scholarly journals The Kodaira dimension and singularities of moduli of stable sheaves on some elliptic surfaces

2020 ◽  
pp. 2050125
Author(s):  
Kimiko Yamada
Keyword(s):  
Kodaira Dimension ◽  
The Other ◽  
Generic Fiber ◽  
Elliptic Surfaces ◽  
Canonical Singularity ◽  
Dimension Formula ◽  
Rank One ◽  
Canonical Singularities ◽  
Stable Sheaves ◽  
Multiple Fibers

Let [Formula: see text] be an elliptic surface over [Formula: see text] with [Formula: see text], and [Formula: see text] be the moduli scheme of rank-two stable sheaves [Formula: see text] on [Formula: see text] with [Formula: see text] in [Formula: see text]. We look into defining equations of [Formula: see text] at its singularity [Formula: see text], partly because if [Formula: see text] admits only canonical singularities, then the Kodaira dimension [Formula: see text] can be calculated. We show the following: (A) [Formula: see text] is at worst canonical singularity of [Formula: see text] if the restriction of [Formula: see text] to the generic fiber of [Formula: see text] has no rank-one subsheaf, and if the number of multiple fibers of [Formula: see text] is a few. (B) We obtain that [Formula: see text] and the Iitaka program of [Formula: see text] can be described in purely moduli-theoretic way for [Formula: see text], when [Formula: see text], [Formula: see text] has just two multiple fibers, and one of its multiplicities equals [Formula: see text]. (C) On the other hand, when [Formula: see text] has a rank-one subsheaf, it may be insufficient to look at only the degree-two part of defining equations to judge whether [Formula: see text] is at worst canonical singularity or not.

scholarly journals Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Junyan Cao ◽  
Henri Guenancia ◽  
Mihai Păun
Keyword(s):  
General Type ◽  
Einstein Metrics ◽  
Kodaira Dimension ◽  
Einstein Metric ◽  
Canonical Bundle ◽  
Fiber Space ◽  
Generic Fiber ◽  
Lelong Numbers ◽  
Kähler Einstein Metrics

Abstract Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric induces a semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p. We also propose a conjectural generalization of this result for relative twisted Kähler–Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).

scholarly journals Towards a classification of rigid product quotient varieties of Kodaira dimension 0

2021 ◽  
Author(s):  
Ingrid Bauer ◽  
Christian Gleissner
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Structure Theorem ◽  
Complete Classification ◽  
Kodaira Dimension ◽  
The Other ◽  
Diagonal Action ◽  
Quotient Varieties ◽  
A Finite Group

AbstractIn this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group G. It is shown that only for $$G = {{\,\mathrm{He}\,}}(3), {\mathbb {Z}}_3^2$$ G = He ( 3 ) , Z 3 2 , and only for dimension $$\ge 4$$ ≥ 4 such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension 4 is given. For the other finite groups a strong structure theorem for rigid quotients is proven.

Moon Lake’s Orphans and “The Other Way to Live”

2019 ◽  
pp. 79-102
Author(s):  
Jean C. Griffith
Keyword(s):  
Twentieth Century ◽  
Child Welfare ◽  
Welfare Policy ◽  
The Other ◽  
American Democracy ◽  
Child Welfare Policy ◽  
Early Twentieth Century ◽  
Rank One ◽  
Poor Children ◽  
Moon Lake

This essay examines the roles the character Easter in “Moon Lake” plays in the context of early-twentieth-century debates about the roots of poverty and society’s level of responsibility to poor children. By placing the focus of the story not on Easter but on the genteel Morgana girls’ shifting attitudes about her, Welty illustrates the ways child welfare policy was shaped by conflicting attitudes, whereby sympathy for innocent children coexisted with scorn for their parents. Assuming that Easter lives outside the boundaries that mark their own places in Morgana’s gendered, class-bound, and racially-segregated society, Jinny Love Stark and Nina Carmichael imagine the “orphan” to embody a womanhood untethered by race or rank, one, perhaps, more representative of American democracy. Ultimately, though, the girls come to see that Easter’s status as an orphan makes her more marked by and vulnerable to the violence and oppression that shape the South’s racial patriarchy.

scholarly journals On some dimension formula for automorphic forms of weight one I

1982 ◽  
Vol 85 ◽  
pp. 213-221 ◽  
Author(s):  
Toyokazu Hiramatsu
Keyword(s):  
Fuchsian Group ◽  
Automorphic Forms ◽  
Galois Representations ◽  
The Other ◽  
Dimension Formula ◽  
Linearly Independent ◽  
Other Hand ◽  
Algebraic Properties ◽  
Lecture Notes ◽  
Weight One

Let Γ be a fuchsian group of the first kind not containing the element . We shall denote by d0 the number of linearly independent automorphic forms of weight 1 for Γ. It would be interesting to have a certain formula for d0. But, Hejhal said in his Lecture Notes 548, it is impossible to calculate d0 using only the basic algebraic properties of Γ. On the other hand, Serre has given such a formula of d0 recently in a paper delivered at the Durham symposium ([7]). His formula is closely connected with 2-dimensional Galois representations.

The Savoyard Cousins: A Comparison of the Careers and Relative Success of the Grandson (Grandison) and Champvent (Chavent) Families in England

2006 ◽  
Vol 86 ◽  
pp. 148-178
Author(s):  
Michael Ray
Keyword(s):  
Thirteenth Century ◽  
The Other ◽  
Royal Patronage ◽  
Relative Success ◽  
Rank One ◽  
Family Based

In the mid-thirteenth century members of two branches of a family based in Savoy came to England and, through royal service, they reached baronial rank. One family, the Grandsons, thoroughly embedded itself in England and its members are recalled even today while the other, the Champvents, lapsed into obscurity, the name disappearing from the records after 1410. To discover why, this article looks at the significance of royal service to the families, the amount of royal patronage they received, their marriage strategies, how they related to the localities into which they were implanted, the extent to which religious loyalties and family piety illustrated their attitudes and whether they cut their ties with their former home lands.

scholarly journals On the -purity of isolated log canonical singularities

2013 ◽  
Vol 149 (9) ◽  
pp. 1495-1510 ◽  
Author(s):  
Osamu Fujino ◽  
Shunsuke Takagi
Keyword(s):  
Characteristic Zero ◽  
Three Dimensional ◽  
Canonical Singularity ◽  
Pure Type ◽  
Log Canonical ◽  
Canonical Singularities ◽  
Log Canonical Singularity

AbstractA singularity in characteristic zero is said to be of dense $F$-pure type if its modulo $p$ reduction is locally Frobenius split for infinitely many $p$. We prove that if $x\in X$ is an isolated log canonical singularity with $\mu (x\in X)\leq 2$ (where the invariant $\mu $ is as defined in Definition 1.4), then it is of dense $F$-pure type. As a corollary, we prove the equivalence of log canonicity and being of dense $F$-pure type in the case of three-dimensional isolated $ \mathbb{Q} $-Gorenstein normal singularities.

scholarly journals RANK ONE CONNECTIONS ON ABELIAN VARIETIES, II

2012 ◽  
Vol 23 (12) ◽  
pp. 1250125
Author(s):  
INDRANIL BISWAS ◽  
JACQUES HURTUBISE ◽  
A. K. RAINA
Keyword(s):  
Line Bundle ◽  
Exact Sequence ◽  
Abelian Varieties ◽  
Explicit Construction ◽  
The Other ◽  
Holomorphic Line Bundle ◽  
Canonical Isomorphism ◽  
Complex Torus ◽  
Rank One

Given a holomorphic line bundle L on a compact complex torus A, there are two naturally associated holomorphic ΩA-torsors over A: one is constructed from the Atiyah exact sequence for L, and the other is constructed using the line bundle [Formula: see text], where α is the addition map on A × A, and p1 is the projection of A × A to the first factor. In [I. Biswas, J. Hurtvbise and A. K. Raina, Rank one connections on abelian varieties, Internat. J. Math.22 (2011) 1529–1543], it was shown that these two torsors are isomorphic. The aim here is to produce a canonical isomorphism between them through an explicit construction.

scholarly journals Osserman pseudo-Riemannian manifolds of signature (2,2)

2001 ◽  
Vol 71 (3) ◽  
pp. 367-396 ◽  
Author(s):  
Novica Blažić ◽  
Neda Bokan ◽  
Zoran Rakić
Keyword(s):  
Riemannian Manifold ◽  
Characteristic Polynomial ◽  
Riemannian Manifolds ◽  
Jacobi Operator ◽  
Tangent Vector ◽  
Jordan Form ◽  
The Other ◽  
Jacobi Operators ◽  
Osserman Manifold ◽  
Rank One

AbstractA pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.

Chern-Yamabe problem and Chern-Yamabe soliton

2021 ◽  
Vol 32 (03) ◽  
pp. 2150016
Author(s):  
Pak Tung Ho ◽  
Jinwoo Shin
Keyword(s):  
Scalar Curvature ◽  
Complex Manifold ◽  
The Other ◽  
Compact Complex Manifold ◽  
Yamabe Problem ◽  
Conformal Metric ◽  
Hermitian Metric ◽  
Dimension Formula ◽  
Complex Dimension ◽  
Yamabe Soliton

Let [Formula: see text] be a compact complex manifold of complex dimension [Formula: see text] endowed with a Hermitian metric [Formula: see text]. The Chern-Yamabe problem is to find a conformal metric of [Formula: see text] such that its Chern scalar curvature is constant. In this paper, we prove that the solution to the Chern-Yamabe problem is unique under some conditions. On the other hand, we obtain some results related to the Chern-Yamabe soliton.

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