Complex Surface Of General Type Research Articles

Unchaining surgery and topology of symplectic 4-manifolds

We study a symplectic surgery operation we call unchaining, which effectively reduces the second Betti number and the symplectic Kodaira dimension at the same time. Using unchaining, we give novel constructions of symplectic Calabi–Yau surfaces from complex surfaces of general type and completely resolve a conjecture of Stipsicz on the existence of exceptional sections in Lefschetz fibrations. Combining the unchaining surgery with others, which all correspond to certain monodromy substitutions for Lefschetz pencils, we provide further applications, such as new constructions of exotic symplectic 4-manifolds, and inequivalent pencils of the same genera and the same number of base points on families of symplectic 4-manifolds. Meanwhile, we present a handy criterion for determining from the monodromy of a pencil whether its total space is spin or not.

Rigid surfaces arbitrarily close to the Bogomolov–Miyaoka–Yau line

We prove the existence of rigid compact complex surfaces of general type whose Chern slopes are arbitrarily close to the Bogomolov--Miyaoka--Yau bound of $3$. In addition, each of these surfaces has first Betti number equal to $4$.

Covering-monopole map and higher degree in non-commutative geometry

In this paper we develop analysis of the monopole maps over the universal covering space of a compact four manifold. We induce a property on local properness of the covering monopole map under the condition of closeness of the AHS complex. In particular we construct a higher degree of the covering monopole map when the linearized equation is isomorphic, which induces a homomorphism between K group of the group C^* algebras. It involves non linear analysis on the covering space, which is related to L^p cohomology. We also obtain various Sobolev estimates on the covering spaces. As a possible application, we propose an aspherical version of 10/8 inequality, combining with Singer conjecture on L^2 cohomology. It is satisfied for a large class of four manifolds which includes some complex surfaces of general type.

Explicit equations of a fake projective plane

Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the two-dimensional ball by explicitly written arithmetic subgroups. In this paper we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order seven automorphism.

Complex surfaces of general type with K2 = 3,4 and pg = q = 0

We construct complex surfaces of general type with pg=0 and K2=3,4 as double covers of Enriques surfaces (called Keum–Naie surfaces) with a different way to the original constructions of Keum and Naie. As a result, we show that there is a (−4)-curve on the example with K2=3, which might imply a special relation between Keum–Naie surfaces with K2=3 and K2=4.

A characterization of Inoue surfaces with $$p_g=0$$ p g = 0 and $$K^2=7$$ K 2 = 7

Inoue constructed the first examples of smooth minimal complex surfaces of general type with \(p_g=0\) and \(K^2=7\). These surfaces are finite Galois covers of the 4-nodal cubic surface with the Galois group, the Klein group \(\mathbb {Z}_2\times \mathbb {Z}_2\). For such a surface S, the bicanonical map of S has degree 2 and it is composed with exactly one involution in the Galois group. The divisorial part of the fixed locus of this involution consists of two irreducible components: one is a genus 3 curve with self-intersection number 0 and the other is a genus 2 curve with self-intersection number \(-\,1\). Conversely, assume that S is a smooth minimal complex surface of general type with \(p_g=0\), \(K^2=7\) and having an involution \(\sigma \). We show that, if the divisorial part of the fixed locus of \(\sigma \) consists of two irreducible components \(R_1\) and \(R_2\), with \(g(R_1)=3, R_1^2=0, g(R_2)=2\) and \(R_2^2=-\,1\), then the Klein group \(\mathbb {Z}_2\times \mathbb {Z}_2\) acts faithfully on S and S is indeed an Inoue surface.

CANONICALLY FIBERED SURFACES OF GENERAL TYPE

In this paper, we construct the first examples of complex surfaces of general type with arbitrarily large geometric genus whose canonical maps induce non-hyperelliptic fibrations of genus $g=4$, and on the other hand, we prove that there is no complex surface of general type whose canonical map induces a hyperelliptic fibrations of genus $g\geqslant 4$ if the geometric genus is large.

New surfaces with $K^2 = 7$ and $p_g = q \leq 2$

We construct smooth minimal complex surfaces of general type with $K^2=7$ and: $p_g=q=2,$ Albanese map of degree $2$ onto a $(1,2)$-polarized abelian surface; $p_g=q=1$ as a double cover of a quartic Kummer surface; $p_g=q=0$ as a double cover of a numerical Campedelli surface with $5$ nodes.

Families of Riemann surfaces, uniformization and arithmeticity

A consequence of the results of Bers and Griffiths on the uniformization of complex algebraic varieties is that the universal cover of a family of Riemann surfaces, with base and fibers of finite hyperbolic type, is a contractible 2 − 2- dimensional domain that can be realized as the graph of a holomorphic motion of the unit disk. In this paper we determine which holomorphic motions give rise to these uniformizing domains and characterize which among them correspond to arithmetic families (i.e. families defined over number fields). Then we apply these results to characterize the arithmeticity of complex surfaces of general type in terms of the biholomorphism class of the 2 − 2- dimensional domains that arise as universal covers of their Zariski open subsets. For the important class of Kodaira fibrations this criterion implies that arithmeticity can be read from the universal cover. All this is very much in contrast with the corresponding situation in complex dimension one, where the universal cover is always the unit disk.

NOTES ON AUTOMORPHISMS OF SURFACES OF GENERAL TYPE WITH AND

Let$S$be a smooth minimal complex surface of general type with$p_{g}=0$and$K^{2}=7$. We prove that any involution on$S$is in the center of the automorphism group of$S$. As an application, we show that the automorphism group of an Inoue surface with$K^{2}=7$is isomorphic to$\mathbb{Z}_{2}^{2}$or$\mathbb{Z}_{2}\times \mathbb{Z}_{4}$. We construct a$2$-dimensional family of Inoue surfaces with automorphism groups isomorphic to$\mathbb{Z}_{2}\times \mathbb{Z}_{4}$.

Geometric Convergence of the Kähler–Ricci Flow on Complex Surfaces of General Type

Geometric Convergence of the Kähler–Ricci Flow on Complex Surfaces of General Type

Surfaces on the Severi line

Let S be a minimal complex surface of general type and of maximal Albanese dimension; by the Severi inequality one has KS2≥4χ(OS). We prove that the equality KS2=4χ(OS) holds if and only if q(S):=h1(OS)=2 and the canonical model of S is a double cover of the Albanese surface branched on an ample divisor with at most negligible singularities.

Chern slopes of simply connected complex surfaces of general type are dense in [2,3

We prove that for any number r 2 [2;3], there are spin (resp. nonspin and minimal) simply connected complex surfaces of general type X with c 2(X)=c2(X) arbitrarily close to r. In particular, this shows the existence of simply connected surfaces of general type arbitrarily close to the Bogomolov-Miyaoka-Yau line. In addition, we prove that for any r2 [1;3] and any integer q 0, there are minimal complex surfaces of general type X with c 2(X)=c2(X) arbitrarily close to r and 1(X) isomorphic to the fundamental group of a compact Riemann surface of genus q.

On numerically pluricanonical cyclic coverings

We investigate some properties of cyclic coverings (where is a complex surface of general type) branched along smooth curves that are numerically equivalent to a multiple of the canonical class of . Our main results concern coverings of surfaces of general type with and Miyaoka–Yau surfaces. In particular, such coverings provide new examples of multi-component moduli spaces of surfaces with given Chern numbers and new examples of surfaces that are not deformation equivalent to their complex conjugates.

Fake quadrics from irreducible lattices acting on the product of upper half planes

In the present article, we provide examples of fake quadrics, that is, minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in $$\mathbb P^3$$ which are quotients of the bidisc by an irreducible lattice of automorphisms. Moreover we list classes of arithmetic lattices over a real quadratic number field which define a fake quadric and give general results towards a classification of all such fake quadrics.

Twisted Kodaira-Spencer classes and the geometry of surfaces of general type

We study the cohomology groups H 1 ( X , Θ X ( − m K X ) ) H^1(X,\Theta _X(-mK_X)) , for m ≥ 1 m\geq 1 , where X X is a smooth minimal complex surface of general type, Θ X \Theta _X its holomorphic tangent bundle, and K X K_X its canonical divisor. One of the main results is a precise vanishing criterion for H 1 ( X , Θ X ( − K X ) ) H^1(X,\Theta _X (-K_X)) (Theorem 1.1). The proof is based on the geometric interpretation of non-zero cohomology classes of H 1 ( X , Θ X ( − K X ) ) H^1(X,\Theta _X (-K_X)) . This interpretation in turn uses higher rank vector bundles on X X . We apply our methods to the long standing conjecture saying that the irregularity of surfaces in P 4 \mathbb {P}^4 is at most 2 2 . We show that if X X has bounded holomorphic Euler characteristic, no irrational pencil, and is embedded in P 4 \mathbb {P}^4 with a sufficiently large degree, then the irregularity of X X is at most 3 3 .

SURFACES OF GENERAL TYPE WITH pg= 1 AND q = 0

We construct a new family of simply connected minimal complex surfaces of general type with <TEX>$p_g$</TEX> = 1, <TEX>$q$</TEX> = 0, and <TEX>$K^2$</TEX> = 3, 4, 5, 6, 8 using a <TEX>$\mathbb{Q}$</TEX>-Gorenstein smoothing theory.

A complex surface of general type with 𝑝_{𝑔}=0, 𝐾²=2 and 𝐻₁=ℤ/4ℤ

We construct a new minimal complex surface of general type with p g = 0 p_g=0 , K 2 = 2 K^2=2 and H 1 = Z / 4 Z H_1=\mathbb {Z}/4\mathbb {Z} (in fact, π 1 alg = Z / 4 Z \pi _1^{\text {alg}}=\mathbb {Z}/4\mathbb {Z} ), which settles the existence question for numerical Campedelli surfaces with all possible algebraic fundamental groups. The main techniques involved in the construction are a rational blow-down surgery and a Q \mathbb {Q} -Gorenstein smoothing theory.

A simply connected numerical Campedelli surface with an involution

We construct a simply connected minimal complex surface of general type with \(p_g=0\) and \(K^2=2\) which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with \(p_g=0\) and \(K^2=1\). In order to construct the example, we combine a double covering and \(\mathbb Q \)-Gorenstein deformation. Especially, we develop a method for proving unobstructedness for deformations of a singular surface by generalizing a result of Burns and Wahl which characterizes the space of first order deformations of a singular surface with only rational double points. We describe the stable model in the sense of Kollár and Shepherd-Barron of the singular surfaces used for constructing the example. We count the dimension of the invariant part of the deformation space of the example under the induced \(\mathbb Z /{2}\mathbb Z \)-action.

A uniform bound on the canonical degree of Albanese defective curves on surfaces

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. 641 (2010), 163-175].