Decomposition of Lagrangian classes on K3 surfaces

We study the decomposability of a Lagrangian homology class on a K3 surface into a sum of classes represented by special Lagrangian submanifolds, and develop criteria for it in terms of lattice theory. As a result, we prove the decomposability on an arbitrary K3 surface with respect to the Kähler classes in dense subsets of the Kähler cone. Using the same technique, we show that the Kähler classes on a K3 surface which admit a special Lagrangian fibration form a dense subset also. This implies that there are infinitely many special Lagrangian 3-tori in any log Calabi-Yau 3-fold.https://arxiv.org/abs/2001.00202Othe

On the irrationality of moduli spaces of projective hyperk\"ahler manifolds

The aim of this paper is to estimate the irrationality of moduli spaces of hyperk\"ahler manifolds of types K3$^{[n]}$, Kum$_{n}$, OG6, and OG10. We prove that the degrees of irrationality of these moduli spaces are bounded from above by a universal polynomial in the dimension and degree of the manifolds they parametrize. We also give a polynomial bound for the degrees of irrationality of moduli spaces of $(1,d)$-polarized abelian surfaces.Comment: 29 pages. Comments are welcome

On the irrationality of moduli spaces of K3 surfaces

We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genus. We prove that the growth is bounded by a polynomial function of degree $14+\varepsilon$ for any $\varepsilon>0$ and, for three sets of infinitely many genera, the bounds can be improved to degree 10. The main ingredients in our proof are the modularity of the generating series of Heegner divisors due to Borcherds and its generalization to higher codimensions due to Kudla, Millson, Zhang, Bruinier, and Westerholt-Raum. For special genera, the proof is also built upon the existence of K3 surfaces associated with certain cubic fourfolds, Gushel-Mukai fourfolds, and hyperkaehler fourfolds.Comment: v2: Results substantially improved. We use Kudla's modularity conjecture to obtain a uniform polynomial bound of the degrees of irrationality for all moduli spaces of polarized K3s. 20 page

Controlled Heterogeneous Nucleation and Growth of Germanium Quantum Dots on Nanopatterned Silicon Dioxide and Silicon Nitride Substrates

Controlled heterogeneous nucleation and growth of Ge quantum dots (QDs) are demonstrated on SiO_2/Si_3N_4 substrates by means of a novel fabrication process of thermally oxidizing nanopatterned SiGe layers. The otherwise random self-assembly process for QDs is shown to be strongly influenced by the nanopatterning in determining both the location and size of the QDs. Ostwald ripening processes are observed under further annealing at the oxidation temperature. Both nanopattern oxidation and Ostwald ripening offer additional mechanisms for lithography for controlling the size and placement of the QDs

New rational cubic fourfolds arising from Cremona transformations

Are Fourier--Mukai equivalent cubic fourfolds birationally equivalent? We obtain an affirmative answer to this question for very general cubic fourfolds of discriminant 20, where we produce birational maps via the Cremona transformation defined by the Veronese surface. Moreover, by studying how these maps act on the cubics known to be rational, we found new rational examples.Comment: 42 pages. Comments are welcome