Related papers: Moduli of Coassociative Submanifolds and Semi-Flat…

Let $G$ be a simple complex Lie group, $\alg{g}$ be its Lie algebra, $K$ be a maximal compact form of $G$ and $\alg{k}$ be a Lie algebra of $K$. We denote by $X\rightarrow \overline{X}$ the anti-involution of $\alg{g}$ which singles out the…

Let $X$ be a compact Riemann surface of genus $g \geq 2$ and $D\subset X$ be a fixed finite subset. Let $\xi$ be a line bundle of degree $d$ over $X$. Let $\mathcal{M}(\alpha, r, \xi)$ (respectively, $\mathcal{M}_{\mathrm{conn}}(\alpha, r,…

In this paper we construct various moduli spaces of K3 surfaces $M$ equipped with a surjective holomorphic map $\pi:M\to\Pb^1$ with generic fiber a complex torus (e.g., an elliptic fibration). Examples include moduli spaces of such maps…

We use techniques of Alper-Hall-Rydh to prove a local structure theorem for smooth morphisms between smooth stacks around points with linearly reductive stabilizers. This implies that the good moduli space of a smooth stack over a base has…

We construct canonical semi-orthogonal decompositions for derived categories of smooth projective surfaces. These decompositions are compatible with the operations in the minimal model program, such as blow-ups and conic bundles. Therefore…

Let $M^7$ be a smooth manifold equipped with a $G_2$-structure $\phi$, and $Y^3$ be an closed compact $\phi$-associative submanifold. In \cite{McL}, R. McLean proved that the moduli space $\bm_{Y,\phi}$ of the $\phi$-associative…

We study the moduli space $\mathcal{F}_{T_1}$ of quasi-trielliptic K3 surfaces of type I, whose general member is a smooth bidegree $(2,3)$-hypersurface of $\mathbb{P}^1\times \mathbb{P}^2$. Such moduli space plays an important role in the…

We study D3 branes at orbifolded conifold singularities in the presence of discrete torsion. The vacuum moduli space of open strings becomes non-commutative due to a deformation of the superpotential and is studied via the representation…

We compute the dimensions of some moduli spaces of left-invariant closed and coclosed $\mathrm{G}_2$-structures on 7-dimensional nilmanifolds, showing that they are not related to the third Betti number. We also prove that, in contrast to…

Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed,…

Let $X$ be an algebraic K3 surface, $v=(r,H,s)$ a primitive isotropic Mukai vector on $X$ and $M_X(v)$ the moduli of sheaves over $X$ with $v$. Let $N(X)$ be Picard lattice of $X$. In math.AG/0309348 and math.AG/0606289, all divisors in…

We discuss the moduli space of flat connections of Yang-Mills theories formulated on T^3 x R, with periodic boundary conditions. When the gauge group is SO(N>=7), G_2, F_4, E_6, E_7 or E_8, the moduli space consists of more than one…

In this paper we construct certain moduli spaces, which we call moduli spaces of (principal) $F$-bundles, and study their basic properties. These spaces are associated to triples consisting of a smooth projective geometrically connected…

We give infinite series of groups Gamma and of compact complex surfaces of general type S with fundamental group Gamma such that 1) Any surface S' with the same Euler number as S, and fundamental group Gamma, is diffeomorphic to S. 2) The…

A new method is presented for solving the Gauss-Codazzi equations for a compact Riemann surface to be immersed in a 3-manifold of constant curvature. In the negative curvature case, the moduli for such embeddings are cohomology classes of…

We study the infinitesimal deformations of a proper nearly parallel G_2-structure and prove that they are characterized by a certain first order differential equation. In particular we show that the space of infinitesimal deformations…

The endpoint of this series of papers is to construct the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $\omega$ is a suitable closed 2-form. In the first paper, we…

We show that the Lie potential on the minimal semisimple adjoint orbit $\mathcal{O}_n$ of $\mathfrak{sl}(n+1,\mathbb{C})$ coincides with toric potential on $T^*\mathbb P^{n}$. We then study the corresponding Landau-Ginzburg models in…

This is the second of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being…

The main goal of this paper is to study the formal geometry of dg manifolds \`a la Fedosov. For any dg manifold $(\mathcal{M}, Q)$, we construct a Fedosov dg foliation (or dg Lie algebroid) $\mathcal{F}_Q \to \mathcal{N}_Q$. We establish…