A report on the hypersymplectic flow

Fine, Joël; Yao, Chengjian

doi:10.48550/arxiv.2001.11755

Cited by 2 publications

(3 citation statements)

References 40 publications

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“…On the other hand, concerning characterizing diffeomorphism type of a symplectic Calabi-Yau 4-manifold, Theorem 1.1 is the first result of such kind (under a finite symmetry condition). Finally, for connections of this question with hypersymplectic structures and Donaldson's conjecture, we refer the readers to the recent article [14].…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Finite group actions on symplectic Calabi-Yau $4$-manifolds with $b_1>0$

Chen¹

2020

Preprint

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This is the first of a series of papers devoted to the topology of symplectic Calabi-Yau 4-manifolds endowed with certain symplectic finite group actions. We completely determined the fixed-point set structure of a finite cyclic action on a symplectic Calabi-Yau 4-manifold with b1 > 0. As an outcome of this fixed-point set analysis, the 4-manifold was shown to be a T 2 -bundle over T 2 in some circumstances, e.g., in the case where the group action is an involution which fixes a 2-dimensional surface in the 4-manifold. Our project on symplectic Calabi-Yau 4-manifolds is based on an analysis of existence and classification of disjoint embeddings of certain configurations of symplectic surfaces in a rational 4-manifold. This paper laid the ground work for such an analysis at the homological level. Some other results which are of independent interest, concerning the maximal number of disjointly embedded symplectic (−2)-spheres in a rational 4-manifold, were also obtained.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Finite group actions on symplectic Calabi-Yau $4$-manifolds with $b_1>0$

Chen¹

2020

Preprint

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“…The triple ω is called torsion-free hypersymplectic if φ is a torsion-free G 2 structure. Locally, this is a weaker condition than being hyperkähler, see examples in [14] or [18]. Donaldson observed in [14] that the boundary value problem for torsionfree G 2 structures can also be reduced to dimension 4.…”

Section: Introductionmentioning

confidence: 99%

“…Let us begin with some arguments and results in [18]. Let P denotes the set of symmetric positive-definite 3 by 3 matrices.…”

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confidence: 99%

A compactness theorem for hyperkaehler 4-manifolds with boundary

Liu¹

2022

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In this paper, we study the compactness of a boundary value problem for hyperkähler 4-manifolds. We show that under certain topological conditions and the positive mean curvature condition on the boundary, a sequence of hyperkähler triples converges smoothly up to diffeomorphisms if and only if their restrictions to the boundary converge smoothly up to diffeomorphisms. We also generalize this result to torsion-free hypersymplectic triples.

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A report on the hypersymplectic flow

Cited by 2 publications

References 40 publications

Finite group actions on symplectic Calabi-Yau $4$-manifolds with $b_1>0$

Finite group actions on symplectic Calabi-Yau $4$-manifolds with $b_1>0$

A compactness theorem for hyperkaehler 4-manifolds with boundary

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