The deformed Hermitian-Yang-Mills equation on the blowup of \(\mathbb{P}^n\) (Q6160107)

This article concerns the so-called \textit{deformed Hermitian-Yang-Mills (dHYM) equation}, which can be formulated on a compact Kähler manifold \((X,J,\omega)\). A class \([\alpha]\in H^{1,1}(X,\mathbb R)\) is said to solve the dHYM equation with \textit{phase} \(e^{i\hat\theta}\in S^1\) if \([\alpha]\) admits a representative \(\beta\in [\alpha]\) such that \(\mathrm{Im}(e^{i\hat\theta}(\omega+i\beta)^n)=0\). Using the \(\partial\bar\partial\)-lemma, this equation is to be thought of as an elliptic nonlinear PDE for a real-valued function \(\phi\) on \(X\), which arises naturally in certain supersymmetric physical theories as well as in the setting of SYZ mirror symmetry. The relationship between the existence of solutions to natural PDEs and algebraic stability conditions is one of the central themes of the past decades in complex (algebraic) geometry, and the main aim of the authors is to investigate this question for the dHYM equation in a simple setting, namely the case where \(X\) is the blow-up of \(\mathbb P^n\) at a point. The proofs of the main theorems rely strongly on the special structure afforded by this simplified setting, such as the simple second-degree cohomology of \(X\). To formulate the main results, we need to introduce some notation. For any irreducible, analytic subvariety \(V\subset X\), we can consider the number \[ Z_{[\alpha],[\omega]}(V)= - \int_V e^{-i\omega+\alpha} \] where the exponential is interpreted as a formal power series and we take the convention that integration is done only over the term of this power series of degree \(\dim_{\mathbb R} V\). Now fix \(X\) as the blowup of \(\mathbb P^n\) at a point, let \(\omega\) be any Kähler class and \(\alpha\) any real cohomology class of type \((1,1)\). Then the first theorem asserts the following: If \(Z_{[\alpha],[\omega]}\neq 0\) and there exists a sequence \((\epsilon_1,\dots,\epsilon_{n-1})\), \(\epsilon_i\in \pm 1\), such that for each \(k\)-dimensional subvariety \(V^k\subset X\) we have \[ \mathrm{sign}\,\bigg(\mathrm{Im}\bigg(\frac{Z_{[\alpha],[\omega]}(V^k)}{Z_{[\alpha],[\omega]}(X)}\bigg)\bigg) = \epsilon_k, \] then \(\alpha\) admits a solution to the dHYM equation. This solution moreover satisfies the Calabi ansatz, meaning that it takes on a specific form on \(X\setminus (H\cup E)\cong \mathbb C^n \setminus\{0\}\), where \(H\) and \(E\) are the hyperplane and exceptional divisor, respectively. While the above condition is sufficient, it is not clear that it is necessary. In fact, the authors prove that it is not, and formulate a pair of necessary and sufficient conditions for the existence of a solution to the dHYM equation. Defining the so-called average angle \(\hat\Theta_{V^k}\) as the argument of \(\int_{V^k}(\omega+i\alpha)^k\), these conditions are expressed in terms of (i) \(\hat\theta=\hat\Theta_X\) and (ii) the collection of all \(\hat\Theta_{V^{n-1}}\), where \(V^{n-1}\) is a divisor.