Related papers: The period-index problem for hyperk\"ahler manifol…

Let $X$ be a hyperk\"ahler variety, and assume that $X$ admits a non-symplectic automorphism $\sigma$ of order $k>{1\over 2}\dim X$. Bloch's conjecture predicts that the quotient $X/<\sigma>$ should have trivial Chow group of $0$-cycles. We…

Hilbert-Arnold (HA) problem, motivated by Hilbert 16-th problem, is to prove that for a generic k-parameter family of smooth vector fields {\dot x=v(x,\eps)}_{\eps\in B^k} on the 2-dimensional sphere S^2 has uniformly bounded number of…

We give a negative answer to a question posed by Severi in 1951, whether the Abelian Varieties are the only projective manifolds with trivial Chern classes. By Yau' s celebrated result, compact K\"ahler manifolds with trivial Chern classes…

We adapt Bost's algebraicity characterization to the situation of a germ in a compact K\"ahler manifold. As a consequence, we extend the algebraic integrability criteria of Campana-P\u{a}un and of Druel to foliations on compact K\"ahler…

We show that for complex analytic K3 surfaces any torsion class in H^2(X,O_X^*) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to…

This note gives an explicit example of transcendental Brauer-Manin obstruction to weak approximation. It has two features which the only previously known example of such obstruction did not have: the class in the Brauer group which is…

Let $k$ be a number field and let $\pi \colon X \rightarrow\mathbb{P}_k^1$ be a smooth conic bundle. We show that if $X/k$ has four geometric singular fibers and either $X(\mathbb{A}_k)\neq \emptyset$ or $X/k$ has non-trivial Brauer group,…

The ``complex Plateau problem'' (or boundary problem) in a complexe manifold X is the problem of characterizing the real submanifolds $\Gamma$ of X which are boundaries of analytic sub-varieties of $X \backslash \Gamma$. Our principal…

We show how to evaluate the periods in Seiberg-Witten theories and in K3-fibered Calabi-Yau manifolds by using fibrations of the theories. In the Seiberg-Witten theories, it is shown that the dual pair of fields can be constructed from the…

We study weakly symmetric special biserial algebras of infinite representation type. We show that usually some socle deformation of such an algebra has non-periodic bounded modules. The exceptions are precisely the algebras whose Brauer…

Let $C$ be a smooth projective curve of genus 2 over a number field $k$ with a rational point. We prove that the index and exponent coincide for elements in the 2-torsion of $\Sha(Br(C))$. In the appendix, an isomorphism of the moduli space…

Given a smooth function $U(t,x)$, $T$-periodic in the first variable and satisfying $U(t,x) = \mathcal{O}(\vert x \vert^{\alpha})$ for some $\alpha \in (0,2)$ as $\vert x \vert \to \infty$, we prove that the forced Kepler problem $$ \ddot x…

We prove a theorem which asserts that the Lie algebra of all holomorphic vector fields on a compact K\"ahler manifold with a perturbed extremal metric has the structure similar to the case of an unperturbed extremal K\"ahler metric proved…

After reformulate the incompressible Euler-$\alpha$ equations in 3D smooth domain with Drichlet data, we obtain the unique classical solutions to Euler-$\alpha$ equations exist in uniform time interval independent of $\alpha$. We also show…

In this note, we discuss unpolarized, complex variation of Hodge structures for non-K\"ahler manifolds. In particular, given a holomorphic family of compact complex manifolds whose central fiber satisfies: the inclusions…

Given a Lagrangian fibration $\pi : X \to \mathbb{P}^n$ of a compact hyper-K\"ahler manifold of $\text{K3}^{[n]}$, $\text{Kum}_n$, $\text{OG10}$ or $\text{OG6}$-type, we construct a natural compactification of its dual torus fibration.…

We introduce the notion of a Hyper-K\"{a}hler manifold $X$ induced by a Hodge structure of K3-type. We explore this notion for the known deformation types of hyper-K\"{a}hler manifolds studying those that are induced by a K3 or abelian…

Given a compact K\"ahler manifold, the Infinitesimal Torelli problem asks whether the differential of the period map of a Kuranishi family is injective. Unlike the classical Torelli theorem for curves, there is a negative answer for example…

Sometime ago, we showed that a pure Artin braid group is not K\"ahler, i.e. it is not the fundamental group of a compact K\"ahler manifold. This used a result of Bressler, Ramachandran and the author that K\"ahler groups cannot be too…

We prove that the class of Brauer graph algebras coincides with the class of indecomposable idempotent algebras of biserial weighted surface algebras. These algebras are associated to triangulated surfaces with arbitrarily oriented…