Related papers: On the abundance theorem in the case $\nu=0$

In this paper, we use canonical bundle formulas to prove some generalizations of an old theorem of Kawamata on the semiampleness of nef and abundant log canonical divisors. In particular, we show that for klt pairs $(X,B)$ with $K_X+B$…

We show that a suitable quantitative Fatou Theorem characterizes uniform rectifiability in the codimension 1 case.

In the recent preprint [3], Goldston, Pintz, and Y{\i}ld{\i}r{\i}m established, among other things, $$ \liminf_{n\to\infty}{p_{n+1}-p_n\over\log p_n}=0,\leqno(0) $$ with $p_n$ the $n$th prime. In the present article, which is essentially…

We prove the classical Nakano vanishing theorem with H\"ormander $L^2$-estimates on a compact K\"ahler manifold using Siu's so called $\partial\dbar$-Bochner-Kodaira method, thereby avoiding the K\"ahler identities completely. We then…

A famous theorem of Szemer\'edi asserts that given any density $0 < \delta \leq 1$ and any integer $k \geq 3$, any set of integers with density $\delta$ will contain infinitely many proper arithmetic progressions of length $k$. For general…

Control properties of the Kawahara equation are considered when the equation is posed on an unbounded domain. Precisely, the paper's main results are related to an approximation theorem that ensures the exact (internal) controllability in…

In this paper, we establish a logarithmic vanishing theorem on weakly pseudoconvex K\"ahler manifolds, where the divisor may have infinitely many irreducible components. This result serves as a generalization of Norimatsu's findings on…

The main goal of this note is to prove the following theorem. If $A_n$ is a sequence of measurable sets in a $\sigma$-finite measure space $(X, \mathcal{A}, \mu)$ that covers $\mu$-a.e. $x \in X$ infinitely many times, then there exists a…

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n,$ where the coefficients $a_j,$ $j \in \{0,1,2,\cdots n\},$ are real numbers. We impose some restriction on the coefficients and then prove some extensions and…

For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional…

In this short note we prove the Iitaka C_nm conjecture for algebraic fiber spaces over surfaces

We use the canonical bundle formula for parabolic fibrations to give an inductive approach to the generalized abundance conjecture using nef reduction. In particular, we observe that generalized abundance holds for a klt pair $(X,B)$ if the…

We consider an extended variant of the classical coupon collector's problem with infinite number of collections. An arriving coupon is placed in the $r^{th}$ collection, $r\ge0$, if $r$ is the smallest index such that the corresponding…

We prove the discrete analogue of Kakeya conjecture over $\mathbb{R}^n$. This result suggests that a (hypothetically) low dimensional Kakeya set cannot be constructed directly from discrete configurations. We also prove a generalization…

Optimal upper bounds are provided for the dominant dimensions of Nakayama algebras and more generally algebras $A$ with an idempotent $e$ such that there is a minimal faithful injective-projective module $eA$ and such that $eAe$ is a…

Let $A$ be a Nakayama algebra with $n$ simple modules and a simple module $S$ of even projective dimension $m$. Choose $m$ minimal such that a simple $A$-module with projective dimension $2m$ exists, then we show that the global dimension…

We will prove a Kodaira-Nakano type of vanishing theorem for the logarithmic de Rham complex of unitary local system. We will then study the weight filtration on the logarithmic de Rham complex, and prove a stronger statement for the…

On smooth projective variety, for a reduced effective divisor which is weakly ample in the sense of cohomology, we introduce a Kadaira--Saito vanishing theorem for it.

An analogue of the theory of integral closure and reductions is developed for a more general class of closures, called Nakayama closures. It is shown that tight closure is a Nakayama closure by proving a ``Nakayama lemma for tight…

We generalize Coincidence theorem due to Walsh for symmetric and linear polynomial in n complex variables, that is linear in each of them having total degre n. We discuss case when total degree is smaller then n. This case has been already…