scholarly journals On numerically effective log canonical divisors

2002 ◽  
Vol 30 (9) ◽  
pp. 521-531 ◽  
Author(s):  
Shigetaka Fukuda

Let(X,Δ)be a4-dimensional log variety which is proper over the field of complex numbers and with only divisorial log terminal singularities. The log canonical divisorKX+Δis semiample, if it is numerically effective (NEF) and the Iitaka dimensionκ(X,KX+Δ)is strictly positive. For the proof, we use Fujino's abundance theorem for semi-log canonical threefolds.

Author(s):  
SALVATORE CACCIOLA

AbstractWe study graded rings associated to big divisors on LC pairs whose difference with the log-canonical divisor is nef. For divisors that are positive enough at the LC centers of the pair, we prove the finite generation of such rings if the pair is DLT or the dimension is low, given that a Zariski decomposition exists.


2020 ◽  
Vol 156 (8) ◽  
pp. 1517-1559 ◽  
Author(s):  
Junho Peter Whang

AbstractWe show that every coarse moduli space, parametrizing complex special linear rank-2 local systems with fixed boundary traces on a surface with nonempty boundary, is log Calabi–Yau in that it has a normal projective compactification with trivial log canonical divisor. We connect this to a novel symmetry of generating series for counts of essential multicurves on the surface.


2020 ◽  
Vol 2020 (767) ◽  
pp. 109-159
Author(s):  
Kenta Hashizume ◽  
Zheng-Yu Hu

AbstractUnder the assumption of the minimal model theory for projective klt pairs of dimension n, we establish the minimal model theory for lc pairs {(X/Z,\Delta)} such that the log canonical divisor is relatively log abundant and its restriction to any lc center has relative numerical dimension at most n. We also give another detailed proof of results by the second author, and study termination of log MMP with scaling.


2014 ◽  
Vol 150 (4) ◽  
pp. 593-620 ◽  
Author(s):  
Osamu Fujino ◽  
Yoshinori Gongyo

AbstractWe prove the finiteness of log pluricanonical representations for projective log canonical pairs with semi-ample log canonical divisor. As a corollary, we obtain that the log canonical divisor of a projective semi log canonical pair is semi-ample if and only if the log canonical divisor of its normalization is semi-ample. We also treat many other applications.


2006 ◽  
Author(s):  
Stephen C. Roy
Keyword(s):  

2021 ◽  
Vol 148 ◽  
pp. 111044
Author(s):  
Can Kızılateş ◽  
Tiekoro Kone
Keyword(s):  

2021 ◽  
Vol 20 (5) ◽  
Author(s):  
Paweł J. Szabłowski

AbstractWe analyze the mathematical structure of the classical Grover’s algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one ‘chosen’ element (sometimes called a ‘solution’) of the dataset, but a set of m such ‘chosen’ elements (out of $$n>m)$$ n > m ) . Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that ‘marks,’ by a suitable phase change $$\varphi $$ φ , all these ‘chosen’ elements. In the first part of the paper, we construct a unique unitary operator that selects all ‘chosen’ elements in one step. The constructed operator is uniquely defined by the numbers $$\varphi $$ φ and $$\alpha $$ α which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on $$\alpha $$ α . In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, ‘convenient’ phase change $$\varphi ,$$ φ , and by sequentially applying the so-constructed operator, we find the number of steps to find these ‘chosen’ elements with great probability. We apply this knowledge to study the generalizations of Grover’s algorithm ($$m=1,\phi =\pi $$ m = 1 , ϕ = π ), which are of the form, the found previously, unitary operators.


2019 ◽  
Vol 7 (1) ◽  
pp. 230-245
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.


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