An application of exceptional bundles to the moduli of stable sheaves on a K3 surface

Let M(v) be the moduli of stable sheaves on K3 surfaces X of Mukai vector v. If v is primitive, than it is expected that M(v) is deformation equivalent to some Hilbert scheme and weight two hogde structure can be described by H^*(X,Z). These are known by Mukai, O'Grady and Huybrechts if rank is 1 or 2, or the first Chern class is primitive. Under some conditions on the dimension of M(v), we shall show that these assertion are true. For the proof, we shall use Huybrechts's results on symplectic manifolds.Comment: 12 pages, AMS-Late

The volume growth of hyperkaehler manifolds of type A_{\infty}

We study the volume growth of hyperkaehler manifolds of type $A_{\infty}$ constructed by Anderson-Kronheimer-LeBrun and Goto. These are noncompact complete 4-dimensional hyperkaehler manifolds of infinite topological type. These manifolds have the same topology but the hyperkaehler metrics are depends on the choice of parameters. By taking a certain parameter, we show that there exists a hyperkaehler manifold of type $A_{\infty}$ whose volume growth is r^a for each 3<a<4

Construction of a one-dimensional set which asymptotically and omnidirectionally contains arithmetic progressions

In this paper, we construct a subset of $\mathbb{R}^d$ which asymptotically and omnidirectionally contains arithmetic progressions but has Assouad dimension 1. More precisely, we say that $F$ asymptotically and omnidirectionally contains arithmetic progressions if we can find an arithmetic progression of length $k$ and gap length $\Delta>0$ with direction $e\in S^{d-1}$ inside the $\epsilon \Delta$ neighbourhood of $F$ for all $\epsilon>0$, $k\geq 3$ and $e\in S^{d-1}$. Moreover, the dimension of our constructed example is the lowest-possible because we prove that a subset of $\mathbb{R}^d$ which asymptotically and omnidirectionally contains arithmetic progressions must have Assouad dimension greater than or equal to 1. We also get the same results for arithmetic patches, which are the higher dimensional extension of arithmetic progressions.Comment: 7 page

Liquid Xenon detector R&D for 0ν2β0\nu2\beta search (KamXP)

The R&D for a new type of liquid xenon (LXe) detector is ongoing to search for the neutrinoless double-beta decay ($0\nu2\beta$). As a result of the KamLAND-Zen experiment, it is very important to realize the all active region detector for the energy deposition of the radiation. Newly developed additional BG reduction techniques will improve the sensitivity of $0\nu2\beta$ search in the future. Our detector concept is LXe stored in a new type of plastic scintillator vessel. The wavelength of LXe scintillation light is shifted to visible light from 175nm (VUV) on the inner surface of the vessel. Therefore, the LXe scintillation light can be detected by photon sensors which are far away from the $0\nu2\beta$ target nuclei. The pulse shape difference between LXe and plastic scintillator is used for the additional BG reduction. In the future, $^{8}$B solar neutrino events will be one of the dominant BGs in $0\nu2\beta$ search. To reduce the $^{8}$B solar neutrino BG, the directional information of the Cherenkov light might be useful. The status of LXe detector R&D to improve the sensitivity of $0\nu2\beta$ search is reported.Comment: 4 pages, 6 figures, Proceedings of the 16th International Conference on Topics in Astroparticle and Underground Physics (TAUP 2019), September 9-13, 2019, Toyama, Japa

A note on Fourier-Mukai transform

In this note, we consider the problem on the preservation of stability under the Fourier-Mukai transforms. We first show that the Fourier-Mukai transform on an abelian surface or a K3 surface does not always preserve the stability, even for a $\mu$-stable vector bundle. We next provide some positive results on this problem. Finally we discuss the birational map of moduli spaces on an abelian surface.Comment: 19 pages, some results adde

A note of moduli of vector bundles on rational surfaces

We discuss some relations of moduli of sheaves on rational surfaces by using universal extensions. These are a generalization of Maruyama's method to construct Uhlenbeck compactification of moduli of vector bundles.Comment: 11 page

Twisted stability and Fourier-Mukai transform

In this paper, we consider the preservation of stability by using the notion of Twisted stability. As applications, (1) we show that moduli spaces of vector bundles on K3 and abelian surfaces are irreducible, (2) we compute Hodge polynomials of some moduli spaces of vector bundles on Enriques surfaces and on elliptic surfaces with a section.Comment: 28 page

Albanese map of moduli of stable sheaves on abelian surfaces

Periods of moduli spaces of stable sheaves on K3 surfaces were computed by Mukai, O'Grady and the author. In this paper, we shall treat moduli spaces of stable sheaves on abelian surfaces.Comment: 10 page

The nonuniqueness of the tangent cones at infinity of Ricci-flat manifolds

It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume growth in the above result is essential. More precisely, we construct a complete Ricci-flat manifold of dimension 4 with non-Euclidean volume growth who has at least two distinct tangent cones at infinity and one of them has a smooth cross section

A fractal proof of the infinitude of primes

This short paper gives another proof of the infinitude of primes by using upper box dimension, which is one of fractal dimensions.Comment: 4 page