arXiv 1002.06982010-02math.AG

Duality and integrability on contact Fano manifolds

Jarosław Buczyński

备注：: Dedicated in memory of Marcin Hauzer. 23 pages. v2 consolidates the published article and its erratum available at https://www.mimuw.edu.pl/~jabu/CV/publications/contact_errata.pdf
发表信息：: Documenta Math. 15 (2010) 821--841

英文摘要：

We address the problem of classification of contact Fano manifolds. It is conjectured that every such manifold is necessarily homogeneous. We prove that the Killing form, the Lie algebra grading and parts of the Lie bracket can be read from geometry of an arbitrary contact manifold. Minimal rational curves on contact manifolds (or contact lines) and their chains are the essential ingredients for our constructions. Along the way we collect several facts about bundles of projective lines admitting a contractible section, which might be of independent interest.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

The Sp3-grassmannian and duality for prime Fano threefolds of genus 9

By a result of Mukai, the non-abelian Brill-Noether locus X = M_C(2,K:3F) of type II, defined by a stable rank 2 vector bundle F of invariant 3 over a plane quartic curve C, is a prime Fano 3-fold X of degree 16. The associate ruled surface S^X = P(F) is uniquely defined by X, and we see that for the general X = X_{16}, S^X is isomorphic to the Fano surface of conics on X. The argument uses the geometry of the Sp_3-grassmannian and the double projection from a line on X_{16}.

2002-11-03math.AG

Duality Related with Key Varieties of $\mathbb{Q}$-Fano 3-folds. I

Abstract. In our previous paper arXiv:2210.16008, we show that any prime $\mathbb{Q}$-Fano 3-folds $X$ with only $1/2(1,1,1)$-singularities in certain 5 classes can be embedded as linear sections into bigger dimensional $\mathbb{Q}$-Fano varieties called key varieties, where each of the key varieties is constructed from certain data of the Sarkisov link staring from the blow-up at one $1/2(1,1,1)$-singularity of $X$. In this paper, we introduce varieties associated with the key varieties which are dual in a certain sense. As an application, we interpret a fundamental part of the Sarkisov link …

2022-11-13math.AG

A numerical ampleness criterion via Gale duality

The main object of the present paper is a numerical criterion determining when a Weil divisor of a $\Q$--factorial complete toric variety admits a positive multiple Cartier divisor which is either numerically effective (nef) or ample. It is a consequence of $\Z$--linear interpretation of Gale duality and se\-con\-dary fan as developed in several previous papers of us. As a byproduct we get a computation of the Cartier index of a Weil divisor and a numerical characterization of weak $\Q$--Fano, $\Q$--Fano, Gorenstein, weak Fano and Fano toric varieties. Several examples are then given and studi…

2016-11-16math.AG

Mori 梦想空间的重数

The multiplicity of a Mori Dream Space: applications to polar duality and the classification of toric $\mathbb{Q}$-Fano varieties

Università di Milano Bicocca（米兰比可卡大学）

本文将伪加权射影空间的重数概念推广到Mori梦想空间，引入完备toric簇的权群$G_Q$和权模$|Q|$，为Fano类toric簇分类提供框架，并给出Batyrev极对偶性的代数解释。

2026-09-29math.AG

Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties

We propose a combinatorical duality for lattice polyhedra which conjecturally gives rise to the pairs of mirror symmetric families of Calabi-Yau complete intersections in toric Fano varieties with Gorenstein singularities. Our construction is a generalization of the polar duality proposed by Batyrev for the case of hypersurfaces.

1993-10-02alg-geommath.AG

Classification problems and mirror duality

We propose a general approach to classification problems in algebraic geometry via mirror duality. For Fano threefolds, a modularity conjecture describes small quantum cohomology and predicts the values of certain Gromov-Witten invariants.

2005-10-13math.AG

更多相关论文收起更多论文

On the Frobenius integrability of certain holomorphic p-forms

The goal of this note is to exhibit the integrability properties (in the sense of the Frobenius theorem) of holomorphic p-forms with values in certain line bundles with seminegative curvature on a compact Kaehler manifold. There are in fact very strong restrictions, both on the holomorphic form and on the curvature of the seminegative line bundle. As a consequence, these observations provide interesting information on the structure of projective manifolds which admit a contact structure: either they are Fano manifolds or, thanks to results of Kebekus-Peternell-Sommese-Wisniewski, they are biho…

2000-05-01math.AG

五维中的Bondal猜想

Bondal's conjecture in dimension five

CNRS/Université Claude Bernard Lyon 1（法国国家科学研究中心/里昂第一大学）；IMPA（巴西纯数学和应用数学研究所）；McGill University（麦吉尔大学）；Université de Rennes（雷恩大学）

本文证明了五维Fano流形上的Bondal猜想，并给出了所有奇数维Fano流形的部分结果，证明使用了弱Fano流形上余维一叶状结构的代数可积性准则、Poisson结构的模留数以及Pfaff域不变子簇的上同调约束。

2026-06-02math.AGmath.SG

Mirror symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifolds

We investigate a potential relationship between mirror symmetry for Calabi-Yau manifolds and the mirror duality between quasi-Fano varieties and Landau-Ginzburg models. More precisely, we show that if a Calabi-Yau admits a so-called Tyurin degeneration to a union of two Fano varieties, then one should be able to construct a mirror to that Calabi-Yau by gluing together the Landau-Ginzburg models of those two Fano varieties. We provide evidence for this correspondence in a number of different settings, including Batyrev-Borisov mirror symmetry for K3 surfaces and Calabi-Yau threefolds, Dolgachev…

2016-11-07math.AG

Abel-Jacobi maps for hypersurfaces and non commutative Calabi-Yau's

It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed p-form with p=2n-4 on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y of degree n. We provide several definitions of this form - via the Abel-Jacobi map, via Hochschild homology, and via the linkage class, and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Y we show that the Fano scheme is birational to a certain moduli space of sheaves on a p-dimensional Calabi--Yau variety X arising naturally in the context…

2009-02-23math.AG

An Involution on the Quantum Cohomology Ring of the Grassmannian

For a Fano manifold M, complex conjugation defines a real involution on the quantum cohomology ring. For the Grassmannian we identify this involution with an explicit transformation on Schubert classes defined over the integers. It is a composition of a power of the cyclic action recently defined by Agnihotri and Woodward with Poincare duality. The existence of the involution has been observed independently and from a different point of view by Postnikov (math.CO/0205165).

2002-05-24math.AG