arXiv research

The paper shows how to destabilize unstable Fano varieties using stability thresholds.

Equivalence proven between divisorial stability and quotient log divisorial stability.

We prove that among all Kollár components obtained by plt blow ups of a klt singularity $o \in (X, D)$, there is at most one that is (log-)K-semistable. We achieve this by showing that if such a Kollár component exists, it uniquely minimizes the normalized volume function introduced in [Li15a] among all divisorial valu…

For any $\mathbb{Q}$-Gorenstein klt singularity $(X,o)$, we introduce a normalized volume function $\widehat{\rm vol}$ that is defined on the space of real valuations centered at $o$ and consider the problem of minimizing $\widehat{\rm vol}$. We prove that the normalized volume has a uniform positive lower bound by pro…

This is a continuation to the paper [arXiv:1511.08164] in which a problem of minimizing normalized volumes over $\mathbb{Q}$-Gorenstein klt singularities was proposed. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler-Einstein metrics on Fano varieties. In particul…

Researchers compute limits of Kähler-Einstein forms on degenerating manifolds.

Proves properness of K-moduli spaces for Fano varieties.

We introduce a new effective stability named "divisorial stability" for Fano manifolds which is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume function. As a corollary, we prove that the first coordinate of the barycenter of the …

Kähler-Ricci flow smooths out positive closed currents with divisorial singularities

Introduces new hyperbolicity concepts for complex manifolds.

New hyperbolicity concepts expand manifold study.

We study the behaviour of Chern numbers of three dimensional terminal varieties under divisorial contractions.

We discuss a technique to construct Ricci-flat hermitian metrics on complements of (some) anticanonical divisors of almost homogeneous manifolds and discuss when this metric is complete and Kähler. This construction has a strong interplay with invariance groups of the same dimension as the manifold acting with an open …

Establishes K"ahler-Ricci flow on log canonical varieties.

Paper rigorously defines Feynman graph integrals on Kähler manifolds.

We compute a closed formula for the class of the closure of the locus of curves in $\overline{\mathcal{M}}_g$ that admit an abelian differential of signature $κ=(k_1,...,k_{g-2})$.

We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimens…

Proves existence of Kähler-Einstein metrics in big cohomology classes.

The paper explores positivity and irreducibility in Hurwitz spaces related to differentials of the second kind.

We prove the existence and uniqueness of the weak Kahler-Ricci flow on projective varieties with log terminal singularities. It is also shown that the weak Kahler-Ricci flow can be uniquely continued through divisorial contractions and flips if they exist. We then propose an analytic version of the Minimal Model Progra…

Study of K-moduli of prime Fano threefolds of genus twelve, proving boundary purely divisorial.

Study shows hypercomplex twistor spaces lack divisors and special metrics.

Paper introduces new actuarial-consistent valuations for insurance liabilities.

Paper recovers uncertainty from dynamic valuation rules.

Study convolution of invariant valuations on Lie groups.

Study shows Kähler-Einstein metric singularities linked to curvature.

In this note we propose to show that the Kähler-Ricci flow fits naturally within the context of the Minimal Model Program for projective varieties. In particular we show that the flow detects, in finite time, the contraction theorem of any extremal ray and we analyze the singularities of the metric in the case of divis…

SL(n) covariant valuations on Orlicz spaces are represented and characterized.

Market valuation duration is 175 years, but drops to 46 years during crises.

Paper simplifies default process modeling and credit valuation.

Business cycles affect startup valuations, both directly and indirectly.

Classification of SL(n) covariant valuations on Orlicz spaces.

We prove new kinematic formulas for tensor valuations and simplify previously known Crofton formulas by using the recently developed algebraic theory of translation invariant valuations. The heart of the paper is the computation of the Alesker-Fourier transform on the large class of spherical valuations, which is achie…

Classifies contravariant matrix-valued valuations on polytopes without continuity assumptions.

Let $\mathrm{SO}^+(p,q)$ denote the identity connected component of the real orthogonal group with signature $(p,q)$. We give a complete description of the spaces of continuous and generalized translation- and $\mathrm{SO}^+(p,q)$-invariant valuations, generalizing Hadwiger's classification of Euclidean isometry-invari…

Study evaluates valuation models for UK companies using case studies.

This paper proposes a paradigm shift in the valuation of long term annuities, away from classical no-arbitrage valuation towards valuation under the real world probability measure. Furthermore, we apply this valuation method to two examples of annuity products, one having annual payments linked to a mortality index and…

This paper provides intuition on the relationship of accrual and mark-to-market valuation for cash and forward interest rate trades. Discounted cashflow valuation is compared to spread-based valuation for forward trades, which explains the trader's view on valuation. This is followed by Taylor series approximation for …

The classification of continuous, translation invariant Minkowski valuations which are contravariant (or covariant) with respect to the complex special linear group is established in a 2-dimensional complex vector space. Every such valuation is given by the sum of a valuation of degree of homogeneity 1 and 3. In dimens…

Computes tube formulas for valuations in complex space forms.

This paper addresses credit valuation adjustment with a new closeout convention.

Classifies SL(n) covariant matrix-valued valuations on Lp-spaces.

Existence of smooth valuations on subspaces is shown for certain conditions.

Develops a new method to study algebraic tangent cones of sheaves using valuations.

We study the properties of the multiplicative structure on valuations on convex sets. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss related problems of integral geometry. Then we formulate a conjectural analogue of this result for odd valuations.

We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.

Fair market valuations ignore future worker profits in employee-owned firms.

A description of continuous rigid motion compatible Minkowski valuations is established. As an application, we present a Brunn-Minkowski type inequality for intrinsic volumes of these valuations.