Related papers: A note on Tsuji's criterion for numerical triviali…

We apply Tsuji's theory of numerically trivial fibrations to the abundance problem.

The Reduction Map Theorem in H. Tsuji's work on numerical trivial fibrations is corrected and proven. To this purpose various definitions of Tsuji's new intersection numbers for pseudo-effective line bundles equipped with a positive…

Let f: X \to Z be a surjective morphism of smooth complex projective varieties with connected fibers. Suppose that L is a pseudo-effective divisor on X that is f-numerically trivial. We show that there is a divisor D on Z such that L is…

We prove the abundance theorem for numerically trivial log canonical divisors of log canonical pairs and semi-log canonical pairs.

Consider a one-parameter family of algebraic varieties degenerating to a reducible one. Our main result is a formula for the fundamental cycle of the limit subscheme of any family of effective Cartier divisors. The formula expresses this…

In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…

Using ideas from algebraic $K$-theory, we prove that a simple and naturally applicable criterion of Kitaev suffices to trivialize the Fredholm determinant of a multiplicative commutator.

We give a short proof -- not relying on ideal classes or the geometry of numbers -- of a known criterion for quadratic orders to possess unique factorization.

For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\to S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D \subset X$. We show,…

We classify the "quotients" of a tannakian category in which the objects of a tannakian subcategory become trivial, and we examine the properties of such quotient categories.

We study torsors for groups defined by algebraic difference equations. Our main result provides necessary and sufficient conditions on the base difference field for all such torsors to be trivial. We also present an application to the…

We present a short proof, relaying on the divergence theorem, verifying that minimal sets in the plane are trivial.

Explicit expression for quasi-triviality of scalar non-linear PDE is under consideration.

We make the simple, and yet deep, observation that a regular conditional distribution (rcd) almost surely trivialises the conditioning $\sigma$-algebra if and only if there exists a "measurable selection" of regular conditional…

We translate Uchimura's identity for the divisor function and whose generalizations into combinatorics of partitions, and give a combinatorial proof of them. As a by-product of their proofs, we obtain some combinatorial results.

We show the integrality of the simple Hurwitz numbers. The main tool is the cut-and-join operator, and our proof is a purely combinatorial one.

In this paper, we give a simple proof of a triviality criterion due to I.Biswas and J.Pedro and P.Dos Santos. We also prove a vector bundle on a homogenous space is trivial if and only if the restrictions of the vector bundle to Schubert…

We consider subtorus actions on divisorial toric varieties. Here divisoriality means that the variety has many Cartier divisors like quasiprojective and smooth ones. We characterize when a subtorus action on such a toric variety admits a…

We prove that if the moduli $\mathbb Q$-b-divisor of a basic slc-trivial fibration is b-numerically trivial then it is $\mathbb Q$-b-linearly trivial. As a consequence, we prove that the moduli part of a basic slc-trivial fibration is…

We construct some families of quadratic fields whose class numbers are divisible by $3.$ The main tools used are a trinomial introduced by Kishi and a parametrization of Kishi and Miyake of a family of quadratic fields whose class numbers…