arXiv research

The paper establishes a correspondence for projective bundles over curves using test configurations and extremal metrics.

The paper classifies equivariant test configurations for spherical varieties.

A machine learning configuration refers to a combination of preprocessor, learner, and hyperparameters. Given a set of configurations and a large dataset randomly split into training and testing set, we study how to efficiently select the best configuration with approximately the highest testing accuracy when trained f…

For smooth test configurations, there always exist C^{1,1} geodesic rays in Kahler metric space parallel to the algebraic ray. The $\yen$ invariant agrees with Futaki invariant, at least under nice assumptions. Explicit examples in Toric cases are calculated. On simple test configurations, Donaldson's correspondence be…

The paper classifies and computes limits of equivariant compactifications of groups.

Characterizes K-semistability for log Fano cone singularities.

Given a polarized complex manifold, projection of a torus-equivariant test configuration to holomorphic vector fields was introduced by G. Székelyhidi, as the limit of the associated $\mathbb{C}^*$-actions. We show that there actually holds the moment convergence of the weight distributions. Our analytic approach at th…

Geodesic rays and chordal distances link algebraic and geometric properties of positive metrics.

New extremal metrics found on Kähler manifolds.

We give a parametrization of test configurations in the sense of Donaldson via spherical buildings, and show the existence of "optimal" destabilizing test configurations for unstable varieties, in the wake of Mumford and Kempf. We also give an account of the recent slight amendment to definition of K-stability after Li…

Formula proves invariant matches for smooth and orbifold test configurations.

Adaptive RL optimizes testing resource allocation for dynamic software environments.

Characterizes closures of test configurations and algebraic singularity types.

For any flat projective family $(\mX,\mL)\rightarrow C$ such that the generic fibre $\mX_η$ is a klt Q-Fano variety and $\mL|_{\mX_η}\sim_{Q}-K_{X_η}$, we use the techniques from the minimal model program (MMP) to modify the total family. The end product is a family such that every fiber is a klt Q-Fano variety. Moreov…

In this note, we consider a sequence of test configurations compatible with a Kaehler metric in $c_1(L)$ on a polarized algebraic manifold $(X,L)$. Then an explicit formula for the Donaldson-Futaki invariant $F_1$ for the sequence will be given.

Ensuring that all supposedly valid configurations of a software product line (SPL) lead to well-formed and acceptable products is challenging since it is most of the time impractical to enumerate and test all individual products of an SPL. Machine learning classifiers have been recently used to predict the acceptabilit…

For test configurations, the Donaldson-Futaki invariant F_1 is well-known. In this note, its refinement will be discussed. Then we see that Li-Xu's pathology doesn't occur, since their example of a non-normal test configuration, with trivial normalization, actually has non-vanishing F_1 in this refined sense.

We apply the integral formula of volumes to the family of graded linear series constructed from any test configuration. This solves the conjecture raised by Witt--Nyström so that the sequence of spectral measures for the induced $\mathbb{C}^*$-action on the central fiber converges to the canonical Duistermatt--Heckman …

The paper classifies test configurations and derives a criterion for uniform K-stability of certain algebraic varieties.

Paper computes stability of Q-Fano spherical varieties using test configurations and Futaki invariants.

Geodesic rays of class C^{1,1} are constructed for any test configuration of a positive line bundle L on X using resolution of singularities. The construction reduces to finding a subsolution of the corresponding Monge-Ampere equation. Geometrically, this is accomplished by the use a positive line bundle on the resolut…

In this paper, we give a new version of the modified Futaki invariant for a test configuration associated to the soliton action on a Fano manifold. Our version will naturally come from toric test configurations defined by Donaldson for toric manifolds. As an application, we show that the modified $K$-energy is proper f…

We introduce a strengthening of K-stability, based on filtrations of the homogeneous coordinate ring. This allows for considering certain limits of families of test-configurations, which arise naturally in several settings. We prove that if a manifold with no automorphisms admits a cscK metric, then it satisfies this s…

We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano variety degenerates to a uniquely determined twisted K-polystable Fan…

New approach to $Z$-stability and critical metrics on Kähler manifolds.

Researchers introduce new energies to study constant scalar curvature metrics.

In this note, given a polarized algebraic manifold $(X,L)$, we define the Donaldson-Futaki invariant for a sequence of test configurations for $(X,L)$ with exponents tending to infinity. This then allows us to define a strong version of K-stability or K-semistability for $(X,L)$. In particular, $(X,L)$ will be shown to…

Let $X$ be a compact complex manifold, $L\to X$ an ample line bundle over $X$, and ${\cal H}$ the space of all positively curved metrics on $L$. We show that a pair $(h_0,T)$ consisting of a point $h_0\in {\cal H}$ and a test configuration $T=({\cal L}\to {\cal X}\to {\bf C})$, canonically determines a weak geodesic ra…

We consider the problem of configuring general-purpose solvers to run efficiently on problem instances drawn from an unknown distribution. The goal of the configurator is to find a configuration that runs fast on average on most instances, and do so with the least amount of total work. It can run a chosen solver on a r…

This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic ray itself is C^{1,1} and no better. The \kahler metrics associated to the geodesi…

Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in Kähler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as…

On a K-unstable toric variety we show the existence of an optimal destabilising convex function. We show that if this is piecewise linear then it gives rise to a decomposition into semistable pieces analogous to the Harder-Narasimhan filtration of an unstable vector bundle. We also show that if the Calabi flow exists f…

Study Mabuchi rays on toric Kähler manifolds to understand quantization.

Proves properness of K-moduli spaces for Fano varieties.

The study connects K-stability and large complex structure limits in mirror symmetry.

The paper introduces $μ$K-stability for polarized schemes and develops equivariant calculus.

We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature Kähler metric. This notion of uniform K-stability is analogous to coercivity of the Mabuchi functional. …

Introduces non-Archimedean metrics for pseudoeffective classes on Kähler manifolds.

Proves uniform K-stability is open in Kähler cone.

We prove and test an efficient series representation for the European Black-Scholes call, which generalizes and refines previously known approximations, and works in every market configuration.

Electric vehicles (EVs) have been gaining popularity due to their environmental friendliness and efficiency. EV charging station networks are scalable solutions for supporting increasing numbers of EVs within modern electric grid constraints, yet few tools exist to aid the physical configuration design of new networks.…

In this short note, we give a new proof of a theorem of Arezzo-Tian on the existence of smooth geodesic rays tamed by a special degeneration.

Solves a long-standing problem in Kähler geometry.

We translate the problem of calculating the entropy of a set of binary configurations/signals into a sequence of supervised classification tasks. Subsequently, one can use virtually any machine learning classification algorithm for computing entropy. This procedure can be used to compute entropy, and consequently the f…

It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this conjecture. We also show that it would give a new proof of the K-polystability of constant…

The paper simplifies K-stability conditions for spherical varieties.

Proves Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.

Graph neural networks fail to distinguish certain 3D atom configurations.