infinity are irregular and have smooth links. In the process, we demonstrate (and provide the relevant computer code) how to explicitly compute the Reeb field, the Minkowski summand cone, and its maximal decompositions for a given toric Calabi--Yau cone with smooth link from the data of its toric diagram. In complex dimension three, the component corresponding to a maximal Minkowski decomposition into lattice summands is a smoothing component if and only if every summand is either a primitive lattice segment or a unimodular lattice triangle. Furthermore, we propose an effective strategy to generate smoothable Calabi--Yau cones from a given non-smoothable one by taking Minkowski sums of certain toric diagrams, and provide an example to illustrate the method.

math.DG↗
arXiv2025

Calabi-Yau metrics on rank two symmetric spaces with horospherical tangent cone at infinity

We show that on every non-$G_2$ complex symmetric space of rank two, there are complete Calabi-Yau metrics of Euclidean volume growth with prescribed horospherical singular tangent cone at infinity, providing the first examples of affine Calabi-Yau smoothings of singular and irregular tangent cone. As a corollary, we obtain infinitely many examples of Calabi-Yau manifolds degenerating to the tangent cone in a single step, supporting a recent conjecture by Sun-Zhang, which was only proved when the tangent cone at infinity has only an isolated singularity.

math.DG↗
arXiv2024

K-stable valuations and Calabi-Yau metrics on affine spherical varieties

After providing an explicit K-stability condition for a $\mathbb{Q}$-Gorenstein log spherical cone, we prove the existence and uniqueness of an equivariant K-stable degeneration of the cone, and deduce uniqueness of the asymptotic cone of a given complete $K$-invariant Calabi-Yau metric in the trivial class of an affine $G$-spherical manifold, $K$ being the maximal compact subgroup of $G$. Next, we prove that the valuation induced by $K$-invariant Calabi-Yau metrics on affine $G$-spherical manifolds is in fact $G$-invariant. As an application, we point out an affine smoothing of a Calabi-Yau cone that does not admit any $K$-invariant Calabi-Yau metrics asymptotic to the cone. Another corollary is that on $\mathbb{C}^3$, there are no other complete Calabi-Yau metrics with maximal volume growth and spherical symmetry other than the standard flat metric and the Li-Conlon-Rochon-Székelyhidi metrics with horospherical asymptotic cone. This answers the question whether there is a nontrivial asymptotic cone with smooth cross section on $\mathbb{C}^{3}$ raised by Conlon-Rochon when the symmetry is spherical.

math.AG↗
arXiv2023

On the regularity of conical Calabi-Yau potentials

Using pluripotential theory on degenerate Sasakian manifolds, we show that a locally bounded conical Calabi-Yau potential on a Fano cone is actually smooth on the regular locus. This work is motivated by a similar result obtained by R. Berman in the case where the cone is toric. Our proof is purely pluripotential and independent of any extra symmetry imposed on the cone.

math.DG↗
arXiv2023

Spherical cones: classification and a volume minimization principle

Using a variational approach, we establish the equivalence between a weighted volume minimization principle and the existence of a conical Calabi-Yau structure on horospherical cones with mild singularities. This allows us to do explicit computations on the examples arising from rank-two symmetric spaces, showing the existence of many irregular horospherical cones.

math.DG↗
arXiv2020

On a rationality problem for fields of cross-ratios II

Let $k$ be a field, $x_1, \dots, x_n$ be independent variables and $L_n = k(x_1, \dots, x_n)$. The symmetric group $Σ_n$ acts on $L_n$ by permuting the variables, and the projective linear group $\text{PGL}_2$ acts by \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \colon x_i \mapsto \frac{a x_i + b}{c x_i + d} \] for each $i = 1, \ldots, n$. The fixed field $L_n^{\text{PGL}_2}$ is called "the field of cross-ratios". Given a subgroup $S \subset Σ_n$, H. Tsunogai asked whether $L_n^S$ rational over $K_n^S$. When $n \geqslant 5$ the second author has shown that $L_n^S$ is rational over $K_n^S$ if and only if $S$ has an orbit of odd order in $\{ 1, \dots, n \}$. In this paper we answer Tsunogai's question for $n \leqslant 4$.

math.AG↗