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Register a new userVarieties of general type with many vanishing plurigenera, and optimal sine and sawtooth inequalities
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We construct smooth projective varieties of general type with the smallest known volumes in high dimensions. Among other examples, we construct varieties of general type with many vanishing plurigenera, more than any polynomial function of the dimension. As part of the construction, we solve exactly an optimization problem about equidistribution on the unit circle in terms of the sawtooth (or signed fractional part) function. We also solve exactly the analogous optimization problem for the sine function. Equivalently, we determine the optimal inequality of the form $sum_{k=1}^m a_ksin kxleq 1$, in the sense that $sum_{k=1}^m a_k$ is maximal.
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Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show that the Lang-Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded.
We continue our study on smooth complex projective varieties $X$ of maximal Albanese dimension and of general type satisfying $chi(X, omega_X)=0$. We formulate a conjectural characterization of such varieties and prove this conjecture when the Albanese variety has only three simple factors.
Let $X$ be a nonsingular projective $n$-fold $(nge 2)$ of Fano or of general type with ample canonical bundle $K_X$ over an algebraic closed field $kappa$ of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes $c_1, c_2, cdots, c_n$ by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. Moreover, we show that if the characteristic of $kappa$ is $0$, then the Chern ratios $(frac{c_{2,1^{n-2}}}{c_{1^n}}, frac{c_{2,2,1^{n-4}}}{c_{1^n}}, cdots, frac{c_{n}}{c_{1^n}})$ are contained in a convex polyhedron for all $X$. So we give an affirmative answer to a generalized open question, that whether the region described by the Chern ratios is bounded, posted by Hunt (cite{Hun}) to all dimensions. As a corollary, we can get that there exist constants $d_1$, $d_2$, $d_3$ and $d_4$ depending only on $n$ such that $d_1K_X^nlechi_{top}(X)le d_2 K_X^n$ and $d_3K_X^nlechi(X, mathscr{O}_X)le d_4 K_X^n$. If the characteristic of $kappa$ is positive, $K_X$ (or $-K_X$) is ample and $mathscr{O}_X(K_X)$ ($mathscr{O}_X(-K_X)$, respectively) is globally generated, then the same results hold.
Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a rational variation of Hodge structures of weight one on Y with Higgs bundle E(1,0) + E(0,1), coming from a family of Abelian varieties. If Y is a curve the Arakelov inequality says that the difference of the slope of E(1,0) and the one of E(0,1) is is smaller than or equal to the degree of the canonical sheaf. We prove a similar inequality in the higher dimensional case. If the latter is an equality, as well as the Bogomolov inequality for E(1,0) or for E(0,1), one hopes that Y is a Shimura variety, and V a uniformizing variation of Hodge structures. This is verified, in case the universal covering of Y does not contain factors of rank >1. Part of the results extend to variations of Hodge structures over quasi-projective manifolds. The revised version corrects several mistakes and ambiguities, pointed out by the referee. Following suggestions of the referee the presentation of the results was improved.
We study quantum geometry of Nakajima quiver varieties of two different types - framed A-type quivers and ADHM quivers. While these spaces look completely different we find a surprising connection between equivariant K-theories thereof with a nontrivial match between their equivariant parameters. In particular, we demonstrate that quantum equivariant K-theory of $A_n$ quiver varieties in a certain $ntoinfty$ limit reproduces equivariant K-theory of the Hilbert scheme of points on $mathbb{C}^2$. We analyze the correspondence from the point of view of enumerative geometry, representation theory and integrable systems. We also propose a conjecture which relates spectra of quantum multiplication operators in K-theory of the ADHM moduli spaces with the solution of the elliptic Ruijsenaars-Schneider model.
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