scholarly journals LOGARITHMIC DE RHAM COMPARISON FOR OPEN RIGID SPACES

2019 ◽  
Vol 7 ◽  
Author(s):  
SHIZHANG LI ◽  
XUANYU PAN
Keyword(s):  
Comparison Theorem ◽  
Local Systems ◽  
Analytic Variety ◽  
Normal Crossing ◽  
Analytic Varieties ◽  
De Rham ◽  
Simple Normal Crossing ◽  
Ramified Coverings

In this note, we prove the logarithmic $p$ -adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally $K(\unicode[STIX]{x1D70B},1)$ (in a certain sense) with respect to $\mathbb{F}_{p}$ -local systems and ramified coverings along the divisor. We follow Scholze’s method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After introducing period sheaves in our setting, we prove aforesaid comparison theorem.

Green's Forms and Meromorphic Functions on Compact Analytic Varieties

1951 ◽  
Vol 3 ◽  
pp. 108-128 ◽  
Author(s):  
Kunihiko Kodaira
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Zero Point ◽  
Positive Definite ◽  
Analytic Surface ◽  
Analytic Variety ◽  
Complex Dimension ◽  
Analytic Varieties ◽  
Complex Analytic Variety ◽  
Zero Points

Let be a compact complex analytic variety of the complex dimension n with a positive definite Kâhlerian metric [4] ; the local analytic coordinates on will be denoted by z = (z 1 z 2, … , zn). Now, suppose a meromorphic function f(z) defined on as given. Then the poles and zero-points of f(z) constitute an analytic surface in consisting of a finite number of irreducible closed analytic surfaces Γ1, Γ2, … , Γk, each of which is a polar or a zero-point variety of f(z).

scholarly journals Asymptotically conical Calabi-Yau metrics with cone singularities along a compact divisor

2019 ◽  
Author(s):  
Martin de Borbon ◽  
Cristiano Spotti
Keyword(s):  
Finite Subgroup ◽  
Minimal Resolution ◽  
Gravitational Instantons ◽  
Normal Crossing ◽  
Three Sphere ◽  
Asymptotically Locally Euclidean ◽  
Quotient Singularities ◽  
Simple Normal Crossing

Abstract We construct Asymptotically Locally Euclidean (ALE) and, more generally, asymptotically conical Calabi–Yau metrics with cone singularities along a compact simple normal crossing divisor. In particular, this includes the case of the minimal resolution of 2D quotient singularities for any finite subgroup $\Gamma \subset U(2)$ acting freely on the three-sphere, hence generalizing Kronheimer’s construction of smooth ALE gravitational instantons.

Sheets of Real Analytic Varieties

1960 ◽  
Vol 12 ◽  
pp. 51-67
Author(s):  
Andrew H. Wallace
Keyword(s):  
General Idea ◽  
Algebraic Varieties ◽  
Analytic Variety ◽  
Analytic Case ◽  
Real Algebraic Varieties ◽  
Analytic Varieties ◽  
Local Properties ◽  
Real Analytic ◽  
Real Analytic Variety

In a previous paper (4) the author worked out some results on the analytic connectivity properties of real algebraic varieties, that is to say, properties associated with the joining of points of the variety by analytic arcs lying on the variety. It is natural to ask whether these properties can be carried over to analytic varieties, since the proofs in the algebraic case depend mainly on local properties. But although this generalization can be carried out to a large extent, there are, nevertheless, difficulties in the analytic case, owing mainly to the fact (cf. 2, § 11) that a real analytic variety may not be definable by means of a set of global equations. Thus, although the general idea of the treatment given here is the same as in (4), some variation in the details of the method has proved to be necessary, and some of the final results are slightly weaker in form.

Higher Order Tangents to Analytic Varieties along Curves. II

2008 ◽  
Vol 60 (1) ◽  
pp. 33-63 ◽  
Author(s):  
Rüdiger W. Braun ◽  
Reinhold Meise ◽  
B. A. Taylor
Keyword(s):  
Finite Number ◽  
Algebraic Variety ◽  
Higher Order ◽  
Algebraic Varieties ◽  
Analytic Variety ◽  
Open Set ◽  
Local Case ◽  
Analytic Varieties ◽  
Real Analytic ◽  
Analytic Curves

AbstractLet V be an analytic variety in some open set in ℂn. For a real analytic curve γ with γ(0) = 0 and d ≥ 1, define Vt = t−d(V − γ(t)). It was shown in a previous paper that the currents of integration over Vt converge to a limit current whose support Tγ,δV is an algebraic variety as t tends to zero. Here, it is shown that the canonical defining function of the limit current is the suitably normalized limit of the canonical defining functions of the Vt. As a corollary, it is shown that Tγ,δV is either inhomogeneous or coincides with Tγ,δV for all δ in some neighborhood of d. As another application it is shown that for surfaces only a finite number of curves lead to limit varieties that are interesting for the investigation of Phragmén-Lindelöf conditions. Corresponding results for limit varieties Tσ,δW of algebraic varieties W along real analytic curves tending to infinity are derived by a reduction to the local case.

scholarly journals -ADIC HODGE THEORY FOR RIGID-ANALYTIC VARIETIES

2013 ◽  
Vol 1 ◽  
Author(s):  
PETER SCHOLZE
Keyword(s):  
Hodge Theory ◽  
Analytic Varieties ◽  
De Rham ◽  
Perfectoid Spaces

AbstractWe give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is the pro-étale site, which makes all constructions completely functorial.

scholarly journals Positivity properties of the bundle of logarithmic tensors on compact Kähler manifolds

2016 ◽  
Vol 152 (11) ◽  
pp. 2350-2370 ◽  
Author(s):  
Frédéric Campana ◽  
Mihai Păun
Keyword(s):  
Positive Integer ◽  
Cotangent Bundle ◽  
Short Note ◽  
Intersection Curve ◽  
Normal Crossing ◽  
Injective Morphism ◽  
Image Position ◽  
Compact Kähler Manifold ◽  
Simple Normal Crossing ◽  
Formula Type

Let $X$ be a compact Kähler manifold, endowed with an effective reduced divisor $B=\sum Y_{k}$ having simple normal crossing support. We consider a closed form of $(1,1)$-type $\unicode[STIX]{x1D6FC}$ on $X$ whose corresponding class $\{\unicode[STIX]{x1D6FC}\}$ is nef, such that the class $c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\}\in H^{1,1}(X,\mathbb{R})$ is pseudo-effective. A particular case of the first result we establish in this short note states the following. Let $m$ be a positive integer, and let $L$ be a line bundle on $X$, such that there exists a generically injective morphism $L\rightarrow \bigotimes ^{m}T_{X}^{\star }\langle B\rangle$, where we denote by $T_{X}^{\star }\langle B\rangle$ the logarithmic cotangent bundle associated to the pair $(X,B)$. Then for any Kähler class $\{\unicode[STIX]{x1D714}\}$ on $X$, we have the inequality $$\begin{eqnarray}\displaystyle \int _{X}c_{1}(L)\wedge \{\unicode[STIX]{x1D714}\}^{n-1}\leqslant m\int _{X}(c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\})\wedge \{\unicode[STIX]{x1D714}\}^{n-1}.\end{eqnarray}$$ If $X$ is projective, then this result gives a generalization of a criterion due to Y. Miyaoka, concerning the generic semi-positivity: under the hypothesis above, let $Q$ be the quotient of $\bigotimes ^{m}T_{X}^{\star }\langle B\rangle$ by $L$. Then its degree on a generic complete intersection curve $C\subset X$ is bounded from below by $$\begin{eqnarray}\displaystyle \biggl(\frac{n^{m}-1}{n-1}-m\biggr)\int _{C}(c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\})-\frac{n^{m}-1}{n-1}\int _{C}\unicode[STIX]{x1D6FC}.\end{eqnarray}$$ As a consequence, we obtain a new proof of one of the main results of our previous work [F. Campana and M. Păun, Orbifold generic semi-positivity: an application to families of canonically polarized manifolds, Ann. Inst. Fourier (Grenoble) 65 (2015), 835–861].

scholarly journals Simple normal crossing Fano varieties and log Fano manifolds

2014 ◽  
Vol 214 ◽  
pp. 95-123
Author(s):  
Kento Fujita
Keyword(s):  
Fano Varieties ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Normal Crossing ◽  
Fano Index ◽  
Simple Normal Crossing

AbstractA projective log variety (X, D) is called alog Fano manifoldifXis smooth and ifDis a reduced simple normal crossing divisor onΧwith − (KΧ+D) ample. Then-dimensional log Fano manifolds (X, D) with nonzeroDare classified in this article when the log Fano indexrof (X, D) satisfies eitherr≥n/2withρ(X) ≥ 2 orr≥n− 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.

Sign in / Sign up

Export Citation Format

Share Document