\(J\)-holomorphic curves from closed \(J\)-anti-invariant forms (Q6043433)

This paper is devoted to the relation between anti-invariant 2-forms and pseudoholomorphic curves. Let \((M^{2n},J)\) be an almost complex manifold. The almost complex structure acts on the bundle of real 2-forms \(\Lambda^2\) as the involution \(\alpha(\cdot,\cdot)\to\alpha(J\cdot,J\cdot)\). This involution induces the splitting \(\Lambda^2=\Lambda_J^+\oplus\Lambda_J^-\) corresponding to the eigenspaces of eigenvalues \(\pm1\) respectively. The sections are called \(J\)-invariant and \(J\)-anti-invariant 2-forms respectively. The spaces of these sections are denoted by \(\Omega^{\pm}\). The first main result is given in the following theorem. Theorem 1.1. Suppose \((M,J)\) is a compact connected almost complex 4-manifold and \(\alpha\) is a non-trivial \(J\)-anti-invariant 2-form. Then the zero set \(Z\) of \(\alpha\) supports a \(J\)-holomorphic 1-subvariety \(\Theta_{\alpha}\) in the canonical class \(K_J\). The authors establish a higher-dimensional analogue. Also, the relation between $J$-anti-invariant forms and birational geometry of almost complex manifolds is under consideration. The cohomology groups \[ H_J^{\pm}(M)=\{\omega\in H^2(M,\mathbb R)|\; \exists\alpha\in\mathcal Z_J^{\pm}\; \text{such that}\; [\alpha]=\omega\} \] generalize the real Hodge cohomology groups, where \(Z_J^{\pm}\) are the spaces of closed 2-forms in \(\Omega^{\pm}\). The dimensions of the vector spaces \(H_J^{\pm}(M)\) are denoted by \(h_J^{\pm}(M)\). In the following theorem, it is shown that the dimension of the \(J\)-anti-invariant cohomology is a birational invariant. Theorem 1.2. Let \(\Psi:(M_1,J_1)\to(M_2,J_2)\) be a degree 1 pseudoholomorphic map between closed, connected almost complex 4-manifolds. Then \(h_{J_1}^-(M_1)=h_{J_2}^-(M_2)\).