Comparing numerical dimensions

By Brian Lehmann. 2011

72 citations recorded by the source corpus.

The numerical dimension is a numerical measure of the positivity of a pseudo-effective divisor $L$. There are several proposed definitions of the numerical dimension due to Nakayama (2004) and Boucksom et al. (2004). We prove the equality of these notions and give several additional characterizations. We also prove some new properties of the numerical dimension.

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/31ce0d2d97e12614637f21b027bedb2f869f4e1f

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In Algebra and Number Theory

1. Yoshinori Gongyo
2. Mihai Fulger
3. Brian Lehmann
4. Yuri Tschinkel
5. Brendan Hassett
6. D. Huybrechts
7. Franccois Charles

In Applied Mathematics

1. Yoshinori Gongyo
2. Sho Tanimoto
3. Mihai Fulger
4. Brian Lehmann
5. Yuri Tschinkel
6. Brendan Hassett

In Geometry and Topology

1. Brian Lehmann
2. Mihai Fulger
3. Yuri Tschinkel
4. Brendan Hassett
5. Sho Tanimoto
6. Yoshinori Gongyo
7. Giovanni Mongardi
8. Nguyen-Bac Dang

Citations in the corpus

• Log pluricanonical representations and abundance conjecture (2011, unknown): “y 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 conjecture for pro”
• Balanced line bundles and equivariant compactifications of homogeneous spaces (2013, background): “ of Φ, so Φ has the nontrivial kernel. We conclude that b(Y,L|Y ) ≤ codim(V′) <b(X,L). Remark 37. Conditions (1) and (2) can be replaced by the condition: the numerical dimension ν(D) is zero (see [Leh11] for definitions). Corollary 38. Let H⊂ M⊂ Gbe connected linear algebraic groups and X a smoot”
• Zariski decompositions of numerical cycle classes (2013, background): “hat the basepoint free classes constructed in this way are also \almost pullbacks." ZARISKI DECOMPOSITIONS OF NUMERICAL CYCLE CLASSES 37 The proof uses the following statement. Proposition 6.24 ([Leh13a], Proposition 3.7). Let Xbe a smooth projective variety over C and let Lbe a big divisor wit”
• Reduction maps and minimal model theory (2011, result): “on X. Set ν(D,A) = max{k∈ Z≥0|limsup m→∞ m−kdimH0(X,xmDy+A) >0} if H0(X,xmDy+A) 6= 0 for infinitely many m∈ Nor σ(D,A) = −∞ otherwise. Define ν(D) = max Aample ν(D,A). Remark 2.11. By the results of [Leh2], this definition coincides with the notions of κν(D) from [N, V, 2.20, Definition] and ν(D) fro”