Almost complex geometry of symplectic log Calabi-Yau pairs with applications to almost toric fibrations

<p> Given a symplectic log Calabi-Yau pair, we study the almost complex structures that are adapted to it. For these almost complex structures, we study the behavior of the curve cone, prove a Nakai-Moishezon criterion and describe the almost Kähler cone for the generic ones. As an application, we prove a decomposition theorem for Seiberg-Witten non-trivial classes. This leads to an alternative proof of the existence of almost toric fibrations on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 1"> <mml:semantics> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">c_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -positive rational surfaces [ <italic>Almost toric presentations of symplectic log Calabi-Yau pairs</italic> , arXiv:2303.09964, 2023]. </p>