Marco Manetti
We determine the base space of the Kuranishi family of some complete intersection in the product of an abelian variety and a projective space. As a consequence we obtain new examples of obstructed irregular surfaces with ample canonical bundle and maximal Albanese dimension.
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We exhibit examples of pairs $(X,D)$ where $X$ is a smooth projective variety and $D$ is an anticanonical reduced simple normal crossing divisor such that the deformations of $(X,D)$ are obstructed. These examples are constructed via toric geometry.
In this paper, we study obstructed and unobstructed (holomorphic) Poisson deformations with classical examples in deformation theory.
The tangent bundle of a generalized Kummer fourfold is obstructed
针对广义库默尔型超凯勒四元簇，构造了切丛的无穷小形变，得到首个具有阻碍形变的稳定超全纯丛例子
I construct some smooth Calabi-Yau threefolds in characteristic two and three that do not lift to characteristic zero. These threefolds are pencils of supersingular K3-surfaces. The construction depends on Moret-Bailly's pencil of abelian surfaces and Katsura's analysis of generalized Kummer surfaces. The threefold in characteristic two turns out to be nonrigid.
This paper gives a simple example of a family of Calabi-Yaus of any dimension with canonical singularities of dimension one, whose Kuranishi space is singular. Thus the Bogomolov-Tian-Todorov unobstructedness theorem is not true for Calabi-Yaus with canonical singularities.
Nodal surfaces with obstructed deformations
本文研究节点曲面的变形空间，证明次数 d≤7 或 d≥8 且节点数至多 4d-5 时变形空间光滑且维数符合预期，并构造 d≥8 时具有 4d-4 个节点且切空间维数偏大的显式反例。
Rigid and obstructed tangent bundles on Calabi--Yau threefolds
本文在不假设单连通性的条件下，通过构造自由商和Igusa商例子，回答了Huybrechts关于Calabi-Yau三维流形切丛刚性与奇异形变空间的两个问题，并给出了三维反例及形变芽的显式结构。
A tetragonal canonical curve is the complete intersection of two divisors on a scroll. The equations can be written in `rolling factors' format. For such homogeneous ideals we give methods to compute infinitesimal deformations. Deformations can be obstructed. For the case of quadratic equations on the scroll we derive explicit base equations. They are used to study extensions of tetragonal curves.
We determine obstructedness or unobstructedness of (holomorphic) Poisson deformations of ruled surfaces over an elliptic curve.
We study the deformations of a smooth curve $C$ on a smooth projective threefold $V$, assuming the presence of a smooth surface $S$ satisfying $C \subset S \subset V$. Generalizing a result of Mukai and Nasu, we give a new sufficient condition for a first order infinitesimal deformation of $C$ in $V$ to be primarily obstructed. In particular, when $V$ is Fano and $S$ is $K3$, we give a sufficient condition for $C$ to be (un)obstructed in $V$, in terms of $(-2)$-curves and elliptic curves on $S$. Applying this result, we prove that the Hilbert scheme $\operatorname{Hilb}^{sc} V_4$ of smooth con…
The Kuranishi space of joint deformations of Calabi--Yau manifolds and holomorphic vector bundles
本文研究Calabi-Yau流形与全纯向量丛的联合形变，在Fujiki流形等条件下证明无阻碍性，并构造有阻碍的例子，推广了Li-Pan和Iacono-Manetti定理，回答了Felten的两个问题。
Let $X$ be a complete $\mathbb{Q}$-factorial toric variety. We explicitly describe the space $H^2(X,T_X)$ and the cup product map $H^1(X,T_X)\times H^1(X,T_X)\to H^2(X,T_X)$ in combinatorial terms. Using this, we give an example of a smooth projective toric threefold for which the cup product map does not vanish, showing that in general, smooth complete toric varieties may have obstructed deformations.
Inspired by an article of Cotti, Dubrovin and Guzzetti arXiv:1706.04808, we extend to a degenerate case a result of Malgrange on integrable deformations of irregular singularities. We give an application to integrable deformations of the solution of some Birkhoff problem and apply it to the construction of Frobenius manifolds.
In this paper, we consider the generalized isomonodromic deformations of rank two irregular connections on the Riemann sphere. We introduce Darboux coordinates on the parameter space of a family of rank two irregular connections by apparent singularities. By the Darboux coordinates, we describe the generalized isomonodromic deformations as Hamiltonian systems.
In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying $K_X^2 = 4p_g(X)-8$, for any even integer $p_g\geq 4$. These surfaces also have unbounded irregularity $q$. We carry out our study by investigating the deformations of the canonical morphism $φ:X\to \mathbb{P}^N$, where $φ$ is Galois of degree 4. These canonical covers are classified in by the first two authors into four distinct families. We show that any deformation of $φ$ factors through a double cover of a ruled surface and, hence, is never birational. More interestin…
Let $G$ be a reductive affine algebraic group defined over $\mathbb C$, and let $\nabla_0$ be a meromorphic $G$-connection on a holomorphic $G$-bundle $E_0$, over a smooth complex curve $X_0$, with polar locus $P_0 \subset X_0$. We assume that $\nabla_0$ is irreducible in the sense that it does not factor through some proper parabolic subgroup of $G$. We consider the universal isomonodromic deformation $(E_t\to X_t, \nabla_t, P_t)_{t\in \mathcal{T}}$ of $(E_0\to X_0, \nabla_0, P_0)$, where $\mathcal{T}$ is a certain quotient of a certain framed Teichmüller space we describe. We show that if th…
We introduce an unfolded moduli space of connections, which is an algebraic relative moduli space of connections on complex smooth projective curves, whose generic fiber is a moduli space of regular singular connections and whose special fiber is a moduli space of unramified irregular singular connections. On the moduli space of unramified irregular singular connections, there is a subbundle of the tangent bundle defining the generalized isomonodromic deformation produced by the Jimbo-Miwa-Ueno theory. On an analytic open subset of the unfolded moduli space of connections, we construct a non-c…
Obstructions to deforming space curves lying on a del Pezzo surface
本文研究了位于度为4的光滑 del Pezzo 曲面上的空间曲线的变形，给出了曲线无障碍或有障碍的充分条件，并构造了无穷多个 Hilbert 概形的非约化不可约分支。
We introduce the notion of factorized ramified structure on a generic ramified irregular singular connection on a smooth projective curve. By using the deformation theory of connections with factorized ramified structure, we construct a canonical 2-form on the moduli space of ramified connections. Since the factorized ramified structure provides a duality on the tangent space of the moduli space, the 2-form becomes nondegenerate. We prove that the 2-form on the moduli space of ramified connections is d-closed via constructing an unfolding of the moduli space. Based on the Stokes data, we intro…
In this paper, we investigate the moduli of surfaces of general type admitting genus 2 fibrations with irregularity q = g_b + 1, where g_b >= 2 is the genus of the base. We prove that smooth fibrations are parametrized by a unique component in the moduli space. The same result applies to nonsmooth fibrations with special values of g_b. In the general case, we give a bound on the dimension of the corresponding connected components.
We present a local computation of deformations of the tangent bundle for a resolved orbifold singularity C^d/G. These correspond to (0,2)-deformations of (2,2)-theories. A McKay-like correspondence is found predicting the dimension of the space of first-order deformations from simple calculations involving the group. This is confirmed in two dimensions using the Kronheimer-Nakajima quiver construction. In higher dimensions such a computation is subject to nontrivial worldsheet instanton corrections and some examples are given where this happens. However, we conjecture that the special crepant …
Monodromy groups, i.e. the groups of isometries of the intersection lattice L_X:=H_2/torsion generated by the monodromy action of all deformation families of a given surface, have been computed in math.AG/0006231 for any minimal elliptic surface with p_g>q=0. New and refined methods are now employed to address the cases of minimal elliptic surfaces with p_g+1>q>0. To this end we find explicit families such that any isometry is in the group generated by their monodromies or does not respect the invariance of the canonical class or the spinor norm. The monodromy action is moreover shown to act b…
Topology of irregular isomonodromy times on a fixed pointed curve
该研究定义并研究黎曼曲面非正则类的形变（模）空间，为非正则等单值化系统“时间”提供内在视角，旨在探究作为其基本群的G-辫群的深层推广，尤其关注全G-辫群的推广。
Deformation of moduli spaces of meromorphic $G$-connections on $\mathbb{P}^{1}$ via unfolding of irregular singularities
本文通过展开不规则奇点，研究了 meromorphic $G$-connections 在 $\mathbb{P}^{1}$ 上的模空间变形，并证明了与 Deligne-Simpson 问题的解的存在性相关联。
We study a generalization of the isomonodromic deformation to the case of connections with irregular singularities. We call this generalization Isostokes Deformation. A new deformation parameter arises: one can deform the formal normal forms of connections at irregular points. We study this part of the deformation, giving an algebraic description. Then we show how to use loop groups and hypercohomology to write explicit hamiltonians. We work on an arbitrary complete algebraic curve, the structure group is an arbitrary semisimiple group.
In this paper, we study the deformations of curves in the projective 3-space $\mathbb P^3$ (space curves), one of the most classically studied objects in algebraic geometry. We prove a conjecture due to J. O. Kleppe (in fact, a version modified by Ph. Ellia) concerning maximal families of space curves lying on a smooth cubic surface, assuming the quadratic normality of its general members. We also give a sufficient condition for curves lying on a cubic surface to be obstructed in $\mathbb P^3$ in terms of lines on the surface. For the proofs, we use the Hilbert-flag scheme of $\mathbb P^3$ as …
This paper deals with moduli spaces of framed principal bundles with connections with irregular singularities over a compact Riemann surface. These spaces have been constructed by Boalch by means of an infinite-dimensional symplectic reduction. It is proved that the symplectic structure induced from the Atiyah--Bott form agrees with the one given in terms of hypercohomology. The main results of this paper adapt work of Krichever and of Hurtubise to give an interpretation of some Hitchin Hamiltonians as yielding Hamiltonian vector fields on moduli spaces of irregular connections that arise from…
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 modu…
This note explains a construction of a Poisson manifold whose symplectic foliation describes a deformation of a moduli space of meromorphic connections with unramified irregular singularities. In particular, this deformation of the moduli space corresponds to the unfolding of irregular singularities of the meromorphic connections. This is an announcement of some results in the forthcoming paper.
Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids
本文研究了类型A的任意形式结构下的野映射类群，通过构造配置空间和分裂森林，揭示了复杂反射群作为野映射类群的出现，并计算了野黎曼曲面模空间的维度。
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