Related papers: Stability of the tangent bundle through conifold t…

Let $\Omega$ be a complex manifold, and let $X\subset \Omega$ be an open submanifold whose closure $\bar X$ is a (not necessarily compact) submanifold with smooth boundary. Let $G$ be a complex Lie group, $\Pi$ be a differentiable principal…

We consider N=1, d=4 vacua of heterotic theories in the large radius limit in which alpha' << 1. We construct a real differential operator $\mathcal{D}= D+\bar{D}$ on an extension bundle $(Q, \mathcal{D})$ with underlying topology…

Hyperholomorphic bundle is a bundle with connection defined over a hyperkaehler manifold such that this connection is holomorphic with respect to all complex structures induced by a hyperkaehler structure. A hyperholomorphic connection is…

Let $X$ be a Riemann surface, $K_X \rightarrow X$ the canonical bundle, and $T_X\rightarrow X$ the dual bundle of the canonical bundle. For each integer $r \geq 2$, each $q \in H^0(K_X^r)$, and each choice of the square root $K_X^{1/2}$ of…

We consider the Yang-Mills flow on hyperbolic 3-space. The gauge connection is constructed from the frame-field and (not necessarily compatible) spin connection components. The fixed points of this flow include zero Yang-Mills curvature…

Numerical approximations to Ricci-flat Calabi--Yau metrics make it possible to move beyond the topological and holomorphic data that have traditionally dominated explicit string compactifications. This article explains what new physics and…

Consider a measure-preserving transition kernel $T$ on an arbitrary probability space $(\mathbb X,\mathcal cA,\pi)$. In this level of generality, we prove that a one-step hyper-contractivity estimate of the form $\|T\|_{p\to q}\le 1$ with…

We revisit moduli stabilization on Calabi-Yau manifolds with a discrete symmetry. Invariant fluxes allow for a truncation to a symmetric locus in complex structure moduli space and hence drastically reduce the moduli stabilization problem…

We supply a detailed proof of the result by P.S. Green and T. H$\ddot{\text{u}}$bsch that all complete intersection Calabi--Yau 3-folds in product of projective spaces are connected through projective conifold transitions (known as the…

We use a recently proposed formulation of stable holomorphic vector bundles $V$ on elliptically fibered Calabi--Yau n-fold $Z_n$ in terms of toric geometry to describe stability conditions on $V$. Using the toric map $f: W_{n+1} \to…

We study the conditions under which the tangent bundle $(TM,G)$ of an $n$-dimensional Riemannian manifold $(M,g)$ is conformally flat, where $G$ is a general natural lifted metric of $g$. We prove that the base manifold must have constant…

Given a principal bundle on an orientable closed surface with compact connected structure group, we endow the space of based gauge equivalence classes of smooth connections relative to smooth based gauge transformations with the structure…

We compute the (1,1)-Aeppli cohomology of compact simply-connected Lie groups. From this, we deduce that the Bismut flat metrics on the compact Bismut flat manifolds with finite fundamental group are globally stable for the pluriclosed…

We study the deformation theory of the Einstein-Yang-Mills system on a principal bundle with a compact structure group over a compact manifold. We first construct, as an application of the general slice theorem of Diez and Rudolph, a smooth…

We prove an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter, i.e., $\hat{\theta} \in (\frac{\pi}{2},\frac{3\pi}{2})$, on compact complex three-folds conditioned on a…

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…

We analyze three aspects of N=1 heterotic string compactifications on elliptically fibered Calabi-Yau threefolds: stability of vector bundles, five-brane instanton transitions and chiral matter. First we show that relative Fourier-Mukai…

We show the stability of certain syzygies of line bundles on curves, which we call transforms, and are kernels of the evaluation map on subspaces of the space of global sections. For the transforms constructed, we prove the existence of…

We consider Higgs bundles satisfying a notion of numerical flatness (H-nflatness) that was previously introduced, and show that they have Jordan-H\"older filtrations whose quotients are stable, locally free and H-nflat. This is applied to…

These lecture notes introduce conifold transitions between complex threefolds with trivial canonical bundle from the differential geometric point of view, and with a particular view towards aspects of mathematical physics and string theory.…