Bott manifolds with vanishing Futaki invariants for all K\"ahler classes
We prove that the only Bott manifolds such that the Futaki invariant vanishes for any K\"ahler class are isomorphic to the products of the projective lines.
math.AG↗
arXiv subjects
Yuji Sano
Publications and source records attributed to Yuji Sano.
Characteristic vectors for the Hurwitz polytopes of toric varieties
We introduce a characteristic vector with respect to a regular triangulation of the momentum polytope to compute the Hurwitz polytope of a given smooth toric variety. As a result, we prove that the convex hull of such vectors of all regular triangulations is included in the Hurwitz polytope of a smooth toric surface. In addition, we discuss the relations of such vectors to K-stability of pairs by Paul and toric K-stability by Donaldson.
math.AG↗
Weight polytopes and energy functionals of toric varieties
We prove that the weight polytope of the Hurwitz form of a polarized smooth toric variety coincides with the convex hull of the characteristic vectors introduced in the previous work of Ogusu and the author with respect to all regular triangulations of the momentum polytope. Our proof relies on the combination of the two slope formulas of K-energy in the toric setting.
math.AG↗
Experimentally shock-induced melt veins in basalt: Improving the shock classification of eucrites
Basaltic rocks occur widely on the terrestrial planets and differentiated asteroids, including the asteroid 4 Vesta. We conducted a shock recovery experiment with decaying compressive pulses on a terrestrial basalt at Chiba Institute of Technology, Japan. The sample recorded a range of pressures, and shock physics modeling was conducted to add a pressure scale to the observed shock features. The shocked sample was examined by optical and electron microscopy, electron back-scattered diffractometry, and Raman spectroscopy. We found that localized melting occurs at a lower pressure (~10 GPa) than previously thought (>20 GPa). The shocked basalt near the epicenter represents shock degree C of a recently proposed classification scheme for basaltic eucrites and, as such, our results provide a pressure scale for the classification scheme. Finally, we estimated the total fraction of the basaltic eucrites classified as shock degree C to be ~15% by assuming the impact velocity distribution onto Vesta.
astro-ph.EP↗
Laboratory measurements of stretching band strengths of deuterated Quenched Carbonaceous Composites (D-QCC)
The observed large variation in the abundance of deuterium (D) in the interstellar medium (ISM) suggests that a significant fraction of D may be depleted into polycyclic aromatic hydrocarbons (PAHs). Signatures of deuteration of PAHs are expected to appear most clearly through C-D stretching modes at 4.4--4.7 micron, whose strengths in emission spectra relative to those of C-H stretching modes at 3.3--3.5 micron provide the relative abundance of D to hydrogen (H) in PAHs once we have accurate relative band strengths of both stretching modes. We report experimental results of the band strength of C-D stretching modes relative to C-H. We employ a laboratory analogue of interstellar carbonaceous dust, Quenched Carbonaceous Composite (QCC), and synthesize deuterated QCC (D-QCC) by replacing the starting gas of CH$_4$ of QCC by mixtures of CH$_4$ and CD$_4$ with various ratios. Infrared spectra of D-QCC are taken to estimate the relative band strengths of the stretching modes, while the D/H ratios in the D-QCC samples are measured with a nano-scale secondary ion mass spectrometer (NanoSIMS). We obtain that the relative strength of aromatic and aliphatic C-D to C-H stretches is 0.56 +/- 0.04 and 0.38 +/- 0.01 per D/H, respectively. The ratio for the aromatic stretches is in good agreement with the results of theoretical calculations, while that of aliphatic stretches is smaller than that of aromatic. The present results do not significantly change the D/H ratios in the interstellar PAHs previously estimated from observed spectra.
astro-ph.GA↗
Shock recovery with decaying compressive pulses: Shock effects in calcite (CaCO$_3$) around the Hugoniot elastic limit
Shock metamorphism of minerals in meteorites provides insights into the ancient Solar System. Calcite is an abundant aqueous alteration mineral in carbonaceous chondrites. Return samples from the asteroids Ryugu and Bennu are expected to contain calcite-group minerals. Although shock metamorphism in silicates has been well studied, such data for aqueous alteration minerals are limited. Here, we investigated the shock effects in calcite with marble using impact experiments at the Planetary Exploration Research Center of Chiba Institute of Technology. We produced decaying compressive pulses with a smaller projectile than the target. A metal container facilitates recovery of a sample that retains its pre-impact stratigraphy. We estimated the peak pressure distributions in the samples with the iSALE shock physics code. The capability of this method to produce shocked grains that have experienced different degrees of metamorphism from a single experiment is an advantage over conventional uniaxial shock recovery experiments. The shocked samples were investigated by polarizing microscopy and X-ray diffraction analysis. We found that more than half of calcite grains exhibit undulatory extinction when peak pressure exceeds 3 GPa. This shock pressure is one order of magnitude higher than the Hugoniot elastic limit (HEL) of marble, but it is close to the HEL of a calcite crystal, suggesting that the undulatory extinction records dislocation-induced plastic deformation in the crystal. Finally, we propose a strategy to re-construct the maximum depth of calcite grains in a meteorite parent body, if shocked calcite grains are identified in chondrites and/or return samples from Ryugu and Bennu.
astro-ph.EP↗
Toric Fano manifolds of dimension at most eight with positive second Chern characters
We show that any toric Fano manifold of dimension at most eight with the positive second Chern character is isomorphic to the projective space by using polymake.
math.AG↗
Migration of D-type asteroids from the outer Solar System inferred from carbonate in meteorites
Recent dynamical models of Solar System evolution and isotope studies of rock-forming elements in meteorites have suggested that volatile-rich asteroids formed in the outer Solar System beyond Jupiter's orbit, despite being currently located in the main asteroid belt. The ambient temperature under which asteroids formed is a crucial diagnostic to pinpoint the original location of asteroids and is potentially determined by the abundance of volatiles they contain. In particular, abundances and 13C/12C ratios of carbonates in meteorites record the abundances of carbon-bearing volatile species in their parent asteroids. However, the sources of carbon for these carbonates remain poorly understood. Here we show that the Tagish Lake meteorite contains abundant carbonates with consistently high 13C/12C ratios. The high abundance of 13C-rich carbonates in Tagish Lake excludes organic matter as their main carbon source. Therefore, the Tagish Lake parent body, presumably a D-type asteroid, must have accreted a large amount of 13C-rich CO2 ice. The estimated 13C/12C and CO2/H2O ratios of ice in Tagish Lake are similar to those of cometary ice. Thus, we infer that at least some D-type asteroids formed in the cold outer Solar System and were subsequently transported into the inner Solar System owing to an orbital instability of the giant planets.
astro-ph.EP↗
A moment map picture of relative balanced metrics on extremal K\"ahler manifolds
We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature K\"ahler metrics to the case of extremal metrics. Namely, we show that a given extremal metric is the limit of some specific relatively balanced metrics. As a corollary, we recover uniqueness and splitting results for extremal metrics in the polarized case.
math.DG↗
Extremal metrics and lower bound of the modified K-energy
We provide a new proof of a result of X.X.Chen and G.Tian : for a polarized extremal K\"ahler manifold, an extremal metric attains the minimum of the modified K-energy. The proof uses an idea of C.Li adapted to the extremal metrics using some weighted balanced metrics.
math.DG↗
Alpha invariant and K-stability of Q-Fano varieties
We give a purely algebro-geometric proof that if the alpha-invariant of a Q-Fano variety X is greater than dim X/(dim X+1), then (X,O(-K_X)) is K-stable. The key of our proof is a relation among the Seshadri constants, the alpha-invariant and K-stability. It also gives applications concerning the automorphism group.
math.AG↗
Lower diameter bounds for compact shrinking Ricci solitons
It is shown that the diameter of a compact shrinking Ricci soliton has a universal lower bound. This is proved by extending universal estimates for the first non-zero eigenvalue of Laplacian on compact Riemannian manifolds with lower Ricci curvature bound to a twisted Laplacian on compact shrinking Ricci solitons.
math.DG↗
Multiplier ideal sheaves and geometric problems
In this expository article we first give an overview on multiplier ideal sheaves and geometric problems in Kählerian and Sasakian geometries. Then we review our recent results on the relationship between the support of the subschemes cut out by multiplier ideal sheaves and the invariant whose non-vanishing obstructs the existence of Kähler-Einstein metrics on Fano manifolds.
math.DG↗
An example of asymptotically Chow unstable manifolds with constant scalar curvature
Donaldson proved that if a polarized manifold $(V,L)$ has constant scalar curvature K\"ahler metrics in $c_1(L)$ and its automorphism group Aut$(M,L)$ is discrete, $(V,L)$ is asymptotically Chow stable. In this paper, we shall show an example which implies that the above result does not hold in the case when Aut$(V,L)$ is not discrete.
math.DG↗
Multiplier ideal sheaves and the Kähler-Ricci flow on toric Fano manifolds with large symmetry
The purpose of this paper is to calculate the support of the multiplier ideal sheaves derived from the Kähler-Ricci flow on certain toric Fano manifolds with large symmetry. The early idea of this paper has already been in Appendix of \cite{futaki-sano0711}.
math.DG↗
Hilbert series and obstructions to asymptotic semistability
Given a polarized manifold there are obstructions for asymptotic Chow semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar curvature K\"ahler (cscK) metric. A natural question is whether or not the other obstructions are linearly dependent on the obstruction to the existence of a cscK metric. The purpose of this paper is to see that this is not the case by exhibiting toric Fano threefolds in which these obstructions span at least two dimension. To see this we show that on toric Fano manifolds these obstructions are obtained as derivatives of the Hilbert series. This last observation should be regarded as an extension of the volume minimization of Martelli, Sparks and Yau.
math.DG↗
Multiplier ideal sheaves and integral invariants on toric Fano manifolds
We extend Nadel's results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author. Applying our extension we can determine the multiplier ideal sheaves on toric del Pezzo surfaces which do not admit Kähler-Einstein metrics. We also show that one can define multiplier ideal sheaves for Kähler-Ricci solitons and extend the result of Nadel using the holomorphic invariant defined by Tian and Zhu.
math.DG↗