Taming symplectic forms and the Calabi–Yau equation

By Valentino Tosatti, B. Weinkove, Shing-Tung Yau. 2007

101 citations recorded by the source corpus.

We study the Calabi–Yau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature condition, we show that the conjecture holds.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5ce9b1e4196b6b32dcac8d078336f848ac8c5c33

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Citations in the corpus

On the J-anti-invariant cohomology of almost complex 4-manifolds (2011, background): “At least in the case b+ = 1, the tame-compatible question is known to be a consequence of the symplectic Calabi-Yau problem, also introduced by Donaldson in [10] (see also [31], [29], [28], and section 5 below).”
Adiabatic limits of Ricci-flat Kähler metrics (2009, background): “. The starting point is a precise formula for ∆ω˜tS. This is just Yau’s C 3 estimate [Y1], but without assuming that the metrics ˜ωt and ωX are equivalent, and it is done in a more general setting in [TWY] (see also [PSS]). With the notation of [TWY] we can write S = X i,j,k |ai jk| 2. We then choos”
Interior derivative estimates for the Kähler–Ricci flow (2011, unknown): “There exist now many generalizations of Calabi’s estimate [Cherrier 1987; Tosatti 2010; Tosatti et al. 2008; Zhang and Zhang 2011].”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, methodology): “Evans–Krylov results were proved for this equation in [70] and [62].”
Lectures on Stability and Constant Scalar Curvature (2008, background): “measures are in Dinh, Nguyen, and Sibony [41]. The importance role of an α-invariant for tame symplectic manifolds, if it is strictly positive, has been brought to light by Tosatti, Weinkove, and Yau [150]. 8.2 Nadel’s multiplier ideal sheaves criterion In the case of vector bundles, as proved by Do”
The Monge–Ampère equation for non-integrable almost complex structures (2016, background): “[52]), whose curvature form can be expressed as a skew-Hermitian matrix of 2-forms {Ωj} (1 6 i, j 6 n) in any local unitary frame, and defining Ric(ω) := √ −1 n ∑”
Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds (2007, methodology): “Motivated by a question of Donaldson, we use these groups to relate J-tamed symplectic cones and Jcompatible symplectic cones over a large class of almost complex manifolds, including all Kähler manifolds, almost Kähler 4- manifolds and complex surfaces.”
On the evolution of a Hermitian metric by its Chern-Ricci form (2011, methodology): “This method was employed in the Kähler case in [59], and a related argument was used in the almost complex setting in [55].”
Complex Monge-Ampere equations and totally real submanifolds (2009, background): “In [24], Donaldson proposed to generalize Yau’s theorems in a different direction; see [69,68, 65,66] for some recent developments.”
A general Schwarz Lemma for almost-Hermitian manifolds (2007, methodology): “The proof of the Schwarz lemma employs Cartan’s formalism of moving frames and the canonical connection, as in [24].”