scholarly journals Extension of cohomology classes and holomorphic sections defined on subvarieties

2021 ◽  
Vol 31 (1) ◽  
pp. 137-179
Author(s):  
Xiangyu Zhou ◽  
Langfeng Zhu
Keyword(s):  
Extension Theorem ◽  
Positive Answer ◽  
Extension Theorems ◽  
Content Type ◽  
Plurisubharmonic Functions ◽  
Holomorphic Sections ◽  
Ideal Sheaves ◽  
Multiplier Ideal

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general L 2 L^2 extension theorem obtained by Demailly.

Subadditivity of generalized Kodaira dimensions and extension theorems

2020 ◽  
Vol 31 (12) ◽  
pp. 2050098
Author(s):  
Xiangyu Zhou ◽  
Langfeng Zhu
Keyword(s):  
Extension Theorem ◽  
Kodaira Dimension ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Extension Theorems ◽  
Crucial Step ◽  
Ideal Sheaves ◽  
Multiplier Ideal ◽  
Singular Weights ◽  
Optimal Formula

In this paper, we introduce the notion of generalized Kodaira dimension with multiplier ideal sheaves, and prove the subadditivity of these generalized Kodaira dimensions for certain Kähler fibrations, which was originally obtained for Kodaira dimensions of algebraic fiber spaces by Kawamata and Viehweg. Our method is analytic and based on some new results in recent years. The crucial step in our proof is to prove an [Formula: see text] extension theorem for twisted pluricanonical sections on compact Kähler manifolds. Moreover, we also discuss the relation between two previous optimal [Formula: see text] extension theorems with singular weights on weakly pseudoconvex Kähler manifolds.

Restriction formula and subadditivity property related to multiplier ideal sheaves

2020 ◽  
Vol 2020 (769) ◽  
pp. 1-33
Author(s):  
Qi’an Guan ◽  
Xiangyu Zhou
Keyword(s):  
Necessary Conditions ◽  
Extremal Case ◽  
Complex Singularity ◽  
Plurisubharmonic Functions ◽  
Singularity Exponents ◽  
Ideal Sheaves ◽  
Multiplier Ideal

AbstractWe give a restriction formula on jumping numbers which is a reformulation of Demailly–Ein–Lazarsfeld’s important restriction formula for multiplier ideal sheaves and a generalization of Demailly–Kollár’s important restriction formula on complex singularity exponents, and then we establish necessary conditions for the extremal case in the reformulated formula; we pose the subadditivity property on the complex singularity exponents of plurisubharmonic functions which is a generalization of Demailly–Kollár’s fundamental subadditivity property, and then we establish necessary conditions for the extremal case in the generalization. We also obtain two sharp relations on jumping numbers, introduce a new invariant of plurisubharmonic singularities and get its decreasing property for consecutive differences.

scholarly journals Injectivity theorem for pseudo-effective line bundles and its applications

2021 ◽  
Vol 8 (27) ◽  
pp. 849-884
Author(s):  
Osamu Fujino ◽  
Shin-ichi Matsumura
Keyword(s):  
Type Theorem ◽  
Line Bundles ◽  
Vanishing Theorem ◽  
Harmonic Forms ◽  
Hermitian Metrics ◽  
Content Type ◽  
Generic Vanishing ◽  
Torsion Freeness ◽  
Ideal Sheaves ◽  
Multiplier Ideal

We formulate and establish a generalization of Kollár’s injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár’s torsion-freeness, Kollár’s vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge theoretic but analytic, which enables us to treat singular Hermitian metrics with nonalgebraic singularities. For the proof of the main injectivity theorem, we use L 2 L^{2} -harmonic forms on noncompact Kähler manifolds. For applications, we prove a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems.

scholarly journals Elliptic Theory for Sets with Higher Co-dimensional Boundaries

2021 ◽  
Vol 274 (1346) ◽  
Author(s):  
G. David ◽  
J. Feneuil ◽  
S. Mayboroda
Keyword(s):  
Green Function ◽  
Elliptic Operator ◽  
Harmonic Measure ◽  
Lipschitz Constant ◽  
Extension Theorems ◽  
Content Type ◽  
Elliptic Theory ◽  
Basic Set ◽  
Degenerate Elliptic ◽  
The Green Function

Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let Γ ⊂ R n \Gamma \subset \mathbb {R}^n be an Ahlfors regular set of dimension d > n − 1 d>n-1 (not necessarily integer) and Ω = R n ∖ Γ \Omega = \mathbb {R}^n \setminus \Gamma . Let L = − div ⁡ A ∇ L = - \operatorname {div} A\nabla be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix A A are bounded from above and below by a multiple of dist ⁡ ( ⋅ , Γ ) d + 1 − n \operatorname {dist}(\cdot , \Gamma )^{d+1-n} . We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the Hölder continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or L p L^p estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to L L , establish its doubling property, non-degeneracy, change-of-the-pole formulas, and, finally, the comparison principle for local solutions. In another article to appear, we will prove that when Γ \Gamma is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator L L for which the harmonic measure given here is absolutely continuous with respect to the d d -Hausdorff measure on Γ \Gamma and vice versa. It thus extends Dahlberg’s theorem to some sets of codimension higher than 1.

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