arXiv subjects

Jarosław A. Wiśniewski

Publications and source records attributed to Jarosław A. Wiśniewski.

16 recordsLinked to original sources

arXiv2025

Chow quotients of $\mathbb{C}^*$-actions

Given an action of the one-dimensional torus on a projective variety, the associated Chow quotient arises as a natural parameter space of invariant $1$-cycles, which dominates the GIT quotients of the variety. In this paper we explore the relation between the Chow and the GIT quotients of a variety, showing how to construct explicitly the former upon the latter via successive blowups under suitable assumptions. We also discuss conditions for the smoothness of the Chow quotient, and present some examples in which it is singular.

math.AG↗

arXiv2023

Geometric realizations of birational transformations via $\mathbb{C}^*$-actions

In this paper we study varieties admitting torus actions as geometric realizations of birational transformations. We present an explicit construction of these geometric realizations for a particular class of birational transformations, and study some of their geometric properties, such as their Mori, Nef and Movable cones.

math.AG↗

arXiv2021

Rational homogeneous spaces as geometric realizations of birational transformations

A geometric realization of a birational map $ψ$ among two complex projective varieties is a variety $X$ endowed with a $\mathbb{C}^*$-action inducing $ψ$ as the natural birational map among two extremal geometric quotients. In this paper we study geometric realizations of some classic birational maps --inversion maps, special Cremona transformations, special birational transformations of type $(2,1)$--, by considering $\mathbb{C}^*$-actions on certain rational homogeneous spaces and their subvarieties.

math.AG↗

arXiv2021

Small bandwidth ${\mathbb C}^*$-actions and birational geometry

In this paper we study smooth projective varieties and polarized pairs with an action of a one dimensional complex torus. As a main tool, we define birational geometric counterparts of these actions, that, under certain assumptions, encode the information necessary to reconstruct them. In particular, we consider some cases of actions of low complexity -- measured in terms of two invariants of the action, called bandwidth and bordism rank -- and discuss how they are determined by well known birational transformations, namely Atiyah flips and Cremona transformations.

math.AG↗

arXiv2021

High rank torus actions on contact manifolds

We prove LeBrun--Salamon conjecture in the following situation: if $X$ is a contact Fano manifold of dimension $2n+1$ whose group of automorphisms is reductive of rank $\geq \max(2,(n-3)/2)$ then $X$ is the adjoint variety of a simple group. The rank assumption is fulfilled not only by the three series of classical linear groups but also by almost all the exceptional ones.

math.AG↗

arXiv2021

Small modifications of Mori dream spaces arising from ${\mathbb C}^*$-actions

We link small modifications of projective varieties with a ${\mathbb C}^*$-action to their GIT quotients. Namely, using flips with centers in closures of Białynicki-Birula cells, we produce a system of birational equivariant modifications of the original variety, which includes those on which a quotient map extends from a set of semistable points to a regular morphism. The structure of the modifications is completely described for the blowup along the sink and the source of smooth varieties with Picard number one with a ${\mathbb C}^*$-action which has no finite isotropy for any point. Examples can be constructed upon homogeneous varieties with a ${\mathbb C}^*$-action associated to short grading of their Lie algebras.

math.AG↗

arXiv2020

Algebraic torus actions on contact manifolds

We prove the LeBrun-Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This implies that any positive quaternion-Kahler manifold of real dimension at most 16 is necessarily a symmetric space, one of the Wolf spaces. A similar result about contact Fano manifolds of dimension at most 9 with reductive automorphism group also holds. The main difficulty in approaching the conjecture is how to recognize a homogeneous space in an abstract variety. We contribute to such problem in general, by studying the action of algebraic torus on varieties and exploiting Bialynicki-Birula decomposition and equivariant Riemann-Roch theorems. From the point of view of T-varieties (that is, varieties with a torus action), our result is about high complexity T-manifolds. The complexity here is at most 1/2(dim X+5) with dim X arbitrarily high, but we require this special (contact) structure of X. Previous methods for studying T-varieties in general usually only apply for complexity at most 2 or 3.

math.AG↗

arXiv2020

Adjunction for varieties with a $\mathbb{C}^*$ action

Let $X$ be a complex projective manifold, $L$ an ample line bundle on $X$, and assume that we have a $\mathbb{C}^*$ action on $(X,L)$. We classify such triples $(X,L,\mathbb{C}^*)$ for which the closure of a general orbit of the $\mathbb{C}^*$ action is of degree $\leq 3$ with respect to $L$ and, in addition, the source and the sink of the action are isolated fixed points, and the $\mathbb{C}^*$ action on the normal bundle of every fixed point component has weights $\pm 1$. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension $11$ and $13$ are homogeneous if their group of automorphisms is reductive of rank $\geq 2$.

math.AG↗

arXiv2016

On rigidity of flag varieties

We prove that the variety of complete flags for any semisimple algebraic group is rigid in any smooth family of Fano manifolds.

math.AG↗

arXiv2016

Flag bundles on Fano manifolds

As an application of a recent characterization of complete flag manifolds as Fano manifolds having only ${\mathbb P}^1$-bundles as elementary contractions, we consider here the case of a Fano manifold $X$ of Picard number one supporting an unsplit family of rational curves whose subfamilies parametrizing curves through a fixed point are rational homogeneous, and we prove that $X$ is homogeneous. In order to do this, we first study minimal sections on flag bundles over the projective line, and discuss how Grothendieck's theorem on principal bundles allows us to describe a flag bundle upon some special sections.

math.AG↗

arXiv2016

On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32

We provide a construction of 81 symplectic resolutions of a 4-dimensional quotient singularity obtained by an action of a group of order 32. The existence of such resolutions is known by a result of Bellamy and Schedler. Our explicit construction is obtained via GIT quotient of the spectrum of a ring graded in the Picard group generated by the divisors associated to the conjugacy classes of symplectic reflections of the group in question. As the result we infer the geometric structure of these resolutions and their flops. Moreover, we represent the group in question as a group of automorphisms of an abelian 4-fold so that the resulting quotient has singularities with symplectic resolutions. This yields a new Kummer-type symplectic 4-fold.

math.AG↗

arXiv2016

A very special EPW sextic and two IHS fourfolds

We show that the Hilbert scheme of two points on the Vinberg $K3$ surface has a 2:1 map onto a very symmetric EPW sextic $Y$ in $\mathbb{P}^5$. The fourfold $Y$ is singular along $60$ planes, $20$ of which form a complete family of incident planes. This solves a problem of Morin and O'Grady and establishes that $20$ is the maximal cardinality of such a family of planes. Next, we show that this Hilbert scheme is birationally isomorphic to the Kummer type IHS fourfold $X_0$ constructed in [DW]. We find that $X_0$ is also related to the Debarre-Varley abelian fourfold.

math.AG↗

arXiv2016

Fano manifolds whose elementary contractions are smooth $\mathbb P^1$-fibrations: A geometric characterization of flag varieties

The present paper provides a geometric characterization of complete flag varieties for semisimple algebraic groups. Namely, if $X$ is a Fano manifold whose all elementary contractions are $\mathbb P^1$-fibrations then $X$ is isomorphic to the complete flag manifold $G/B$ where $G$ is a semi-simple Lie algebraic group and $B$ is a Borel subgroup of $G$.

math.AG↗

arXiv2015

A survey on the Campana-Peternell Conjecture

In 1991 Campana and Peternell proposed, as a natural algebro-geometric extension of Mori's characterization of the projective space, the problem of classifying the complex projective Fano manifolds whose tangent bundle is nef, conjecturing that the only varieties satisfying these properties are rational homogeneous. In this paper we review some background material related to this problem, with special attention to the partial results recently obtained by the authors.

math.AG↗

arXiv1996

On Contractions of smooth varieties

Let $\f: X \ra Z$ be a proper surjective map from a smooth complex manifold $X$ onto a normal variety $Z$. If $\f$ has connected fibers and $-K_X$ is $\f$-ample then $\f$ is called a good contraction. In the present paper we study good contractions, fibers of which have dimension less or equal than two: after describing possible two dimensional isolated fibers we discuss their scheme theoretic structure and the geometry of $\f:X\ra Z$ nearby such a fiber. If $dimX=4$ and $\f$ is birational with an isolated 2 dimensional fiber then we obtain a complete description of $\f$. We provide also a description of a 4 dimensional conic fibration with an isolated fiber which is either a plane or a quadric. We construct pertinent examples.

alg-geom↗

arXiv1993

A note on non-vanishing and applications

Let $X$ be a normal variety over the field of complex numbers with log terminal singularities and the canonical divisor $K_X$ being ${\bf Q}$-Gorenstein. Assume that $L$ is an ample line bundle over $X$ and $ϕ: X\to Y$ is a morphism supported by $K_X+rL$ for some positive rational number $r$. In the present paper we study the evaluation $ϕ^*ϕ_*(L)\to L$ and the locus of points where it is not surjective which we call relative base point locus of $L$. In particular, we prove that, if the dimension of a fiber of $ϕ$ is small with respect to $r$ then the relative base point locus does not meet the fiber. Consequently, in this case, we discuss the structure of the map $ϕ$ for a smooth $X$.

alg-geom↗