Related papers: Hodge-index type inequalities, hyperbolic polynomi…
We prove a coherence theorem for invertible objects in a symmetric monoidal category. This is used to deduce associativity, skew-commutativity, and related results for multi-graded morphism rings, generalizing the well-known versions for…
We get a new inequality on the Hodge number $h^{1,1}(S)$ of fibred algebraic complex surfaces $S$, which is a generalization of an inequality of Beauville. Our inequality implies the Arakelov type inequalities due to Arakelov, Faltings,…
Let $Y$ be a projective submanifold of the total space of the inverse of a very ample line bundle $\pi:L^{-1}\rightarrow B$ over a projective manifold $B$. Any section of $L^{-1}\rightarrow B$ is isomorphic to $B$ and the Hodge numbers of…
Motivated by a question of Baldi-Klingler-Ullmo, we provide a general sufficient criterion for the existence and analytic density of typical Hodge loci associated to a polarizable $\mathbb{Z}$-variation of Hodge structures $\mathbb{V}$. Our…
A new hyperelliptic solution class for the hyperbolic Ernst equation is obtained by transforming the regarding solution of the elliptic Ernst equation. Furthermore, a nontrivial way for obtaining general polarized colliding wave solutions…
In this paper, we establish an innovative framework in logarithmic Hodge theory for toroidal varieties, introducing weighted toroidal structures and developing a systematic obstruction theory for Hodge classes. Building upon recent advances…
Majorization inequalities for symmetric polynomials have interested mathematicians for centuries, from the AM-GM inequality for two variables going back at least to Euclid, through classical results of Newton, Muirhead and Gantmacher, to…
We prove the structure theorem of the intersection complexes of toric varieties in the category of mixed Hodge modules. This theorem is due to Bernstein, Khovanskii and MacPherson for the underlying complexes with rational coefficients. As…
We study homomorphisms of Hecke monoids, notably parabolic homomorphisms, which map parabolic elements to parabolic elements, and injective ones. The importance of the first class stems from the fact that parabolic elements form a rather…
Using the $\infty$-categorical enhancement of mixed Hodge modules constructed by the author in a previous paper, we explain how mixed Hodge modules canonically extend to algebraic stacks, together with all the $6$ operations and weights. We…
Invariants of general linear system of two hyperbolic partial differential equations (PDEs) are derived under transformations of the dependent and independent variables by real infinitesimal method earlier. Here a subclass of the general…
In this paper, we explore a notion of nonabelian Hodge structure on the fundamental group of an algebraic variety. This is approach is compared to some alternative approaches due to Morgan, Hain and others. We also give criteria for a…
We consider the Hardy-Littlewood-Sobolev inequality on mixed-norm Lebesgue spaces. We give a complete characterization of indices $\vec p$ and $\vec q$ such that the Riesz potential is bounded from $L^{\vec p}$ to $L^{\vec q}$, including…
We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real-rank-one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic…
In one of the fundamental results of Arakelov's arithmetic intersection theory, Faltings and Hriljac (independently) proved the Hodge Index Theorem for arithmetic surfaces by relating the intersection pairing to the negative of the…
Given a set of inequalities determined by homogeneous forms, the following intertwined results are established: (1) the volume of the real semi-algebraic domain determined by these inequalities is explicitly determined; it is shown to be…
Many combinatorial proofs rely on induction. When these proofs are formulated in traditional language, they can be bulky and unmanageable. Coalgebras provide a language which can reduce reduce many inductive proofs in graded poset theory to…
Building on the nonabelian Hodge theory in positive characteristic developed by Ogus, Vologodsky, and Schepler, we propose a generalization of the decomposition theorem of Deligne and Illusie from the perspective of mixed Hodge modules.…
In this paper we improve a result recently proved by Irshad et al. [On the Inequalities Concerning to the Polar Derivative of a Polynomial with Restricted Zeroes, Thai Journal of Mathematics, 2014 (Article in Press)] and also extend…
Recent discoveries make it possible to compute the K-theory of certain rings from their cyclic homology and certain versions of their cdh-cohomology. We extend the work of G. Corti\~nas et al. who calculated the K-theory of, in addition to…