Blowing up and desingularizing constant scalar curvature Kähler manifolds

Arezzo, Claudio; Pacard, Frank

doi:10.1007/s11511-006-0004-6

Cited by 111 publications

(265 citation statements)

References 57 publications

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“…The main tool we will use is the following remarkable theorems 3.1.1, 3.1.2 by Arezzo-Pacard [1]. Kronheimer [21] has shown that there exist asymptotically locally Euclidean resolutions of singularities of the type C 2 /Γ, where Γ is a finite subgroup of SU (2).…”

Section: Arezzo-pacard Theorem and Some Resultsmentioning

confidence: 99%

“…To construct an elliptic surface with cscK metrics and positive Euler number in each deformation class, we start with the elliptic surface S, which is obtained by applying logarithmic transform along some fibres on the trivial elliptic bundle. Under certain choices of the base curve and the fibres on which logarithmic transform is applied, there exists an holomorphic involution ι : S → S. Although the quotient of S by the action of involution ι is singular, we can show that its resolution is smooth and carries cscK metrics by an application of Arezzo-Pacard's result [1]. Finally, to show elliptic surfaces with positive Euler number of each deformation class can be constructed in this way, we use the deformation theory of elliptic surfaces [11]: in the case of positive Euler number, the deformation class of an elliptic surface is determined by the diffeomorphism type of the base orbifold and the Euler number χ.…”

Section: Introductionmentioning

confidence: 90%

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Compact complex surfaces and constant scalar curvature Kähler metrics

LeBrun

Shu

2008

Geom Dedicata

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Abstract. In this article, I prove the following statement: Every compact complex surface with even first Betti number is deformation equivalent to one which admits an extremal Kähler metric. In fact, this extremal Kähler metric can even be taken to have constant scalar curvature in all but two cases: the deformation equivalence classes of the blow-up of P 2 at one or two points. The explicit construction of compact complex surfaces with constant scalar curvature Kähler metrics in different deformation equivalence classes is given. The main tool repeatedly applied here is the gluing theorem of C. Arezzo and F. Pacard which states that the blow-up/resolution of a compact manifold/orbifold of discrete type, which admits cscK metrics, still admits cscK metrics.

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Section: Arezzo-pacard Theorem and Some Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 90%

Compact complex surfaces and constant scalar curvature Kähler metrics

LeBrun

Shu

2008

Geom Dedicata

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“…By a result of LeBrun and Simanca [16] the cone E of extremal Kähler classes is open in the Kähler cone, and the locus where the Futaki invariant F 1 vanishes is the set C of all cscK classes. By the results of Arezzo and Pacard [2], [3] there is a non-empty open set of cscK classes under mild conditions. Under such conditions we may be able to show that the locus Z where F 2 = · · · = F m = 0 is a Zariski closed subset in C. Then a rational point in C\Z will be a cscK but asymptotically unstable polarization.…”

Section: It Is Easy To See That Chow(m L) Is the Mumford Weight Of Tmentioning

confidence: 90%

Asymptotic Chow polystability in Kähler geometry

Ji¹,

Poon²,

Yang³

et al. 2012

Fifth International Congress of Chinese Mathematicians

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Abstract. It is conjectured that the existence of constant scalar curvature Kähler metrics will be equivalent to K-stability, or K-polystability depending on terminology (Yau-Tian-Donaldson conjecture). There is another GIT stability condition, called the asymptotic Chow polystability. This condition implies the existence of balanced metrics for polarized manifolds (M, L k ) for all large k. It is expected that the balanced metrics converge to a constant scalar curvature metric as k tends to infinity under further suitable stability conditions. In this survey article I will report on recent results saying that the asymptotic Chow polystability does not hold for certain constant scalar curvature Kähler manifolds. We also compare a paper of Ono with that of Della Vedova and Zuddas.

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“…The K-stability has been introduced by Tian [30] and then reformulated in a purely algebraic-geometric way by Donaldson [7]. While the "only if" part of the conjecture has been proved thanks the works of Tian [30], Donaldson [7], Arezzo and Pacard [2], Stoppa [28] and Mabuchi [17,19], the "if" part is widely open in general and has been proved only in the case of toric surfaces by Donaldson [8], and in the case of projective bundles over a curve thanks to the works of Narasimhan and Seshadri [22], Ross and Thomas [25] and Mabuchi [17,19]. The K-stability is not the only GIT stability notion related to the existence of cscK metrics.…”

Section: Introductionmentioning

confidence: 99%

Scalar curvature and asymptotic Chow stability of projective bundles and blowups

Vedova

Zuddas

2012

Trans. Amer. Math. Soc.

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Abstract. The holomorphic invariants introduced by Futaki as obstruction to the asymptotic Chow semistability are studied by an algebraic-geometric point of view and are shown to be the Mumford weights of suitable line bundles on the Hilbert scheme of P n . These invariants are calculated in two special cases. The first is a projective bundle P(E) over a curve of genus g ≥ 2, and it is shown that it is asymptotically Chow polystable (with every polarization) if and only the bundle E is slope polystable. This proves a conjecture of Morrison with the extra assumption that the involved polarization is sufficiently divisible. Moreover it implies that P(E) is asymptotically Chow polystable (with every polarization) if and only if it admits a constant scalar curvature Kähler metric. The second case is a manifold blown-up at points, and new examples of asymptotically Chow unstable constant scalar curvature Kähler classes are given.

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Blowing up and desingularizing constant scalar curvature Kähler manifolds

Cited by 111 publications

References 57 publications

Compact complex surfaces and constant scalar curvature Kähler metrics

Compact complex surfaces and constant scalar curvature Kähler metrics

Asymptotic Chow polystability in Kähler geometry

Scalar curvature and asymptotic Chow stability of projective bundles and blowups

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