In the first part of the paper Beilinson's theorem on the bounded derived category of coherent sheaves on P^n is extended to weighted projective spaces in a rather explicit form. To this purpose the usual category of coherent sheaves is replaced by a suitable category of graded sheaves, and a more general theory of graded schemes is developed. In the second part of the paper the weighted version of Beilinson's theorem is applied to prove a structure theorem for certain canonical projections of surfaces of general type into a 3-dimensional weighted projective space. This result (which generalizes to the weighted case a theorem by Catanese and Schreyer) is mainly interesting for irregular surfaces, and we illustrate it by studying a family of surfaces with p_g=q=2 and K^2=4, whose canonical rings are explicitly computed along the way.
In this note we construct an unlimited family of irregular algebraic surfaces of general type with canonical map of degree $ 8 $, irregularity $ 1 $ and arbitrarily large geometric genus such that the image of the canonical map is not a surface of minimal degree.
Let $X$ be a complex smooth projective threefold of general type. Assume $q(X)>0$. We show that the $m$-canonical map of $X$ is birational for all $m\geq 5$.
Let S be a smooth minimal complex projective surface of maximal Albanese dimension. Under the assumption that the canonical class of S is ample and the irregularity of S, q(S), is greater or equal to 5 we show that K^2>= 4χ(S)+(10/3)q(S)-8, thus improving the well known Severi inequality K^2>=4χ(S). We also give stronger inequalities under extra assumptions on the Albanese map or on the canonical map of S.
We prove that for a smooth projective irregular $3$-fold $X$ with $K_X\equiv 0$ and a nef and big divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\geq 3$ and all $P\in \text{Pic}^0(X)$. We also use the same method to deal with $4$-folds, and prove that for a smooth projective irregular $4$-fold $X$ with $K_X\equiv 0$ and an ample divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\geq 5$ and all $P\in \text{Pic}^0(X)$. These results are also optimal.
We describe some methods to compute fundamental groups, (co)homology, and irregularity of semi-log-canonical surfaces. As an application, we show that there are exactly two irregular Gorenstein stable surfaces with $K^2=1$, both of which have $χ(X) = 0$ and $\mathrm{Pic}^0(X)=\mathbb{C}^*$ but different homotopy type.
Let X be a complex projective n-dimensional manifold of general type, whose canonical system is composite with a pencil. If the Albanese map is generically finite, but not surjective, or if the irregularity is strictly larger than n and the image of X in its Albanese variety of Kodaira dimension one, then the geometric genus of a general fibre of the canonical map is one and the latter factors through the Albanese map. The last part of this result holds true for any threefold with irregularity 5 or larger.
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…
We study minimal surfaces X of general type with $K^2_X=6p_g-14$ and $q(X)>0$ such that $K_X$ is ample, the image of the canonical map is a canonically embedded surface of general type and the canonical map is not birational. The main result states that if X satisfies the above assumptions and $q(X)\ge 2$, then, apart from a finite number of exceptions, X belongs to an infinite series of examples due to Beauville. The exceptions are described in detail and some new examples are given.
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.
We construct a surface with irregularity $q=2,$ geometric genus $p_g=3,$ self-intersection of the canonical divisor $K^2=16$ and canonical map of degree $16.$
We construct a complex algebraic surface with geometric genus $p_g=3$, irregularity $q=0$, self-intersection of the canonical divisor $K^2=24$ and canonical map of degree $24$ onto $\mathbb P^2$.
We give an overview of results on irregular complex surfaces of general type, discussing in particular the distribution of the numerical invariants self-intersection of a canonical divisor and holomorphic Euler characteristic for the minimal ones. This is an expanded version of the talk given by the second author at the workshop "Classical Algebraic Geometry Today", M.S.R.I., January 26--30, 2009.
We treat two different topics on the log minimal model program, especially for four-dimensional log canonical pairs. (a) Finite generation of the log canonical ring in dimension four. (b) Abundance theorem for irregular fourfolds. We obtain (a) as a direct consequence of the existence of four-dimensional log minimal models by using Fukuda's theorem on the four-dimensional log abundance conjecture. We can prove (b) only by using traditional arguments. More precisely, we prove the abundance conjecture for irregular $(n+1)$-folds on the assumption that the minimal model conjecture and the abundan…
In this note, we construct nine families of projective complex minimal surfaces of general type having the canonical map of degree 8 and irregularity 0 or 1. For six of these families the canonical system has a non trivial fixed part.
In this note, we construct a minimal surface of general type with geometric genus p g = 4, self-intersection of the canonical divisor K^2 = 32 and irregularity q = 1 such that its canonical map is an abelian cover of degree 16 of P^1 x P^1.
We introduce a category of possibly irregular holonomic D-modules which can be endowed in a canonical way with an irregular Hodge filtration. Mixed Hodge modules with their Hodge filtration naturally belong to this category, as well as their twist by $\expφ$ for any meromorphic function $φ$. This category is stable by various standard functors, which produce many more filtered objects. The irregular Hodge filtration satisfies the $E_1$-degeneration property by a projective morphism. This generalizes some results proved by Esnault-Sabbah-Yu arxiv:1302.4537 and Sabbah-Yu arxiv:1406.1339. We also…
A good canonical projection of a surface $S$ of general type is a morphism to the 3-dimensional projective space P^3 given by 4 sections of the canonical line bundle. To such a projection one associates the direct image sheaf F of the structure sheaf of S, which is shown to admit a certain length 1 symmetrical locally free resolution. The structure theorem gives necessary and sufficient conditions for such a length 1 locally free symmetrical resolution of a sheaf F on P^3 in order that $Spec (mathcal F)$ yield a canonically projected surface of general type. The result was found in 1983 by the…
In this note, we construct two minimal surfaces of general type with geometric genus p_g= 3, irregularity q = 0, self-intersection of the canonical divisor K^22 =20,24 such that their canonical map is of degree 20. In one of these surfaces, the canonical linear system has a non-trivial fixed part. These surfaces, to our knowledge, are the first examples of minimal surfaces of general type with canonical map of degree 20.
We use the Borisov-Keum equations of a fake projective plane and the Borisov-Yeung equations of the Cartwright-Steger surface to show the existence of a regular surface with canonical map of degree 36 and of an irregular surface with canonical map of degree 27. As a by-product, we get equations (over a finite field) for the $\mathbb Z/3$-invariant fibres of the Albanese fibration of the Cartwright-Steger surface and show that they are smooth.
Canonical coordinates for moduli spaces of rank two irregular connections on curves
University of Hyogo（兵库大学）；Univ Rennes, CNRS, IRMAR - UMR 6625（雷恩第一大学，法国国家科学研究中心，IRMAR实验室-联合研究单位6625）；Kobe Gakuin University（神户学院大学）；Eötvös Loránd University（罗兰大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究与创新网络阿尔弗雷德·雷尼数学研究所）
Let f :S\to B be a non locally trivial fibred surface. We prove a lower bound for the slope of f depending increasingly from the relative irregularity of f and the Clifford index of the general fibres.
The article proves the Infinitesimal Torelli theorem for surfaces subject to the following conditions: 1) the canonical bundle of a surface is ample and generated by its global sections, 2)the geometric genus $p_g \geq 4$, 3) the irregularity $q=0$ . The main novelty is a realization of the Kodaira-Spencer classes lying in the kernel of the cohomology cup-product controlling the derivative of the period map of weight 2 in the category of the coherent sheaves of a surface.
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…
We shall study minimal complex surfaces with $c^2 = 9$ and $χ=5$ whose canonical classes are divisible by $3$ in the integral cohomology groups, where $c_1^2$ and $χ$ denote the first Chern number of an algebraic surface and the Euler characteristic of the structure sheaf, respectively. The main results are a structure theorem for such surfaces, the unirationality of the moduli space, and a description of the behavior of the canonical map. As a byproduct, we shall also rule out a certain case mentioned in a paper by Ciliberto--Francia--Mendes Lopes. Since the irregularity $q$ vanishes for our …
The Hamiltonian theory of isomonodromy equations for meromorphic connections with irregular singularities on algebraic curves is constructed. An explicit formula for the symplectic structure on the space of monodromy and Stokes matrices is obtained. The Whitham equations for the isomonodromy equations are derived. It is shown that they provide a flat connection on the space of the spectral curves of the Hitchin systems.
The present work deals with the canonical map of smooth, compact complex surfaces of general type in a polarization of type $(1,2,2)$ on an abelian threefold. A natural and classical question is whether the canonical system of such surfaces is very ample in the general case. In this work, we provide a positive answer to this question. First, we describe the structure of the canonical map of those smooth ample surfaces of type $(1,2,2)$ in an abelian threefold which are bidouble cover of principal polarizations. Then, we study the general behavior of the canonical map of general ample surfaces …
We give a new construction of the irregular, generalized Lagrangian, surfaces of general type with p_g=5, χ=2, K^2=8, recently discovered by Chad Schoen. Our approach proves that, if S is a general Schoen surface, its canonical map is a finite morphism of degree 2 onto a canonical surface with invariants p_g=5, χ=6, K^2=8, a complete intersection of a quadric and a quartic hypersurface in P^4, with 40 even nodes.
We carry out an analysis of the canonical system of a minimal complex surface of general type with irregularity q>0. Using this analysis we are able to sharpen in the case q>0 the well known Castelnuovo inequality K^2>=3p_g+q-7. Then we turn to the study of surfaces with p_g=2q-3 and no fibration onto a curve of genus >1. We prove that for q>=6 the canonical map is birational. Combining this result with the analysis of the canonical system, we also prove the inequality: K^2>=7χ+2. This improves an earlier result of the first and second author [M.Mendes Lopes and R.Pardini, On surfaces with p_g…
We introduce exponential complexes of sheaves on manifolds. They are resolutions of the (Tate twisted) constant sheaves of the rational numbers, generalising the short exact exponential sequence. There are canonical maps from the exponential complexes to the de Rham complex. Using this, we introduce new complexes calculating the rational Deligne cohomology. We call them exponential Deligne complexes. Their advantage is that, at least at the generic point of a complex variety, one can define Beilinson's regulator map to the rational Deligne cohomology on the level of complexes. Namely, we defin…
Let S be a minimal complex surface of general type with irregularity q>=2 and let C be an irreducible curve of geometric genus g contained in S. Assume that C is "Albanese defective", i.e., that the image of C via the Albanese map does not generate the Albanese variety Alb(S); we obtain a linear upper bound in terms of K^2_S and g for the canonical degree K_SC of C. As a corollary, we obtain a bound for the canonical degree of curves with g<= q-1, thereby generalizing and sharpening the main result of [S.Y. Lu, On surfaces of general type with maximal Albanese dimension, J. Reine Angew. Math. …
Let $X$ be a smooth compact complex surface subject to the following conditions: (i) the canonical line bundle $\mathcal{O}_X(K_X) $ is very ample, (ii) the irregularity $q(X): = h^1(\mathcal{O}_X) =0$, (iii) $X$ contains no rational normal curves of degree $\leq (p_g-1)$, (iv) the multiplication map $m_2: Sym^2(H^0(\mathcal{O}_X(K_X))) \longrightarrow H^0 (\mathcal{O}_X (2K_X))$ is surjective. It is shown that the Infinitesimal Torelli holds for such $X$. Our proof is based on the study of the cup-product $$ H^1 (Θ_X) \longrightarrow (\mathcal{O}_X(K_X))^{\ast} \otimes H^1 (Ω_X) $$ where $Θ_X…
The cohomology algebra of the canonical bundle of a compact Kähler manifold is naturally viewed as a module over an exterior algebra. We use the Bernstein-Gel'fand-Gel'fand correspondence, together with Generic Vanishing theory, in order to understand the regularity properties of this module. We also relate it to the infinitesimal theory of the canonical linear series inside paracanonical space. Finally, we apply vector bundle methods on the polynomial ring side to obtain inequalities for the holomorphic Euler characteristic and the Hodge numbers of compact Kähler manifolds without irregular f…
The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincaré complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more generally, the topological L-class, for closed oriented stratified pseudomanifolds that can be equipped with a middle-perverse Verdier self-dual complex of sheaves, determined by Lagrangian sheaves along strata of odd codimension (so-called L-pseudomanifolds). This class of spaces contains all Witt spaces and thus all pure-dimensional complex algebraic varieties. We ap…
We study the category of G(O)-equivariant perverse coherent sheaves on the affine Grassmannian of G. This coherent Satake category is not semisimple and its convolution product is not symmetric, in contrast with the usual constructible Satake category. Instead, we use the Beilinson-Drinfeld Grassmannian to construct renormalized r-matrices. These are canonical nonzero maps between convolution products which satisfy axioms weaker than those of a braiding. We also show that the coherent Satake category is rigid, and that together these results strongly constrain its convolution structure. In par…
The method of intersection spaces associates rational Poincaré complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA theory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the intersection space complex, on the hypersurface. Moreover, the intersection space complex …
