Related papers: Variation of log canonical thresholds in linear sy…

Shokurov's ACC Conjecture says that the set of all log canonical thresholds on varieties of bounded dimension satisfies the Ascending Chain Condition. This conjecture was proved for log canonical thresholds on smooth varieties in [EM1].…

We present a new relation between an invariant of singularities in characteristic zero (the log canonical threshold) and an invariant of singularities defined via the Frobenius morphism in positive characteristic (the F-pure threshold). We…

The minimal log discrepancy is an invariant of singularities that plays an important role in the birational classification of algebraic varieties. Shokurov conjectured that the minimal log discrepancy can always be bounded from above in…

Minimal log discrepancies (mld's) are related not only to termination of log flips, and thus to the existence of log flips but also to the ascending chain condition (acc) of some global invariants and invariants of singularities in the Log…

We find an explicit upper bound for the anticanonical volume of Fano 4-folds with canonical singularities.

We show that log canonical thresholds for complex analytic spaces satisfy the ACC.

We show that minimal models of log canonical pairs exist, assuming the existence of minimal models of smooth varieties.

An inductive approach to classifying toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are 674,688…

We consider methods for obtaining local lower bounds on characteristics of quantum (correspondingly, classical) systems, i.e. lower bounds valid in the trace norm $\epsilon$-neighborhood of a given state (correspondingly, probability…

We find new lower bounds on the torsion orders of very general Fano hypersurfaces over (uncountable) fields of arbitrary characteristic. Our results imply that unirational parametrizations of most Fano hypersurfaces need to have enormously…

We prove the existence of Fano ladders on weak log Fano varieties of coindex less than 4, as an application of adjunction and nonvanishing.

In this short note, we give an alternative proof of the semipositivity of the Chow-Mumford line bundle for families of K-semistable log-Fano pairs, and of the nefness threeshold for the log-anti-canonical line bundle on families of K-stable…

We estimate an upper bound of the spectral radius of a linear operator on the quantum cohomology of the toric Fano manifolds $\mathbb{P}_{\mathbb{P}^{n}}(\mathcal{O}\oplus\mathcal{O}(3))$. This provides a negative answer to Galkin's lower…

We compare the minimal model of a log canonical pair with the minimal model of its reduced boundary. These results are then used to study the existence of the minimal model of a semi-log-canonical pair using its normalization.

Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as another invariant, the stability threshold. The latter…

We generalize the work of Jian Song to compute the alpha invariant of any (nef and big) toric line bundle in terms of the associated polytope. We use the analytic version of the computation of the log canonical threshold of monomial ideals…

In the paper we compute the global log canonical thresholds of the secondary Burniat surfaces with $K^2 = 5$. Furthermore, we establish optimal lower bounds for the log canonical thresholds of members in pluricanonical sublinear systems of…

We provide a sufficient condition for polarisations of Fano varieties to be K-stable in terms of Tian's alpha invariant, which uses the log canonical threshold to measure singularities of divisors in the linear system associated to the…

We study the Picard rank of smooth toric Fano varieties possessing families of minimal rational curves of given degree. We discuss variants of a conjecture of Chen-Fu-Hwang and prove a version of their statement that recovers the original…

Let $\varphi$ be a plurisubharmonic function defined in a neighborhood of the origin in $\mathbb C^n$. For each real number $t>-n$, we associate to $\varphi$ the weighted log canonical threshold \[ c_t(\varphi):=\sup\Bigl\{c\geq…