Complex Projective Surface Research Articles

Slope inequality for an arbitrary divisor

Let [Formula: see text] be a surjective morphism with connected fibers from a smooth complex projective surface [Formula: see text] to a smooth complex projective curve [Formula: see text] with general fiber [Formula: see text]. In this paper, we develop a more general version of the slope inequality for data [Formula: see text], where [Formula: see text] is an arbitrary relatively effective divisor on [Formula: see text] and [Formula: see text] is a locally free sub-sheaf of [Formula: see text]. We see how the speciality of [Formula: see text], restricted to the general fiber, plays a role in the results. Moreover, we compute some natural examples and provide applications.

Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi–Yau Three-Folds

We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles. The obtained formulae express cohomologies in terms of divisor class intersections, and are adapted to the decomposition of the effective cone into Zariski chambers. In particular, we show this occurs on generalised del Pezzo surfaces, toric surfaces, and K3 surfaces. In the second part we use these surface results to derive formulae for all line bundle cohomology on a simple class of elliptically fibered Calabi–Yau three-folds. Computing such quantities is a crucial step in deriving the massless spectrum in string compactifications.

K3 surface entropy and automorphism groups

We derive a characterization of the complex projective K3 surfaces which have automorphisms of positive entropy in terms of their Néron–Severi lattices. Along the way, we classify the projective K3 surfaces of zero entropy with infinite automorphism groups and we determine the projective K3 surfaces of Picard number at least five with almost abelian automorphism groups, which gives an answer to a long standing question of Nikulin.

Counting elliptic fibrations on K3 surfaces

We solve the problem of counting jacobian elliptic fibrations on an arbitrary complex projective K3 surface up to automorphisms. We then illustrate our method with several explicit examples.

Random dynamics on real and complex projective surfaces

Abstract We initiate the study of random iteration of automorphisms of real and complex projective surfaces, as well as compact Kähler surfaces, focusing on the classification of stationary measures. We show that, in a number of cases, such stationary measures are invariant and provide criteria for uniqueness, smoothness and rigidity of invariant probability measures. This involves a variety of tools from complex and algebraic geometry, random products of matrices, non-uniform hyperbolicity, as well as recent results of Brown and Rodriguez Hertz on random iteration of surface diffeomorphisms.

Geometric description of 〈2〉-polarised Hilbert squares of generic K3 surfaces

A generic K3 surface of degree 2t is a general complex projective K3 surface S2t whose Picard group is generated by the class of an ample divisor H∈Div(S2t) such that H2=2t with respect to the intersection form. We show that if X is the Hilbert square of a generic K3 surface of degree 2t with t≠2 which admits an ample divisor D∈Div(X) with qX(D)=2, where qX is the Beauville–Bogomolov–Fujiki form, then X is a double EPW sextic.

Towards Tropically Counting Binodal Surfaces

We count tropical multinodal surfaces using floor plans, looking at the case when two nodes are tropically close together, i.e., unseparated. We generalize tropical floor plans to recover the count of multinodal curves. We then prove that for $$\delta =2$$ or 3 nodes, tropical surfaces with unseparated nodes contribute asymptotically to the second-order term of the polynomial giving the degree of the family of complex projective surfaces in $${\mathbb {P}}^3$$ of degree d with $$\delta $$ nodes. We classify when two nodes in a surface tropicalize to a vertex dual to a polytope with 6 lattice points, and prove that this only happens for projective degree d surfaces satisfying point conditions in Mikhalkin position when $$d>4$$ .

Smooth complex projective rational surfaces with infinitely many real forms

Abstract We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long-standing open question if a smooth complex projective rational surface has only finitely many non-isomorphic real forms or not.

Savage surfaces

Let G be the topological fundamental group of a given nonsingular complex projective surface. We prove that the Chern slopes c_1^2(S)/c_2(S) of minimal nonsingular surfaces of general type S with \pi_1(S) \simeq G are dense in the interval [1,3] .

The stable cohomology of moduli spaces of sheaves on surfaces

Let $X$ be a smooth, irreducible, complex projective surface, $H$ a polarization on $X$. Let $\gamma = (r, c, \Delta)$ be a Chern character. In this paper, we study the cohomology of moduli spaces of Gieseker semistable sheaves $M_{X,H} (\gamma)$. When the rank $r = 1$, the Betti numbers were computed by Göttsche. We conjecture that if we fix the rank $r \geq 1$ and the first Chern class $c$, then the Betti numbers (and more generally the Hodge numbers) of $M_{X,H} (r, c, \Delta)$ stabilize as the discriminant $\Delta$ tends to infinity and that the stable Betti numbers are independent of $r$ and $c$. In particular, the conjectural stable Betti numbers are determined by Göttsche’s calculation. We present evidence for the conjecture. We analyze the validity of the conjecture under blowup and wall-crossing. We prove that when $X$ is a rational surface and $K_X \cdot H \lt 0$, then the classes $[M_{X,H} (\gamma)]$ stabilize in an appropriate completion of the Grothendieck ring of varieties as $\Delta$ tends to $\infty$. Consequently, the virtual Poincaré and Hodge polynomials stabilize to the conjectural value. In particular, the conjecture holds when $X$ is a rational surface, $H \cdot K_X \lt 0$ and there are no strictly semistable objects in $M_{X,H} (\gamma)$.

Rank-one sheaves and stable pairs on surfaces

Finite Quot schemes were used by Bertram, Johnson, and the first author to study Le Potier's strange duality conjecture on del Pezzo surfaces when one of the moduli spaces is the Hilbert scheme of points. In order to rigorously enumerate the finite Quot scheme, we study the moduli space of limit stable pairs in which the target has rank one on a smooth complex projective surface. We obtain an embedding of this moduli space into a smooth space that induces a perfect obstruction theory. This obstruction theory yields a virtual fundamental class that can be computed explicitly. Because the moduli space coincides with the Quot scheme when they have dimension 0, this makes the desired count of the finite Quot scheme in Bertram et al. rigorous. As another application, we obtain a universality result for tautological integrals on the moduli space of stable pairs.

Curves of maximal moduli on K3 surfaces

Abstract We prove that if X is a complex projective K3 surface and $g>0$ , then there exist infinitely many families of curves of geometric genus g on X with maximal, i.e., g-dimensional, variation in moduli. In particular, every K3 surface contains a curve of geometric genus 1 which moves in a nonisotrivial family. This implies a conjecture of Huybrechts on constant cycle curves and gives an algebro-geometric proof of a theorem of Kobayashi that a K3 surface has no global symmetric differential forms.

Deformations of Polystable Sheaves on Surfaces: Quadraticity Implies Formality

We study relations between the quadraticity of the Kuranishi family of a coherent sheaf on a complex projective scheme and the formality of the DG-Lie algebra of its derived endomorphisms. In particular, we prove that for a polystable coherent sheaf on a smooth complex projective surface the DG-Lie algebra of derived endomorphisms is formal if and only if the Kuranishi family is quadratic.

Tropical floor plans and enumeration of complex and real multi-nodal surfaces

The family of complex projective surfaces in P 3 \mathbb {P}^3 of degree d d having precisely δ \delta nodes as their only singularities has codimension δ \delta in the linear system | O P 3 ( d ) | |{\mathcal O}_{\mathbb {P}^3}(d)| for sufficiently large d d and is of degree N δ , C P 3 ( d ) = ( 4 ( d − 1 ) 3 ) δ / δ ! + O ( d 3 δ − 3 ) N_{\delta ,\mathbb {C}}^{\mathbb {P}^3}(d)=(4(d-1)^3)^\delta /\delta !+O(d^{3\delta -3}) . In particular, N δ , C P 3 ( d ) N_{\delta ,\mathbb {C}}^{\mathbb {P}^3}(d) is polynomial in d d . By means of tropical geometry, we explicitly describe ( 4 d 3 ) δ / δ ! + O ( d 3 δ − 1 ) (4d^3)^\delta /\delta !+O(d^{3\delta -1}) surfaces passing through a suitable generic configuration of n = ( d + 3 3 ) − δ − 1 n=\binom {d+3}{3}-\delta -1 points in P 3 \mathbb {P}^3 . These surfaces are close to tropical limits which we characterize combinatorially, introducing the concept of floor plans for multinodal tropical surfaces. The concept of floor plans is similar to the well-known floor diagrams (a combinatorial tool for tropical curve counts): with it, we keep the combinatorial essentials of a multinodal tropical surface S S which are sufficient to reconstruct S S . In the real case, we estimate the range for possible numbers of real multi-nodal surfaces satisfying point conditions. We show that, for a special configuration w \boldsymbol {w} of real points, the number N δ , R P 3 ( d , w ) N_{\delta ,\mathbb {R}}^{\mathbb {P}^3}(d,\boldsymbol {w}) of real surfaces of degree d d having δ \delta real nodes and passing through w \boldsymbol {w} is bounded from below by ( 3 2 d 3 ) δ / δ ! + O ( d 3 δ − 1 ) \left (\frac {3}{2}d^3\right )^\delta /\delta ! +O(d^{3\delta -1}) . We prove analogous statements for counts of multinodal surfaces in P 1 × P 2 \mathbb {P}^1\times \mathbb {P}^2 and P 1 × P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^1 .

Examples of Surfaces with Canonical Map of Maximal Degree

It was shown by Beauville that if the canonical map $\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then $\operatorname{deg}(\varphi_{|K_M|}) \leq 36$. The first example of a surface with canonical degree $36$ was found by the second author. In this article, we show that for any surface which is a degree four Galois étale cover of a fake projective plane $X$ with the largest possible automorphism group $\operatorname{Aut}(X) = C_7:C_3$ (the unique non-abelian group of order $21$), the base locus of the canonical map is finite, and we verify that $35$ of these surfaces have maximal canonical degree $36$. We also classify all smooth degree four Galois étale covers of fake projective planes, which give possible candidates for surfaces of canonical degree $36$. Finally, we also confirm in this paper the optimal upper bound of the canonical degree of smooth threefolds of general type with sufficiently large geometric genus, related to earlier work of Hacon and Cai.

On topologically trivial automorphisms of compact Kähler manifolds and algebraic surfaces

In this paper, we investigate automorphisms of compact Kähler manifolds with different levels of topological triviality. In particular, we provide several examples of smooth complex projective surfaces X whose groups of C^\infty -isotopically trivial automorphisms, resp. cohomologically trivial automorphisms, have a number of connected components which can be arbitrarily large.

A motivic global Torelli theorem for isogenous K3 surfaces

We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective K3 surfaces are isogenous (i.e. their second rational cohomology groups are Hodge isometric) if and only if their Chow motives are isomorphic as Frobenius algebra objects; this can be regarded as a motivic Torelli-type theorem. We ask whether, more generally, twisted derived equivalent hyper-Kähler varieties have isomorphic Chow motives as (Frobenius) algebra objects and in particular isomorphic graded rational cohomology algebras. In the appendix, we justify introducing the notion of “Frobenius algebra object” by showing the existence of an infinite family of K3 surfaces whose Chow motives are pairwise non-isomorphic as Frobenius algebra objects but isomorphic as algebra objects. In particular, K3 surfaces in that family are pairwise non-isogenous but have isomorphic rational Hodge algebras.

Pull-back of singular Levi-flat hypersurfaces

We study singular real analytic Levi-flat subsets invariant by singular holomorphic foliations in complex projective spaces. We give sufficient conditions for a real analytic Levi-flat subset to be the pull-back of a semianalytic Levi-flat hypersurface in a complex projective surface under a rational map or to be the pull-back of a real algebraic curve under a meromorphic function. In particular, we give an application to the case of a singular real analytic Levi-flat hypersurface. Our results improve previous ones due to Lebl and Bretas -- Fernandez-Perez -- Mol.

On the formal principle for curves on projective surfaces

We prove that the formal completion of a complex projective surface along a rigid smooth curve with trivial normal bundle determines the birational equivalence class of the surface.

A surface in odd characteristic with discrete and non-finitely generated automorphism group

It was proved by Tien-Cuong Dinh and me that there is a smooth complex projective surface whose automorphism group is discrete and not finitely generated. In this paper, after observing finite generation of the automorphism group of any smooth projective surface birational to any K3 surface over any algebraic closure of the prime field of odd characteristic, we will show that there is a smooth projective surface, birational to some K3 surface, such that the automorphism group is discrete and not finitely generated, over any algebraically closed field of odd characteristic of positive transcendental degree over the prime field.