Abstract:Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct … Show more
“…An open question which we have not been able to solve is to establish a more clear relation between the Dorfman bracket [, ] on Q and the orthogonal connection D G , in a way that our new stability condition is formulated more naturally in terms of the triple (Q, , , [, ]). Even though our picture is mostly conjectural, we expect that this stability condition, along with the notion of harmonic metric that we introduce, will lead to new obstructions to the existence of solutions in future studies, similarly as in [16].…”
We investigate stability conditions related to the existence of solutions of the Hull-Strominger system with prescribed balanced class. We build on recent work by the authors, where the Hull-Strominger system is recasted using non-Hermitian Yang-Mills connections and holomorphic Courant algebroids. Our main development is a notion of harmonic metric for the Hull-Strominger system, motivated by an infinite-dimensional hyperKähler moment map and related to a numerical stability condition, which we expect to exist generically for families of solutions. We illustrate our theory with an infinite number of continuous families of examples on the Iwasawa manifold.
“…An open question which we have not been able to solve is to establish a more clear relation between the Dorfman bracket [, ] on Q and the orthogonal connection D G , in a way that our new stability condition is formulated more naturally in terms of the triple (Q, , , [, ]). Even though our picture is mostly conjectural, we expect that this stability condition, along with the notion of harmonic metric that we introduce, will lead to new obstructions to the existence of solutions in future studies, similarly as in [16].…”
We investigate stability conditions related to the existence of solutions of the Hull-Strominger system with prescribed balanced class. We build on recent work by the authors, where the Hull-Strominger system is recasted using non-Hermitian Yang-Mills connections and holomorphic Courant algebroids. Our main development is a notion of harmonic metric for the Hull-Strominger system, motivated by an infinite-dimensional hyperKähler moment map and related to a numerical stability condition, which we expect to exist generically for families of solutions. We illustrate our theory with an infinite number of continuous families of examples on the Iwasawa manifold.
“…Meanwhile, they [17,18,19] prove it is a strictly parabolic systems under the pluriclosed assumption and give the short time existence and some basic regularity results. More results about regularity and long-time existence can be found in [6,9,13,14,15].…”
Section: Preliminarymentioning
confidence: 99%
“…For any selection of (2, 0) form ϕ 0 , it is a flow of Hermitian-symplectic forms with initial data Ω 0 = ϕ 0 + ω 0 + φ0 . The idea of flowing a (2, 0) form by the (2, 0)-part of Bismut-Ricci form along the pluriclosed flow first appears in [13] (can also see [6,7]). They consider the system consisting of pluriclosed flow and an evolution equation of (2, 0) form…”
We show pluriclosed flow preserves the Hermitian-symplectic structures. And we observe that it can actually become a flow of Hermitian-symplectic forms when an extra evolution equation determined by the Bismut-Ricci form is considered. Moreover, we get a topological obstruction to the long-time existence in arbitrary dimension.
“…, ω |t=0 = ω . The pluriclosed flow was deeply studied in literature, see for instance [3,5,6,9,12,19,22,23,24,25,26,27] and the references therein.…”
We investigate the pluriclosed flow on Oeljeklaus-Toma manifolds. We parametrize leftinvariant pluriclosed metrics on Oeljeklaus-Toma manifolds and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution ωt which once normalized collapses to a torus in the Gromov-Hausdorff sense. Moreover the lift of 1 1+t ωt to the universal covering of the manifold converges in the Cheeger-Gromov sense to (H s × C s , ω∞) where ω∞ is an algebraic soliton.
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