Singularities of Hermitian–Yang–Mills connections and Harder–Narasimhan–Seshadri filtrations

Chen, Xuemiao; Sun, Song

doi:10.1215/00127094-2020-0014

Cited by 11 publications

(39 citation statements)

References 25 publications

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“…We also give an algebro-geometric characterization of the bubbling set. This strengthens our previous result in [3].…”

supporting

confidence: 92%

“…This article is a continuation of [3] on studying tangent cones of (isolated) singularities of admissible Hermitian-Yang-Mills connections. The goals are the following • Remove a technical assumption in the main theorem in [3] by using a different argument;…”

Section: Introductionmentioning

confidence: 99%

“…Now we recall the main set-up, following [3]. Let B = {|z| < 1} ⊂ C n be the unit ball endowed with the standard flat Kähler metric ω 0 and let A be an admissible Hermitian-Yang-Mills connection on B.…”

Section: Introductionmentioning

confidence: 99%

“…Throughout this paper, we shall call the triple (A ∞ , Σ, µ) an analytic tangent cone of A at 0. Compared to [3], the definition here includes the extra data of the bubbling set and the limit measure, hence contains more information. A priori (A ∞ , Σ, µ) depends on the choice of subsequences as λ → 0.…”

Section: Introductionmentioning

confidence: 99%

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Analytic tangent cones of admissible Hermitian-Yang-Mills connections

Chen,

Sun

2018

Preprint

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“…We also give an algebro-geometric characterization of the bubbling set. This strengthens our previous result in [3].…”

supporting

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analytic tangent cones of admissible Hermitian-Yang-Mills connections

Chen,

Sun

2018

Preprint

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“…However, something weaker can still be shown: weakly holomorphic and locally approximable maps are energy-minimizing on the competition subclass of locally approximable maps, even though not on all of W 1,2 (M, N ) and this restriction forbids in particular the use of Schoen-Uhlenbeck theory. Theorem 1.1 was already proven when the almost complex structure J on the domain is integrable by S. Sun and X. Chen in [11], thanks also to the previous contributions [17] and [16] who established the optimal bound for the Hausdorff measure of the singular set, namely…”

mentioning

confidence: 90%

The Unique Tangent Cone Property for Weakly Holomorphic Maps into Projective Algebraic Varieties

Caniato¹,

Rivière²

2021

Preprint

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In the present paper, we establish the uniqueness of tangent maps for general weakly holomorphic and locally approximable maps from an arbitrary almost complex manifold into projective algebraic varieties. As a byproduct of the approach and the techniques developed we also obtain the unique tangent cone property for a special class of non-rectifiable positive pseudo-holomorphic cycles. This approach gives also a new proof of the main result by C.Bellettini in [3] on the uniqueness of tangent cones for positive integral (p, p)-cycles in arbitrary almost complex manifolds. R. CANIATO AND T. RIVIÈRE(2) for any k ∈ N, the space W 1,2 M, R k is the real vector space of functions u = (u 1 , ..., u k ) such that u j ∈ W 1,2 (M ), for every j = 1, ..., k;(3) given a closed smooth manifold N and a smooth isometric embedding N ֒→ R k , for some k ∈ N large enough, we letDefinition 1.1. Let M be any even-dimensional closed smooth manifold and let J be a Lipschitz complex structure on M . Let (N, J N ) be any closed smooth almost complex manifold. We say that a map u ∈ W 1,2 (M, N ) is weakly (J, J N )-holomorphic ifIn Lemma 3.1, we will show that any weakly (J, J N )-holomorphic map taking values into a closed smooth almost Kähler manifold N is weakly harmonic, i.e.where π N : W → N is the nearest-point projection into N , defined on a suitable tubular neighbourhood W of N , and Φ : N ֒→ R k denotes a smooth, isometric embedding of N into R k . Nevertheless, it is well-known that no regularity is ensured for weakly harmonic maps when the dimension of the domain is larger than 2 (see [20]). Thus, we will need to prescribe some additional condition in order to get that the map u is at least stationary harmonic, i.e.We will show (see Lemma 3.3) that imposing the following local, strong approximability property with respect to the W 1,2 -norm sufficies to our purposes.Definition 1.2. Let M, N be closed smooth manifolds. We say that a map u ∈ W 1,2 (M, N ) is locally (strongly) approximable with respect to the W 1,2 -norm if for every open set U ⊂ M such that U is diffeomorphic to some euclidean ball there exists a sequence of smooth maps {u j } j∈N ⊂ C ∞ (M, N ) such that u j → u as j → +∞, strongly in W 1,2 (U, N ).As explained in [6], a map u is locally approximable if and only ifwhere ω ∈ Ω 2 (N ) is any smooth and closed 2-form generating H 2 dR (N ).

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Stability of the tangent bundle through conifold transitions

Collins

Picard

Yau

2023

Comm Pure Appl Math

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Let X be a compact, Kähler, Calabi‐Yau threefold and suppose , for , is a conifold transition obtained by contracting finitely many disjoint curves in X and then smoothing the resulting ordinary double point singularities. We show that, for sufficiently small, the tangent bundle admits a Hermitian‐Yang‐Mills metric with respect to the conformally balanced metrics constructed by Fu‐Li‐Yau. Furthermore, we describe the behavior of near the vanishing cycles of as .

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Singularities of Hermitian–Yang–Mills connections and Harder–Narasimhan–Seshadri filtrations

Cited by 11 publications

References 25 publications

Analytic tangent cones of admissible Hermitian-Yang-Mills connections

Analytic tangent cones of admissible Hermitian-Yang-Mills connections

The Unique Tangent Cone Property for Weakly Holomorphic Maps into Projective Algebraic Varieties

Stability of the tangent bundle through conifold transitions

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