scholarly journals Counting Rational Curves on K3 Surfaces With Finite Group Actions

2021 ◽  
Author(s):  
Sailun Zhan
Keyword(s):  
Finite Group ◽  
Hodge Structure ◽  
Rational Curves ◽  
K3 Surface ◽  
Projective Surface ◽  
Hilbert Schemes ◽  
Finite Group Actions ◽  
Family Of Curves ◽  
Compactified Jacobian ◽  
A Finite Group

Abstract Göttsche gave a formula for the dimension of the cohomology of Hilbert schemes of points on a smooth projective surface $S$. When $S$ admits an action by a finite group $G$, we describe the action of $G$ on the Hodge structure. In the case that $S$ is a K3 surface, each element of $G$ gives a trace on $\sum _{n=0}^{\infty }\sum _{i=0}^{\infty }(-1)^{i}H^{i}(S^{[n]},\mathbb{C})q^{n}$. When $G$ acts faithfully and symplectically on $S$, the resulting generating function is of the form $q/f(q)$, where $f(q)$ is a cusp form. We relate the Hodge structure of Hilbert schemes of points to the Hodge structure of the compactified Jacobian of the tautological family of curves over an integral linear system on a K3 surface as $G$-representations. Finally, we give a sufficient condition for a $G$-orbit of curves with nodal singularities not to contribute to the representation.

scholarly journals The Tracial Class Property for Crossed Products by Finite Group Actions

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Xinbing Yang ◽  
Xiaochun Fang
Keyword(s):  
Finite Group ◽  
Group Actions ◽  
Crossed Products ◽  
Finite Group Actions ◽  
Infinite Dimensional ◽  
A Finite Group ◽  
Tracial Rokhlin Property

We define the concept of tracial𝒞-algebra ofC*-algebras, which generalize the concept of local𝒞-algebra ofC*-algebras given by H. Osaka and N. C. Phillips. Let𝒞be any class of separable unitalC*-algebras. LetAbe an infinite dimensional simple unital tracial𝒞-algebra with the (SP)-property, and letα:G→Aut(A)be an action of a finite groupGonAwhich has the tracial Rokhlin property. ThenA  ×α  Gis a simple unital tracial𝒞-algebra.

Extending finite group actions on surfaces to hyperbolic 3-manifolds

1995 ◽  
Vol 117 (1) ◽  
pp. 137-151 ◽  
Author(s):  
Monique Gradolato ◽  
Bruno Zimmermann
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Finite Groups ◽  
Group Actions ◽  
Geodesic Boundary ◽  
Totally Geodesic ◽  
Finite Group Actions ◽  
Closed Riemann Surface ◽  
A Finite Group ◽  
The Given

Let G be a finite group of orientation preserving isometrics of a closed orientable hyperbolic 2-manifold Fg of genus g > 1 (or equivalently, a finite group of conformal automorphisms of a closed Riemann surface). We say that the G-action on Fgbounds a hyperbolic 3-manifold M if M is a compact orientable hyperbolic 3-manifold with totally geodesic boundary Fg (as the only boundary component) such that the G-action on Fg extends to a G-action on M by isometrics. Symmetrically we will also say that the 3-manifold M bounds the given G-action. We are especially interested in Hurwitz actions, i.e. finite group actions on surfaces of maximal possible order 84(g — 1); the corresponding finite groups are called Hurwitz groups. First examples of bounding and non-bounding Hurwitz actions were given in [16].

scholarly journals Conjugacy of free finite group actions on infranilmanifolds

1990 ◽  
Vol 32 (2) ◽  
pp. 239-240 ◽  
Author(s):  
Michał Sadowski
Keyword(s):  
Finite Group ◽  
Group Actions ◽  
Finite Group Actions ◽  
Affine Action ◽  
A Finite Group ◽  
Free Actions

In this note we give the proof of the following result (previously known for homotopically trivial and free actions on infranilmanifolds [3, Theorem 5.6]).Theorem 1. Let G be a finite group acting freely and smoothly on a closed infranilmanifold M. Assume that dim M≠3, 4. Then the action of G is topologically conjugate to an affine action.

Goldie Dimension and Chain Conditions for Modular Lattices with Finite Group Actions

1986 ◽  
Vol 29 (3) ◽  
pp. 274-280 ◽  
Author(s):  
Piotr Grzeszczuk ◽  
Edmund R. Puczyłowski
Keyword(s):  
Finite Group ◽  
Fixed Points ◽  
Modular Lattice ◽  
Group Actions ◽  
Modular Lattices ◽  
Finite Group Actions ◽  
Chain Conditions ◽  
Goldie Dimension ◽  
A Finite Group

AbstractA relation between Goldie dimensions of a modular lattice L and its sublattice LG of fixed points under a finite group G of automorphisms of L is obtained. The method used also gives a relation between ACC (DCC) for L and for LG. The results obtained are applied to rings and modules.

scholarly journals Finite group actions on 4-manifolds

1999 ◽  
Vol 66 (3) ◽  
pp. 287-296 ◽  
Author(s):  
Yong Seung Cho
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Group Actions ◽  
Witten Invariant ◽  
Finite Group Actions ◽  
Finite Fundamental Group ◽  
A Finite Group

AbstractLet X be a closed, oriented, smooth 4-manifold with a finite fundamental group and with a non-vanishing Seiberg-Witten invariant. Let G be a finite group. If G acts smoothly and freely on X, then the quotient X/G cannot be decomposed as X1#X2 with (Xi) > 0, i = 1, 2. In addition let X be symplectic and c1(X)2 > 0 and b+2(X) > 3. If σ is a free anti-symplectic involution on X then the Seiberg-Witten invariants on X/σ vanish for all spinc structures on X/σ, and if η is a free symplectic involution on X then the quotients X/σ and X/η are not diffeomorphic to each other.

scholarly journals Existentially closed fields with finite group actions

2018 ◽  
Vol 18 (01) ◽  
pp. 1850003 ◽  
Author(s):  
Daniel M. Hoffmann ◽  
Piotr Kowalski
Keyword(s):  
Finite Group ◽  
Model Theory ◽  
Group Scheme ◽  
General Context ◽  
Finite Group Actions ◽  
Existentially Closed ◽  
Closed Fields ◽  
A Finite Group ◽  
Theory Of Fields ◽  
Model Theory Of Fields

We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a (finite) group scheme action.

scholarly journals Algebraic K3 surfaces with finite automorphism groups

1989 ◽  
Vol 116 ◽  
pp. 1-15 ◽  
Author(s):  
Shigeyuki Kondō
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Line Bundle ◽  
Algebraic Surface ◽  
Automorphism Groups ◽  
Factor Group ◽  
K3 Surface ◽  
Intersection Form ◽  
Canonical Line Bundle ◽  
A Finite Group

The purpose of this paper is to give a proof to the result announced in [3]. Let X be an algebraic surface defined over C. X is called a K3 surface if its canonical line bundle Kx is trivial and dim H1(X, ϕX) = 0. It is known that the automorphism group Aut (X) of X is isomorphic, up to a finite group, to the factor group O(Sx)/Wx, where O(Sx) is the automorphism group of the Picard lattice of X (i.e. Sx is the Picard group of X together with the intersection form) and Wx is its subgroup generated by all reflections associated with elements with square (–2) of Sx ([11]). Recently Nikulin [8], [10] has completely classified the Picard lattices of algebraic K3 surfaces with finite automorphism groups.

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

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