Related papers: Geodesics on a K3 Surface near the Orbifold Limit

We study the stable geodesics of the QFT special K\"ahler geometry ($\equiv$ Seiberg-Witten geometry of 4d $\mathcal{N}=2$ QFT) using the Myers argument. Complete stable geodesics are quite restricted, and can be described very explicitly.…

We develop numerical algorithms for solving the Einstein equation on Calabi-Yau manifolds at arbitrary values of their complex structure and Kahler parameters. We show that Kahler geometry can be exploited for significant gains in…

Right after Yau's resolution of the Calabi conjecture in the late 1970s, physicists Page and Gibbons-Pope conjectured that one may approximate Ricci-flat K\"{a}hler metrics on the $K3$ surface with metrics having "almost special holonomy"…

This is an expository paper which aims to give a simple proof of the existence of Ricci-flat metrics on certain K3 surfaces, as an illustration of general "glueing" techniques.

We conjecture that a non-flat $D$-real-dimensional compact Calabi-Yau manifold, such as a quintic hypersurface with D=6, or a K3 manifold with D=4, has locally length minimizing closed geodesics, and that the number of these with length…

We present a detailed study of the effective cones of Calabi-Yau threefolds with $h^{1,1}=2$, including the possible types of walls bounding the K\"ahler cone and a classification of the intersection forms arising in the geometrical phases.…

Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat K\"ahler metric) as one approaches a large complex…

This is a r\'esum\'e of an extensive investigation of some examples in which one obtains the rigid limit of N=2 supergravity by means of an expansion around singular points in the moduli space of a Calabi-Yau 3-fold. We make extensive use…

The limiting procedure of special Kahler manifolds to their rigid limit is studied for moduli spaces of Calabi-Yau manifolds in the neighbourhood of certain singularities. In two examples we consider all the periods in and around the rigid…

We prove the existence of a Ricci flat metric on the Kummer K3 surface. The proof follows the general strategy of Donaldson's gluing construction. However, we tackle the analysis without appealing to weighted norms or conformal…

Motivated by some conjectures originating in the Physics literature, we use Foscolo's construction of Ricci-flat Kahler metrics on K3 surfaces to locate, with high precision, several closed geodesics and compute their index (their length is…

Let S be a K3 surface that admits a non-symplectic automorphism $\rho$ of order 3. We divide $S\times \mathbb{P}^1$ by $\rho\times\psi$ where $\psi$ is an automorphism of order 3 of $\mathbb{P}^1$. There exists a threefold ramified cover of…

We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend…

We study threefolds fibred by Kummer surfaces associated to products of elliptic curves, that arise as resolved quotients of threefolds fibred by certain lattice polarized K3 surfaces under a fibrewise Nikulin involution. We present a…

Finding Ricci-flat (Calabi-Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi-Yau metric…

In this technical note we describe a new (to the physics literature) construction of bundles on Calabi-Yaus. We primarily study this construction in the special case of K3 surfaces, for which interesting results can be obtained. For…

We study the stability of compact pseudo-K\"ahler manifolds, i.e. compact complex manifolds $X$ endowed with a symplectic form compatible with the complex structure of $X$. When the corresponding metric is positive-definite, $X$ is K\"ahler…

The conjectural equivalence of curve counting on Calabi-Yau 3-folds via stable maps and stable pairs is discussed. By considering Calabi-Yau 3-folds with K3 fibrations, the correspondence naturally connects curve and sheaf counting on K3…

In this thesis we study the relationship between the existence of canonical metrics on a complex manifold and stability in the sense of geometric invariant theory. We introduce a modification of K-stability of a polarised variety which we…

We construct Bridgeland stability conditions on the the following hyper-K\"ahler or strict Calabi--Yau manifolds: - Generalized Kummer varieties associated to an abelian surface that is isogenous to a product of elliptic curves. - Universal…