A remark on some punctual Quot schemes on smooth projective curves

We show that any Fano fivefold with canonical Gorenstein singularities has an effective anticanonical divisor. Moreover, if a general element of the anticanonical system is reduced, then it has canonical singularities.

If the Hodge conjecture holds for some generic (in the sense of Weil) geometric fibre of an Abelian scheme over a smooth projective curve , then numerical equivalence of algebraic cycles on coincides with homological equivalence. The Hodge conjecture for all complex Abelian varieties is equivalent to the standard conjecture of Lefschetz type on the algebraicity of the Hodge operator for all Abelian schemes over smooth projective curves. We investigate some properties of the Gauss-Manin connection and Hodge bundles associated with Abelian schemes over smooth projective curves, with applications to the conjectures of Hodge and Tate.

We classify three-dimensional Fano varieties with canonical Gorenstein singularities whose degree is greater than 64.Bibliography: 32 titles.

Fano threefolds with canonical Gorenstein singularities are considered. The sharp bound for their degree is proved.

We provide a complete classification of Fano threefolds $$X$$ having canonical Gorenstein singularities and anticanonical degree $$(-K_{X})^{3} = 64$$ .

We classify Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3-folds with canonical Gorenstein singularities which are not intersections of quadrics. We also study rationality questions for most of these varieties.

Equivariant approach to weighted projective curves

Let C be a smooth projective curve over ℂ. Let n,d≥1. Let 𝒬 be the Quot scheme parameterizing torsion quotients of the vector bundle 𝒪 C n of degree d. In this article we study the nef cone of 𝒬. We give a complete description of the nef cone in the case of elliptic curves. We compute it in the case when d=2 and C very general, in terms of the nef cone of the second symmetric product of C. In the case when n≥d and C very general, we give upper and lower bounds for the Nef cone. In general, we give a necessary and sufficient criterion for a divisor on 𝒬 to be nef.

Let X be an irreducible smooth complex projective curve of genus g, with g ≥ 2. Let N be a connected component of the moduli space of semistable principal PGLr (ℂ)-bundles over X; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of N is trivial.

Let X be a normal projective variety over the complex number field C. We call X a Fano variety if X is Q-Gorenstein and the anti-canonical divisor — Kx is ample. A Fano variety X is said to be a log Fano variety if X has only log terminal singularities (cf. [6]). A Fano variety X is called a canonical Fano variety if X has only canonical singularities (cf. [6]). The Cartier index c{X) is the smallest positive integer such that c{X)Kx is a Cartier divisor. The Fano index, denoted by r(X), is the largest positive rational number such that —Kχ~~Q r{X)H (Q-linear equivalence) for a Cartier divisor H. This note consists of two sections. In §1, we shall consider canonical Fano 3-folds and prove the following:

The Langlands program is a vast, loosely connected, collection of theorems and conjectures. At quite different ends, there is the geometric Langlands program, which deals with perverse sheaves on the stack of G -bundles on a smooth projective curve, and the local Langlands program over p -adic fields, which deals with the representation theory of p -adic groups. Recently, inspired by applications to p-adic Hodge theory, Fargues and Fontaine have associated with any p -adic field an object that behaves like a smooth projective curve. Fargues then suggested that one can interpret the geometric Langlands conjecture on this curve, to give a new approach towards the local Langlands program over p -adic fields.

In this paper we construct a coarse moduli scheme of stable unramified irregular singular parabolic connections on a smooth projective curve and prove that the constructed moduli space is smooth and has a symplectic structure. Moreover we will construct the moduli space of generalized monodromy data coming from topological monodromies, formal monodromies, links and Stokes data associated to the generic irregular connections. We will prove that for a generic choice of generalized local exponents, the generalized Riemann-Hilbert correspondence from the moduli space of the connections to the moduli space of the associated generalized monodromy data gives an analytic isomorphism. This shows that differential systems arising from (generalized) isomonodromic deformations of corresponding unramified irregular singular parabolic connections admit geometric Painlev\'e property as in the regular singular cases proved generally in [8].

Let H be the upper half plane and F a discrete subgroup of AutH. When H mo d F is compact, one knows that the moduli space of unitary representations of F has an algebraic interpretation (cf. [7] and [10]); for example, if moreover F acts freely on H, the set of isomorphism classes of unitary representations of F can be identified with the set of equivalence classes of semi-stable vector bundles of degree zero on the smooth projective curve H modF, under a certain equivalence relation. The initial motivation for this work was to extend these considerations to the case when H m od F has finite measure. Suppose then that H modF has finite measure. Let X be the smooth projective curve containing H modF as an open subset and S the finite subset of X corresponding to parabolic and elliptic fixed points under F. Then to interpret algebraically the moduli of unitary representation of F, we find that the problem to be considered is the moduli of vector bundles V on X, endowed with additional structures, namely flags at the fibres of V at PeS. We call these quasi parabolic structures of V at S and, if in addition we attach some weights to these flags, we call the resulting structures parabolic structures on V at S (cf. Definition 1.5). The importance of attaching weights is that this allows us to define the notion of a parabolic degree (generalizing the usual notion of the degree of a vector bundle) and consequently the concept of parabolic semi-stable and stable vector bundles (generalizing Mumford's definition of semi-stable and stable vector bundles). With these definitions one gets a complete generalization of the results of [7, 10, 12] and in particular an algebraic interpretation of unitary representations of F via parabolic semi-stable vector bundles on X with parabolic structures at S (cf, Theorem 4.1). The basic outline of proof in this paper is exactly the same as in [12], however, we believe, that this work is not a routine generalization. There are some new aspects and the following are perhaps worth mentioning. One is of course the idea of parabolic structures; this was inspired by the work of Weil (cf. [16], p. 56). The second is a technical one but took some time to arrive at, namely when one

Let k be an algebraically closed field of characteristic p>0. We study obstructions to lifting to characteristic 0 the faithful continuous action φ of a finite group G on k[[t]]. To each such φ a theorem of Katz and Gabber associates an action of G on a smooth projective curve Y over k. We say that the KGB obstruction of φ vanishes if G acts on a smooth projective curve X in characteristic 0 in such a way that X/H and Y/H have the same genus for all subgroups H⊂G. We determine for which G the KGB obstruction of every φ vanishes. We also consider analogous problems in which one requires only that an obstruction to lifting φ due to Bertin vanishes for some φ, or for all sufficiently ramified φ. These results provide evidence for the strengthening of Oort’s lifting conjecture which is discussed in [8, Conj. 1.2].

We reduce the Hodge conjecture for Abelian varieties to the question of the existence of an algebraic isomorphism for all and all principally polarized complex Abelian schemes of relative dimension over smooth projective curves. If the canonically defined Hodge cycles are algebraic for all integers , then the Grothendieck standard conjecture on the algebraicity of the operators and holds for . We prove for an Abelian scheme under the assumption that for some geometric fibre of non-exceptional dimension.