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Register a new userOn accumulation points of pseudo-effective thresholds
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We characterize a $k$-th accumulation point of pseudo-effective thresholds of $n$-dimensional varieties as certain invariant associates to a numerically trivial pair of an $(n-k)$-dimensional variety. This characterization is applied towards Fujitas log spectrum conjecture for large $k$.
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Let $mathcal Csubset(0,1]$ be a set satisfying the descending chain condition. We show that any accumulation point of volumes of log canonical surfaces $(X, B)$ with coefficients in $mathcal C$ can be realized as the volume of a log canonical surface with big and nef $K_X+B$ and coefficients in $overline{mathcal C}cup{1}$, with at least one coefficient in $Acc(mathcal C)cup{1}$. As a corollary, if $overline{mathcal C}subsetmathbb Q$ then all accumulation points of volumes are rational numbers, solving a conjecture of Blache. For the set of standard coefficients $mathcal C_2={1-frac{1}{n}mid ninmathbb N}cup{1}$ we prove that the minimal accumulation point is between $frac1{7^2cdot 42^2}$ and $frac1{42^2}$.
Given a morphism between complex projective varieties, we make several conjectures on the relations between the set of pseudo-effective (co)homology classes which are annihilated by pushforward and the set of classes of varieties contracted by the morphism. We prove these conjectures for classes of curves or divisors. We also prove that one of these conjectures implies Grothendiecks generalized Hodge conjecture for varieties with Hodge coniveau at least 1.
Let $pi: X to Y$ be a morphism of projective varieties and suppose that $alpha$ is a pseudo-effective numerical cycle class satisfying $pi_*alpha = 0$. A conjecture of Debarre, Jiang, and Voisin predicts that $alpha$ is a limit of classes of effective cycles contracted by $pi$. We establish new cases of the conjecture for higher codimension cycles. In particular we prove a strong version when $X$ is a fourfold and $pi$ has relative dimension one.
We calculate resonances in three-body systems with attractive Coulomb potentials by solving the homogeneous Faddeev-Merkuriev integral equations for complex energies. The equations are solved by using the Coulomb-Sturmian separable expansion approach. This approach provides an exact treatment of the threshold behavior of the three-body Coulombic systems. We considered the negative positronium ion and, besides locating all the previously know $S$-wave resonances, we found a whole bunch of new resonances accumulated just slightly above the two-body thresholds. The way they accumulate indicates that probably there are infinitely many resonances just above the two-body thresholds, and this might be a general property of three-body systems with attractive Coulomb potentials.
Motivated by results of Thurston, we prove that any autoequivalence of a triangulated category induces a filtration by triangulated subcategories, provided the existence of Bridgeland stability conditions. The filtration is given by the exponential growth rate of masses under iterates of the autoequivalence, and only depends on the choice of a connected component of the stability manifold. We then propose a new definition of pseudo-Anosov autoequivalences, and prove that our definition is more general than the one previously proposed by Dimitrov, Haiden, Katzarkov, and Kontsevich. We construct new examples of pseudo-Anosov autoequivalences on the derived categories of quintic Calabi-Yau threefolds and quiver Calabi-Yau categories. Finally, we prove that certain pseudo-Anosov autoequivalences on quiver 3-Calabi-Yau categories act hyperbolically on the space of Bridgeland stability conditions.
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