scholarly journals K3 surfaces with non-symplectic involution and compact irreducible G2-manifolds

2011 ◽  
Vol 151 (2) ◽  
pp. 193-218 ◽  
Author(s):  
ALEXEI KOVALEV ◽  
NAM-HOON LEE
Keyword(s):  
Betti Numbers ◽  
Matching Condition ◽  
K3 Surfaces ◽  
Connected Sum ◽  
Fano Threefolds ◽  
New Class ◽  
Blowing Up ◽  
G2 Manifolds

AbstractWe consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G2 developed by the first named author. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors and the latter K3 surfaces should satisfy a certain ‘matching condition’ intertwining on their periods and Kähler classes. Suitable examples of threefolds were previously obtained by blowing up curves in Fano threefolds.In this paper, we give a large new class of suitable algebraic threefolds using theory of K3 surfaces with non-symplectic involution due to Nikulin. These threefolds are not obtainable from Fano threefolds as above, and admit matching pairs leading to topologically new examples of compact irreducible G2-manifolds. ‘Geography’ of the values of Betti numbers b2, b3 for the new (and previously known) examples of irreducible G2 manifolds is also discussed.

scholarly journals Bubble bag end: a bubbly resolution of curvature singularity

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Ibrahima Bah ◽  
Pierre Heidmann
Keyword(s):  
Black Holes ◽  
Minimal Surface ◽  
Bound States ◽  
Curvature Singularity ◽  
New Class ◽  
Blowing Up ◽  
Geometric Transitions ◽  
A Chain

Abstract We construct a family of smooth charged bubbling solitons in $$ \mathbbm{M} $$ M 4×T2, four-dimensional Minkowski with a two-torus. The solitons are characterized by a degeneration pattern of the torus along a line in $$ \mathbbm{M} $$ M 4 defining a chain of topological cycles. They live in the same parameter regime as non-BPS non-extremal four-dimensional black holes, and are ultracompact with sizes ranging from miscroscopic to macroscopic scales. The six-dimensional framework can be embedded in type IIB supergravity where the solitons are identified with geometric transitions of non-BPS D1-D5-KKm bound states. Interestingly, the geometries admit a minimal surface that smoothly opens up to a bubbly end of space. Away from the solitons, the solutions are indistinguishable from a new class of singular geometries. By taking a limit of large number of bubbles, the soliton geometries can be matched arbitrarily close to the singular spacetimes. This provides the first classical resolution of a curvature singularity beyond the framework of supersymmetry and supergravity by blowing up topological cycles wrapped by fluxes at the vicinity of the singularity.

scholarly journals Projective dimension and regularity of path ideals of cycles

2018 ◽  
Vol 17 (10) ◽  
pp. 1850188 ◽  
Author(s):  
Guangjun Zhu
Keyword(s):  
Betti Numbers ◽  
Projective Dimension ◽  
New Class ◽  
Graded Betti Numbers

In this paper, we study the generalized path ideals, which is a new class of path ideals of cycle graphs. These ideals naturally generalize the standard path ideals of cycles, as studied by Alilooee and Faridi [On the resolution of path ideals of cycles, Comm. Algebra 43 (2015) 5413–5433]. We give some formulas to compute all the top degree graded Betti numbers of these path ideals of cycle graphs. As a consequence, we can give some formulas to compute their projective dimension and regularity.

scholarly journals Projective degenerations of K3 surfaces, Gaussian maps, and Fano threefolds

1993 ◽  
Vol 114 (1) ◽  
pp. 641-667 ◽  
Author(s):  
Ciro Ciliberto ◽  
Angelo Lopez ◽  
Rick Miranda
Keyword(s):  
K3 Surfaces ◽  
Fano Threefolds ◽  
Gaussian Maps

scholarly journals On the existence of certain weak Fano threefolds of Picard number two

2017 ◽  
Vol 120 (1) ◽  
pp. 68 ◽  
Author(s):  
Maxim Arap ◽  
Joseph Cutrone ◽  
Nicholas Marshburn
Keyword(s):  
Picard Number ◽  
Fano Threefolds ◽  
Canonical Map ◽  
Blowing Up ◽  
Numerical Invariants ◽  
Weak Fano

This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds of Picard number $1$ with the exception of $12$ numerical cases.

scholarly journals G2-manifolds from K3 surfaces with non-symplectic automorphisms

2012 ◽  
Vol 62 (11) ◽  
pp. 2214-2226 ◽  
Author(s):  
Max Pumperla ◽  
Frank Reidegeld
Keyword(s):  
K3 Surfaces ◽  
G2 Manifolds ◽  
Symplectic Automorphisms

scholarly journals Threefolds fibred by mirror sextic double planes

2020 ◽  
pp. 1-42
Author(s):  
Remkes Kooistra ◽  
Alan Thompson
Keyword(s):  
Systematic Study ◽  
Betti Numbers ◽  
K3 Surfaces ◽  
Canonical Divisor ◽  
Geometric Properties ◽  
Elliptic Surfaces ◽  
Birational Maps ◽  
Minimal Form ◽  
Birational Model ◽  
Double Planes

Abstract We present a systematic study of threefolds fibred by K3 surfaces that are mirror to sextic double planes. There are many parallels between this theory and the theory of elliptic surfaces. We show that the geometry of such threefolds is controlled by a pair of invariants, called the generalized functional and generalized homological invariants, and we derive an explicit birational model for them, which we call the Weierstrass form. We then describe how to resolve the singularities of the Weierstrass form to obtain the “minimal form”, which has mild singularities and is unique up to birational maps in codimension 2. Finally, we describe some of the geometric properties of threefolds in minimal form, including their singular fibres, canonical divisor, and Betti numbers.

THE BETTI NUMBERS OF THE FREE LOOP SPACE OF A CONNECTED SUM

2001 ◽  
Vol 64 (1) ◽  
pp. 205-228 ◽  
Author(s):  
PASCAL LAMBRECHTS
Keyword(s):  
Exponential Growth ◽  
Loop Space ◽  
Betti Numbers ◽  
Riemannian Metric ◽  
Closed Geodesics ◽  
Free Loop Space ◽  
Connected Sum ◽  
Simply Connected

Let M1 and M2 be two simply connected closed manifolds of the same dimension. It is proved that(1) if k is a coefficient field such that neither M1 nor M2 has the same cohomology as a sphere, then the sequence (bk)k[ges ]1 of Betti numbers of the free loop space on M1 #M2 is unbounded;(2) if, moreover, the cohomology H*(M1;k) is not generated as algebra by only one element, then the sequence (bk)k[ges ]1 has an exponential growth.Thanks to theorems of Gromoll and Meyer and of Gromov, this implies, in case 1, that there exist infinitely many closed geodesics on M1#M2 for each Riemannian metric, and, in case 2, that for a generic metric, the number of closed geodesics of length [les ]t grows exponentially with t.

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