Related papers: The Kawamata-Morrison Cone Conjecture for Generali…

We make precise conjectures relating the genus zero Gromov-Witten theory of a nonabelian GIT quotient X//G to that of the associated abelian quotient X//T by a maximal torus T in G.These conjectures imply in particular closed formulas for…

We give examples over arbitrary fields of rings of invariants that are not finitely generated. The group involved can be as small as three copies of the additive group, as in Mukai's examples over the complex numbers. The failure of finite…

The Bounded Height Conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian $\overline{\mathbb{Q}}$-variety $G$ there is an upper bound on the Weil height of the points…

We propose a geometric and categorical approach to the Hodge Conjecture for all smooth projective complex varieties. By embedding any such variety into a flat family with general fibers smooth complete intersections, we prove the conjecture…

We prove Viehweg's hyperbolicity conjecture over compact bases and over bases with non-uniruled compactification. The most general case of the conjecture states that the the base space of a maximal variation family of smooth projective…

In this article, we study the $G$-autoequivalences of the derived category $\mathbf{D}^b_G(A)$ of $G$-equivariant objects for an abelian variety $A$ with $G$ being a finite subgroup of $\mathrm{Pic}^0(A)$. We provide a result analogue to…

The aim of this survey paper is threefold: (a) to discuss the status of the Morrison-Kawamata cone conjecture, (b) to report on recent developments towards the Abundance Conjecture, and (c) to discuss the nef line bundle version of the…

In two earlier articles, we proved that, if the Hodge conjecture is true for ALL CM abelian varieties over the complex numbers, then both the Tate conjecture and the standard conjectures are true for abelian varieties over finite fields.…

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of…

The study of entire holomorphic curves contained in projective algebraic varieties is intimately related to fascinating questions of geometry and number theory -- especially through the concepts of curvature and positivity which are central…

For a Calabi-Yau manifold $X$, the Kawamata - Morrison movable cone conjecture connects the convex geometry of the movable cone $\overline{\mathrm{Mov}}(X)$ to the birational automorphism group. Using the theory of Coxeter groups, Cantat…

To any finite group G in SL_2(C), and each `t' in the center of the group algebra of G, we associate a category, Coh_t. It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed…

A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic terms of semi-simple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torus-equivariant Gromov -…

McMullen's g-vector is important for simple convex polytopes. This paper postulates axioms for its extension to general convex polytopes. It also conjectures that, for each dimension d, a stated finite calculation gives the formula for the…

In the classical case of irreducible smooth algebraic curves every genus $2$ curve is hyperelliptic, or in other words there is a complete linear series $g_2^1$ on them. On the other hand if $g > 2$, then a generic smooth curve of genus $2$…

In this paper, we show that the GVC (generalized vanishing conjecture) holds for the differential operator $\Lambda=(\partial_x-\Phi(\partial_y))\partial_y$ and all polynomials $P(x,y)$, where $\Phi(t)$ is any polynomial over the base…

In this article we propose a vanishing conjecture for a certain class of $\ell$-adic complexes on a reductive group $G$, which can be regraded as a generalization of the acyclicity of the Artin-Schreier sheaf. We show that the vanishing…

We initiate the study of the asymptotic topology of groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers (these are called here as holomorphically convex…

Let $K$ be a field finitely generated over ${\Q}$, and $A$ an Abelian variety defined over $K$. Then by the Mordell-Weil Theorem, the set of rational points $A(K)$ is a finitely-generated Abelian group. In this paper, assuming Tate's…

We prove the Arveson-Douglas essential normality conjecture for graded Hilbert submodules that consist of functions vanishing on a given homogeneous subvariety of the ball, smooth away from the origin. Our main tool is the theory of…