We introduce tautological system defined by prehomogenous actions of reductive algebraic groups. If the complement of the open orbit is a linear free divisor satisfying a certain finiteness condition, we show that these systems underly mixed Hodge modules. A dimensional reduction is considered and gives rise to one-dimensional differential systems generalizing the quantum differential equation of projective spaces.
We show that the vanishing of the $(g+1)$-st power of the theta divisor in the cohomology and Chow rings of the universal abelian variety implies, by pulling back along a collection of Abel-Jacobi maps, the vanishing results in the tautological ring of $\mathcal{M}_{g,n}$ of Looijenga, Ionel, Graber-Vakil, and Faber-Pandharipande. We also show that Pixton's double ramification cycle relations, which generalize the theta vanishing relations and were recently proved by the first and third authors, imply Theorem $\star$ of Graber and Vakil. Moreover, our proof provides an algorithm for expressing…
Chern Classes Of Logarithmic Vector Fields For Locally-Homogenous Free Divisors
设$X$是非奇异复射影簇，$D$是$X$中的局部拟齐次自由因子.本文研究了$X$上对数导子层关于$D$的Chern类与$X$中$D$的补的Chern-Schwartz-MacPherson类之间的数值关系.我们的结果证实了这些类的一个代数公式，至少在推进到射影空间之后;它证明了$\mathbb P^n$中局部拟齐次自由因子猜想的完整形式。所得结果推广了几个已知的结果。例如，它恢复了M的公式。Mustata和H.自由超平面安排的Schenck for Chern类。我们的主要工具是Riemann-Roch和Calderon-Moreno，Castro-Jimenez，Narvaez-Macarro和David Mond的对数比较定理。作为主要论点的一个子积，我们还获得了局部拟齐次因子的示意性Bertini陈述。
We introduce the concept of directed orbifold, namely triples (X, V, D) formed by a directed algebraic or analytic variety (X, V), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle TX. In this context, we introduce an algebra of orbifold jet differentials and their sections. These jet sections can be seen as algebraic differential operators acting on germs of curves, with meromorphic coefficients, whose poles are supported by D and multiplicities are bounded by the ramification indices of the components of D. We estimate precisely the curvature tensor of the co…
We consider the Hodge filtration on the sheaf of meromorphic functions along free divisors for which the logarithmic comparison theorem holds. We describe the Hodge filtration steps as submodules of the order filtration on a cyclic presentation in terms of a special factor of the Bernstein-Sato polynomial of the divisor and we conjecture a bound for the generating level of the Hodge filtration. Finally, we develop an algorithm to compute Hodge ideals of such divisors and we apply it to some examples.
We prove that any nef b-divisor class on a projective variety defined over an algebraically closed field of characteristic 0 is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef b-divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony. We show that any big and basepoint free curve class is a power of a nef b-divisor, and relate this statement to Zariski decompositions of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces …
Given a finite, flat morphism between embeddable noetherian schemes of pure dimension 1, we define the notion of direct and inverse image for generalized divisors and generalized line bundles. In the case when we deal with (possibly reducible, non-reduced) projective curves over a field and the codomain curve is smooth, we introduce the compactified Jacobians parametrizing torsion-free rank-1 sheaves and we study the Norm and inverse image maps between compactified Jacobians. Finally, we introduce and study the Prym stack defined as the kernel of the Norm map.
We define the principal divisor of a free noncommuatative function. We use these divisors to compare the determinantal singularity sets of free noncommutative functions. We show that the divisor of a noncommutative rational function is the difference of two polynomial divisors. We formulate a nontrivial theory of cohomology, fundamental groups and covering spaces for tracial free functions. We show that the natural fundamental group arising from analytic continuation for tracial free functions is a direct sum of copies of $\mathbb{Q}$. Our results contrast the classical case, where the analogo…
