We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed Hermitian-Yang-Mills equations. That is, the transformation relates supersymmetric A- and B-cycles. In this paper, we assume that the mirror pair are dual torus fibrations with flat tori and that the A-cycle is a section.
We also show that this transformation preserves the (holomorphic) Chern-Simons functional for all connections. Furthermore, on corresponding moduli spaces of supersymmetric cycles it identifies the graded tangent spaces and the holomorphic m-forms. In particular, we verify Vafa's mirror conjecture with bundles in this special case.
The real Fourier-Mukai transform sends a section of a torus fibration to a connection over the total space of the dual torus fibration. By this method, Leung, Yau and Zaslow introduced deformed Hermitian Yang-Mills (dHYM) connections for Kähler manifolds and Lee and Leung introduced deformed Donaldson-Thomas (dDT) connections for $G_2$- and ${\rm Spin}(7)$-manifolds. In this paper, we suggest an alternative definition of a dDT connection for a manifold with a ${\rm Spin}(7)$-structure which seems to be more appropriate by carefully computing the real Fourier-Mukai transform again. We also post…
From string theory, the notion of deformed Hermitian Yang-Mills connections has been introduced by Mariño, Minasian, Moore and Strominger. After that, Leung, Yau and Zaslow proved that it naturally appears as mirror objects of special Lagrangian submanifolds via Fourier-Mukai transform between dual torus fibrations. In their paper, some conditions are imposed for simplicity. In this paper, data to glue their construction on tropical manifolds are proposed and a generalization of the correspondence is proved without the assumption that the Lagrangian submanifold is a section of the torus fibrat…
We provide an introduction to the mathematics and physics of the deformed Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kahler manifolds which plays an important role in mirror symmetry. We discuss the physical origin of the equation, and some recent progress towards its solution. In dimension 3 we prove a new Chern number inequality and discuss the relationship with algebraic stability conditions.
In this paper we give a complete algebro-geometric characterization of analytic tangent cones of admissible Hermitian-Yang-Mills connections over any reflexive sheaves.
We give an example of a homogeneous reflexive sheaf over $\mathbb{C}^3$ which admits a non-conical Hermitian Yang-Mills connection. This is expected to model bubbling phenomenon along complex codimension 2 submanifolds when the Fueter section takes zero value.
In this paper, we consider the deformed Hermitian-Yang-Mills equation on closed almost Hermitian manifolds. In the case of hypercritical phase, we derive a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. As an application, we prove the existence of solutions for the deformed Hermitian-Yang-Mills equation under the condition of existence of a supersolution.
In this paper, we study the hypercritical deformed Hermitian-Yang-Mills equation on compact Kähler manifolds and resolve two conjectures of Collins-Yau.
We survey some recent progress on the deformed Hermitian-Yang-Mills (dHYM) equation. We discuss the role of geometric invariant theory (GIT) in approaching the solvability of the dHYM equation, following work of the first author and S.-T. Yau. We compare the GIT picture with the conjectural picture for dHYM involving Bridgeland stability. In particular, following Arcara-Miles, we show that on the blow-up of $\mathbb{P}^2$ any line bundle admitting a solution of the deformed Hermitian-Yang-Mills equation is Bridgeland stable, but not conversely. Finally, we survey some recent progress on heat f…
In this paper, we analyze the asymptotic behaviour of the Hermitian-Yang-Mills flow over a compact non-Kähler manifold $(X,g)$ with the Hermitian metric $g$ satisfying the Gauduchon and Astheno-Kähler condition.
We introduce a new system of equations coupling Kähler-Einstein and Hermitian-Yang-Mills equations. We provide a moment map interpretation of these equations. We identify a Futaki type invariant as an obstruction to the existence of solutions to these equations. We also prove a Matsushima-Lichnerowicz type theorem. We prove a deformation result that produces nontrivial solutions of these equations under some conditions. We produce examples on some projective bundles using Calabi ansatz.
We construct local examples of singular Hermitian Yang-Mills connections over $B_1\subset \mathbb{C}^3$ with uniformly bounded $L^2$-energy, but the number of essential singular points can be arbitrarily large.
Let $(X, ω)$ be a compact connected Hermitian manifold of dimension $n$. We consider the Bott-Chern cohomology and let $[χ] \in H^{1,1}_{\text{BC}}(X; \mathbb{R})$. We study the deformed Hermitian-Yang-Mills equation, which is the following nonlinear elliptic equation $\sum_{i} \arctan (λ_i) = h(x)$, where $λ_i$ are the eigenvalues of $χ$ with respect to $ω$.
In this work, we study the deformed Hermitian-Yang-Mills equation on compact Kähler manifold. We introduce the notions of coerciveness and properness of the $\mathcal{J}$-functional on the space of almost calibrated $(1,1)$-forms and show that they are both equivalent to the existence of solutions to the hypercritical deformed Hermitian-Yang-Mills equation.
In this paper, we investigate the Chern number inequalities on $4$-dimensional Kähler manifolds admitting the deformed Hermitian-Yang-Mills metrics under the assumption $\hatθ\in (π,2π)$.
In this paper, we study the curvature estimate of the Hermitian-Yang-Mills flow on holomorphic vector bundles. In one simple case, we show that the curvature of the evolved Hermitian metric is uniformly bounded away from the analytic subvariety determined by the Harder-Narasimhan-Seshadri filtration of the holomorphic vector bundle.
The purpose of this paper is to investigate the parabolic deformed Hermitian-Yang-Mills equation with hypercritical phase in a smooth domain $Ω\subset \mathbb{C}^{n}$. By using $J$-functional, we are able to prove the convergence of solutions. As an application, we give an alternative proof of the Dirichlet problem for deformed Hermitian-Yang-Mills equation.
In this paper, we study the non-Hermitian Yang-Mills (NHYM for short) bundles over compact Kähler manifolds. We show that the existence of harmonic metrics is equivalent to the semisimplicity of NHYM bundles, which confirms the Conjecture 8.7 in \cite{KV}.
In this paper, we provide a necessary and sufficient condition for the solvability of the supercritical deformed Hermitian-Yang-Mills equation using integrals on subvarieties. This result confirms the mirror version of the Thomas-Yau conjecture about special Lagrangian submanifolds on Calabi-Yau manifolds.
In this paper, we study the asymptotic behavior of the Hermitian-Yang-Mills flow on a reflexive sheaf. We prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the Harder-Narasimhan-Seshadri filtration, this answers a question by Bando and Siu.
We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections on reflexive sheaves at isolated singularities modelled on $μ$-polystable holomorphic bundles over $\mathbf{P}^{n-1}$.
In this paper, we extend a result of Gao Chen regarding the solvability of the twisted deformed Hermitian Yang-Mills equations on compact Kähler manifolds to allow for the twisting function to be non-constant and slightly negative in all dimensions. Using this result, we prove that the twisted supercritical dHYM equation on compact, projective manifolds can be solved provided certain numerical conditions are satisfied.
In this paper we study the analytic tangent cones of admissible Hermitian-Yang-Mills connections near a homogeneous singularity of a reflexive sheaf, and relate it to the Harder-Narasimhan-Seshadri filtration. We also give an algebro-geometric characterization of the bubbling set. This strengthens our previous result.
Consider a vector bundle over a Kähler manifold which admits a Hermitian Yang-Mills connection. We show that the pullback bundle on the blowup of the Kähler manifold at a collection of points also admits a Hermitian Yang-Mills connection, for Kähler classes on the blowup which make the exceptional divisors small. Our proof uses gluing techniques, and is hence asymptotically explict. This recovers, through the Hitchin-Kobayashi correspondence, algebro-geometric results due to Buchdahl and Sibley.
In this paper, we study the deformed Hermitian-Yang-Mills equation on compact Kähler manifold with non-negative orthogonal bisectional curvature. We prove that the curvatures of deformed Hermitian-Yang-Mills metrics are parallel with respect to the background metric if there exists a positive constant $C$ such that $-\frac{1}{C}ω<\sqrt{-1}F<Cω$. We also study the self-shrinker over $\mathbb{C}^n$ to the corresponding parabolic flow. We prove that the self-shrinker over $\mathbb{C}^n$ is a quadratic polynomial function. We also show the similar rigid theorem for the J-equations and the self-shr…
We give an algebraic criterion for the existence of projectively Hermitian-Yang-Mills metrics on a holomorphic vector bundle $E$ over some complete non-compact Kähler manifolds $(X,ω)$, where $X$ is the complement of a divisor in a compact Kähler manifold and we impose some conditions on the cohomology class and the asymptotic behaviour of the Kähler form $ω$. We introduce the notion of stability with respect to a pair of $(1,1)$-classes which generalizes the standard slope stability. We prove that this new stability condition is both sufficient and necessary for the existence of projectively …
We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey-Lawson. By exploiting the convexity of the level sets of the Lagrangian angle operator in the highest branch, we are able to construct continuous solutions of the associated Dirichlet problem.
We prove an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter, i.e., $\hatθ \in (\fracπ{2},\frac{3π}{2})$, on compact complex three-folds conditioned on a necessary subsolution condition. Our proof hinges on a delicate analysis of a new continuity path obtained by rewriting the equation as a generalised Monge-Ampère equation with mixed sign coefficients.
We address the issue why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two-forms and four-forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies th…
Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths-and even in the dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled.
We prove a priori estimates for a generalised Monge-Ampère PDE with "non-constant coefficients" thus improving a result of Sun in the Kähler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob-Yau to obtain an existence result and a priori estimates for some ranges of the phase angle assuming the existence of a subsolution. We then generalise a theorem of Collins-Szèkelyhidi on toric varieties and use it to address a conjecture of Collins-Jacob-Yau.
We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical Kähler manifolds: If $\mathscr{E}$ is a reflexive sheaf over an ACyl Kähler manifold, which is asymptotic to a $μ$-stable holomorphic vector bundle, then it admits an asymptotically translation-invariant protectively Hermitian Yang-Mills metrics (with curvature in $L^2_{\mathrm{loc}}$ across the singular set). Our proof combines the analytic continuity method of Uhlenbeck and Yau [UY86] with the geometric regularization scheme introduced by Bando and Siu [BS94].
The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background Kähler metric, to be constant. In this paper we introduce and study dHYM equations with variable Kähler metric. These are coupled equations involving both the Lagrangian phase and the radius function, at the same time. They are obtained by using the extended gauge group to couple the moment map interpretation of dHYM connections, due to Collins-Yau and …
We study the deformed Hermitian-Yang-Mills equation on the blowup of complex projective space. Using symmetry, we express the equation as an ODE which can be solved using combinatorial methods if an algebraic stability condition is satisfied. This gives evidence towards a conjecture of the first author, T.C. Collins, and S.-T. Yau on general compact Kahler manifolds.
This paper begins to study the limiting behavior of a family of Hermitian Yang-Mills (HYM for brevity) metrics on a class of rank two slope stable vector bundles over a product of two elliptic curves with Kähler metrics $ω_ε$ when $ε\to 0$. Here $ω_ε$ are flat and have areas $ε$ and $ε^{-1}$ on the two elliptic curves respectively. A family of Hermitian metrics on the vector bundle are explicitly constructed and with respect to them, the HYM metrics are normalized. We then compare the family of normalized HYM metrics with the family of constructed Hermitian metrics by doing estimates. We get t…
