Related papers: Projective contact log varieties

We study deformations of complex projective varieties that are homotopically or homologically trivial. We formulate several conjectures and give some examples and partial answers.

Nous montrons que les seules vari\'et\'es toriques munies d'une structure de contact sont, \`a isomorphisme pr\`es, les espaces projectifs complexes et les vari\'et\'es $\mathbb{P}_{\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}}…

We prove that the projective model structure on the category of unbounded cochain complexes extends naturally to the category of contractions. The proof is completely elementary and we do not assume familiarity with model categories.

We give simple characterizations of contact 1-forms in terms of Dirac structures. We also relate normal almost contact structures to the theory of Dirac structures.

Asymptotic syzygies of a normal crossing variety follow the same vanishing behavior as one of its smooth components, unless there is a cohomological obstruction arising from how the smooth components intersect each other. In that case, we…

Traditionally, Hodge structures are associated with complex projective varieties. In my expository lectures I discussed a non-commutative generalization of Hodge structures in deformation quantization and in derived algebraic geometry.

We introduce the notion of the moduli stack of relations of a quiver. When the quiver with relations is derived-equivalent to an algebraic variety, the corresponding compact moduli scheme can be viewed as a compact moduli of noncommutative…

In this note we propose the generalization of the notion of a holomorphic contact structure on a manifold (smooth variety) to varieties with rational singularities and prove basic properties of such objects. Natural examples of singular…

This survey focuses on the geometric problem of log-surfaces, which are pairs consisting of a smooth projective surface and a reduced non-empty boundary divisor. In the first part, we focus on the geography problem for complex log-surfaces…

We construct an associative ring which is a deformation of the quantum cohomology ring of the projective plane. Just as the quantum cohomology encodes the incidence characteristic numbers of rational plane curves, the contact cohomology…

Nonsingular projective varieties which are both convex and rationally connected are considered. We ask whether such varieties must be algebraic homogeneous spaces G/P. In case X is a complete intersection, an affirmative answer is obtained…

We study complex analytic (possibly singular) projective connections on the plane. We characterize some of them in terms of their families of integral curves. We also give a beginning of classification of second order odes polynomial in the…

We study equivariant contact structures on complex projective varieties arising as partial flag varieties $G/P$, where $G$ is a connected, simply-connected complex simple group of type $ADE$ and $P$ is a parabolic subgroup. We prove a…

The first part of this note contains a review of basic properties of the variety of lines contained in an embedded projective variety and passing through a general point. In particular we provide a detailed proof that for varieties defined…

Contact projective structures have been profoundly studied by D.J.F. Fox. He associated to a contact projective structure a canonical projective structure on the same manifold. We interpret Fox' construction in terms of the equivalent…

These are notes on some algebraic geometry of complex projective curves, together with an application to studying the contact curves in CP^3 and the null curves in the complex quadric Q^3 in CP^4, related by the well-known Klein…

Let $(X,B)$ be a complex projective klt pair, and let $f\colon X\to Z$ be a surjective morphism onto a normal projective variety with maximal albanese dimension such that $K_X+B$ is relatively big over $Z$. We show that such pairs have good…

We provide supplements and open problems related to structure theorems for maximal rationally connected fibrations of certain positively curved projective varieties, including smooth projective varieties with semi-positive holomorphic…

We discuss the Calabi--Yau type structure of normal projective surfaces and Mori dream spaces admitting a non-trivial polarized endomorphism.

We construct smooth varieties admitting small contractions from arbitrary smooth projective varieties. This construction generalizes Kawamata's four-dimensional example. We also give sufficient conditions for divisors on these varieties to…