Zariski-decomposition and abundance

By Noboru Nakayama. 2004

588 citations recorded by the source corpus.

Subfields: Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d47ee4046ed05fac7dbebe7a50d84fcc501f886e

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In Statistical and Nonlinear Physics

Citations in the corpus

Anti-pluricanonical systems on Fano varieties (2016, background): “When D is pseudo-effective, a kind of Zariski decomposition D = Pσ(D) + Nσ(D) is defined in [38] where Pσ(D) is pseudo-effective and Nσ(D) ≥ 0.”
Fundamental theorems for semi log canonical pairs (2012, background): “The basic references of big R-divisors are [L, 2.2] and [N, II. §3 and §5]. Since we have to consider big R-divisors on non-normal varieties, we give supplementary deﬁnitions and arguments to [L] and [N]. First, let us quickly recall the deﬁnition of big Cartier divisors on normal complete varieties”
A TRANSCENDENTAL APPROACH TO KOLL AR'S INJECTIVITY THEOREM II (2007, background): “For related topics, see [25, Section 3], [26, II, Section 5.c, V, Section 3], and [30].”
Numerical character of the effectivity of adjoint line bundles (2010, unknown): “However, the methods of [14] do not seem to apply in the more general context of the previous theorem.”
F-singularities of pairs and Inversion of Adjunction of arbitrary codimension (2003, background): “Nakayama [22] introduced the notion of admissible pairs as an analog of klt pairs in the absence of Q-Gorensteinness.”
Existence of log canonical closures (2011, background): “[Nakayama04] N. Nakayama, Zariski-decomposition and abundance.”
Foliations with positive slopes and birational stability of orbifold cotangent bundles (2015, background): “us argument, without using [3]. 7.1. Numerical dimension. Let Xbe smooth and projective, and let L be a line-bundle (or Q-line-bundle) on X. Let Abe any suﬃciently ample line bundle on X. Recall from [37]: Deﬁnition 7.1. The numerical dimension ν(X,L) of L is deﬁned by: ν(X,L) := max{k∈ Z|lim sup p→”
Log pluricanonical representations and abundance conjecture (2011, unknown): “ 2.8). Thus, by Theorem 4.7, we obtain the desired result. The ﬁnal result is on a generalized abundance conjecture formulated by Nakayama’s numerical Kodaira dimension κσ. For the details of κσ, see [N] (see also [L]). Corollary 5.6 (Generalized abundance conjecture). Assume that the abundance conj”
Building blocks of polarized endomorphisms of normal projective varieties (2016, background): “[29]) Let X be a normal projective variety and D an R-Cartier divisor.”
Extension theorems, non-vanishing and the existence of good minimal models (2010, background): “If κσ(KX+∆) = 0, then by [Nakayama04, V.1.11], we have that KX +∆ is numerically equivalent to Nσ(KX + ∆).”
Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces (2009, background): “ system We propose deﬁnitions of numerical linear systems and numerical Zariski decompositions. They are easier to deal with than R-linear systems as in [1] and sectional decompositions as in [4] and [7]. Let f : X →S be a projective morphism from a Q-factorial normal variety, and D an R-Cartier div”
A simply connected surface of general type with pg=0 and K2=2 (2006, methodology): “faces. Let π : X → ∆ be a Q-Gorenstein smoothing of X. Since the Q-Cartier divisor KX/∆ is π-big over ∆ and π-nef at the point 0, the nefness of KXt is also obtained by shrinking ∆ if it is necessary [15]. Therefore we have Proposition 5.1. Xt is a minimal surface of general type with pg = 0 and K2 ”
Codimension 1 foliations with numerically trivial canonical class on singular spaces (2018, background): “arities. The numerical dimension of Dis deﬁned as ν(D) := ν(β∗D). Then ν(D) is independent of the resolution, and it depends only on the numerical class of Dby [Nak04, Proposition V.2.7]. We refer to [Nak04] for more detailed properties. Remark 2.7. Recall that a Weil Q-divisor Don a normal projecti”
The Minimal Model Program revisited (2012, methodology): “oincide. Nevertheless, this decomposition turned out to be very useful in the recent advances of the Minimal Model Program, e.g. see [BCHM10]. The following result summarises its main properties, see [Nak04]. Lemma 2.4. Let Xbe a smooth projective variety, and let Dbe a pseudo-eﬀective R-divisor. Th”
Differentiability of volumes of divisors and a problem of Teissier (2006, background): “ns of Zariski decompositions that have been proposed in higher dimension (and which all coincide for big classes) is given in [P], and a very complete account on the existence problem can be found in [N]. For a psef class α ∈CN1(X), set P(α) := hαi∈N1(X), so that P(α) ≤α. By deﬁnition, any nef Carti”
A geometric characterization of toric varieties (2016, methodology): “D′)|D′ ∼R D,D′ ≥ 0}. A GEOMETRIC CHARACTERISATION OF TORIC VARIETIES 11 More generally if D is simply pseudo-eﬀective we extend the deﬁnition of σC as follows. Let A be any ample Q-divisor. Following [19], let σC(D) = lim ǫ→0 σC(D +ǫA). Then σC(D)exists andis independent ofthechoice ofA. There areon”
Zariski decompositions of numerical cycle classes (2013, methodology): “ture 5.19 would oer geometric structure to the negative parts of decompositions. Remark 1.7. Several other possible denitions of Zariski decompositions for cycles have been proposed. In particular, [Nak04] explains how the ˙-decomposition for divisors can be generalized to cycles using similar argum”