Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds (Q2781434)

Consider an ample line bundle \(L\) over a symplectic manifold \(M\). The authors study some analogue \(H^0_J(M, L^N)\) of holomorphic sections, which they call `almost-holomorphic'. Using only almost-complex geometry, they construct a parametrix for a Szegö projector \(\Pi_N\) of the same type as in the holomorphic case. They prove that \(\Pi_N(x,y)\) has the same scaling asymptotics as does the holomorphic Szegö kernel. Other analogues with the holomorphic case are derived, related to this scaling asymptotics. Finally, the problem of existence of quantitatively transverse sections in \(H^0_J(M,L^N)\) is considered.