The group F2K(X) as a birational invariant of a surface

1972 ◽  
Vol 11 (1) ◽  
pp. 12-15
Author(s):  
L. R. Gorodetskii
Keyword(s):  
Birational Invariant

The smooth case

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Homotopy Type ◽  
Unit Vector ◽  
Smooth Case ◽  
Open Set ◽  
Birational Invariant

This chapter examines the simplifications occurring in the proof of the main theorem in the smooth case. It begins by stating the theorem about the existence of an F-definable homotopy h : I × unit vector X → unit vector X and the properties for h. It then presents the proof, which depends on two lemmas. The first recaps the proof of Theorem 11.1.1, but on a Zariski dense open set V₀ only. The second uses smoothness to enable a stronger form of inflation, serving to move into V₀. The chapter also considers the birational character of the definable homotopy type in Remark 12.2.4 concerning a birational invariant.

scholarly journals Cycles on Algebraic Varieties

1955 ◽  
Vol 9 ◽  
pp. 173-180
Author(s):  
Hisasi Morikawa
Keyword(s):  
Present Note ◽  
Algebraic Varieties ◽  
Birational Invariant

In the present note, applying the theory of harmonic integrals, we shall show some results on cycles on algebraic varieties and give a new birational invariant.

scholarly journals On Analytic Todd Classes of Singular Varieties

2019 ◽  
Author(s):  
Francesco Bei ◽  
Paolo Piazza
Keyword(s):  
Fundamental Group ◽  
Complex Space ◽  
Homology Class ◽  
Classifying Space ◽  
Projective Varieties ◽  
Singular Varieties ◽  
Push Forward ◽  
Birational Invariant ◽  
Complex Projective

Abstract Let $(X,h)$ be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of $(X,h)$. In the 1st part, assuming either $\dim (\operatorname{sing}(X))=0$ or $\dim (X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\overline{\partial }$ complex, denoted here $\overline{\eth }_{\textrm{rel}}$, induces a class in $K_0 (X)\equiv KK_0(C(X),\mathbb{C})$. A similar result, assuming $\dim (\operatorname{sing}(X))=0$, is proved also for $\overline{\eth }_{\textrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\overline{\partial }$ complex. We then show that when $\dim (\operatorname{sing}(X))=0$ we have $[\overline{\eth }_{\textrm{rel}}]=\pi _*[\overline{\eth }_M]$ with $\pi :M\rightarrow X$ an arbitrary resolution and with $[\overline{\eth }_M]\in K_0 (M)$ the analytic K-homology class induced by $\overline{\partial }+\overline{\partial }^t$ on $M$. In the 2nd part of the paper we focus on complex projective varieties $(V,h)$ endowed with the Fubini–Study metric. First, assuming $\dim (V)\leq 2$, we compare the Baum–Fulton–MacPherson K-homology class of $V$ with the class defined analytically through the rolled-up operator of any $L^2$-$\overline{\partial }$ complex. We show that there is no $L^2$-$\overline{\partial }$ complex on $(\operatorname{reg}(V),h)$ whose rolled-up operator induces a K-homology class that equals the Baum–Fulton–MacPherson class. Finally in the last part of the paper we prove that under suitable assumptions on $V$ the push-forward of $[\overline{\eth }_{\textrm{rel}}]$ in the K-homology of the classifying space of the fundamental group of $V$ is a birational invariant.

Algebraic Analysis for Nonidentifiable Learning Machines

2001 ◽  
Vol 13 (4) ◽  
pp. 899-933 ◽  
Author(s):  
Sumio Watanabe
Keyword(s):  
Learning Curve ◽  
Geometrical Structure ◽  
Generalization Error ◽  
True Parameter ◽  
Stochastic Complexity ◽  
Algebraic Analysis ◽  
Learning Machines ◽  
Training Samples ◽  
Learning Machine ◽  
Birational Invariant

This article clarifies the relation between the learning curve and the algebraic geometrical structure of a nonidentifiable learning machine such as a multilayer neural network whose true parameter set is an analytic set with singular points. By using a concept in algebraic analysis, we rigorously prove that the Bayesian stochastic complexity or the free energy is asymptotically equal to λ1 logn − (m1 − 1) loglogn + constant, where n is the number of training samples and λ1 and m1 are the rational number and the natural number, which are determined as the birational invariant values of the singularities in the parameter space. Also we show an algorithm to calculate λ1 and m1 based on the resolution of singularities in algebraic geometry. In regular statistical models, 2λ1 is equal to the number of parameters and m1 = 1, whereas in nonregular models, such as multilayer networks, 2λ1 is not larger than the number of parameters and m1 ≥ 1. Since the increase of the stochastic complexity is equal to the learning curve or the generalization error, the nonidentifiable learning machines are better models than the regular ones if Bayesian ensemble learning is applied.

scholarly journals RATIONALITY PROPERTIES OF MANIFOLDS CONTAINING QUASI-LINES

2003 ◽  
Vol 14 (10) ◽  
pp. 1053-1080 ◽  
Author(s):  
PALTIN IONESCU ◽  
DANIEL NAIE
Keyword(s):  
Open Subset ◽  
Basic Question ◽  
Projective Manifold ◽  
Rational Varieties ◽  
Direct Sum ◽  
Formal Geometry ◽  
Rationally Connected ◽  
Birational Invariant ◽  
Small Deformations

Let X be a complex, rationally connected, projective manifold. We show that X admits a modification [Formula: see text] that contains a quasi-line, i.e. a smooth rational curve whose normal bundle is a direct sum of copies of [Formula: see text]. For manifolds containing quasi-lines, a sufficient condition of rationality is exploited: there is a unique quasi-line from a given family passing through two general points. We define a numerical birational invariant, e(X), and prove that X is rational if and only if e(X)=1. If X is rational, there is a modification [Formula: see text] which is strongly-rational, i.e. contains an open subset isomorphic to an open subset of the projective space whose complement is at least 2-codimensional. We prove that strongly-rational varieties are stable under smooth, small deformations. The argument is based on a convenient characterization of these varieties. Finally, we relate the previous results and formal geometry. This relies on ẽ(X, Y), a numerical invariant of a given quasi-line Y that depends only on the formal completion [Formula: see text]. As applications we show various instances in which X is determined by [Formula: see text]. We also formulate a basic question about the birational invariance of ẽ(X, Y).

scholarly journals Galois cohomology and birational invariant of algebraic varieties

1969 ◽  
Vol 21 (4) ◽  
pp. 584-603 ◽  
Author(s):  
Megumu MIWA
Keyword(s):  
Galois Cohomology ◽  
Algebraic Varieties ◽  
Birational Invariant

scholarly journals On motivic principal value integrals

2007 ◽  
Vol 143 (3) ◽  
pp. 543-555
Author(s):  
WILLEM VEYS
Keyword(s):  
Minimal Model ◽  
Differential Forms ◽  
Model Program ◽  
Natural Question ◽  
Algebraic Varieties ◽  
Minimal Model Program ◽  
Principal Value ◽  
Birational Invariant

AbstractInspired by p-adic (and real) principal value integrals, we introduce motivic principal value integrals associated to multi-valued rational differential forms on smooth algebraic varieties. We investigate the natural question whether (for complete varieties) this notion is a birational invariant. The answer seems to be related to the dichotomy of the Minimal Model Program.

scholarly journals A birational invariant for algebraic group actions

2002 ◽  
Vol 204 (1) ◽  
pp. 223-246 ◽  
Author(s):  
Z. Reichstein ◽  
B. Youssin
Keyword(s):  
Algebraic Group ◽  
Group Actions ◽  
Birational Invariant

scholarly journals $\overline{M}_{1,n}$ is usually not uniruled in characteristic $p$

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Will Sawin
Keyword(s):  
Modular Forms ◽  
Characteristic P ◽  
Etale Cohomology ◽  
Étale Cohomology ◽  
Birational Invariant

Using etale cohomology, we define a birational invariant for varieties in characteristic $p$ that serves as an obstruction to uniruledness - a variant on an obstruction to unirationality due to Ekedahl. We apply this to $\overline{M}_{1,n}$ and show that $\overline{M}_{1,n}$ is not uniruled in characteristic $p$ as long as $n \geq p \geq 11$. To do this, we use Deligne's description of the etale cohomology of $\overline{M}_{1,n}$ and apply the theory of congruences between modular forms. Comment: 10 pages, published version

scholarly journals On stable conjugacy of finite subgroups of the plane Cremona group, I

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Fedor Bogomolov ◽  
Yuri Prokhorov
Keyword(s):  
Prime Order ◽  
Cremona Group ◽  
Cyclic Groups ◽  
Group I ◽  
Finite Subgroups ◽  
Birational Invariant

AbstractWe discuss the problem of stable conjugacy of finite subgroups of Cremona groups. We compute the stable birational invariant H 1(G, Pic(X)) for cyclic groups of prime order.

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