Smooth Projective Surface Research Articles

Counting twisted sheaves and S-duality

We suggest a construction of Tanaka-Thomas's Vafa-Witten invariants for étale gerbes over smooth projective surface using the moduli space of μr-gerbe twisted sheaves and Higgs sheaves. Twisted sheaves and their moduli space are naturally used to study the period-index theorem for the corresponding μr-gerbe in the Brauer group of the surface. Deformation and obstruction theory of twisted sheaves and Higgs sheaves behave like general sheaves and Higgs sheaves. We define the virtual fundamental class for the moduli space and the twisted Vafa-Witten invariants using virtual localization and Behrend function techniques. As an application, for the Langlands dual group SU(r)/Zr of SU(r) we define the SU(r)/Zr-Vafa-Witten invariants. We prove the S-duality conjecture of Vafa-Witten for the projective plane in the case of rank two and for K3 surface in the case of prime ranks.

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.

Counterexamples to Fujita's conjecture on surfaces in positive characteristic

We present counterexamples to Fujita's conjecture in positive characteristic. More precisely, given any algebraically closed field k of characteristic p>0 and any positive integer m, we show there exists a smooth projective surface S over k admitting an ample Cartier divisor A such that the adjoint linear system |KS+mA| is not free of base points.

π1—smallDivisors and Fundamental Groups of Varieties

AbstractLasell and Ramachandran show that the existence of rational curves of positive self-intersection on a smooth projective surface $X$ implies that all the finite dimensional linear representations of the fundamental group $\pi _1(X)$ are finite. In this article, we generalize Lasell and Ramachandran’s result to the case of $\pi _1-small$ divisors on quasiprojective varieties. We also study $\pi _1-small$ curves and hyperbolicity properties of smooth projective surfaces of general type with infinite fundamental groups.

The generic isogeny decomposition of the Prym Variety of a cyclic branched covering

Let f:S'longrightarrow S be a cyclic branched covering of smooth projective surfaces over {mathbb {C}} whose branch locus Delta subset S is a smooth ample divisor. Pick a very ample complete linear system |{mathcal {H}}| on S, such that the polarized surface (S, |{mathcal {H}}|) is not a scroll nor has rational hyperplane sections. For the general member [C]in |{mathcal {H}}| consider the mu _{n}-equivariant isogeny decomposition of the Prym variety {{,mathrm{Prym},}}(C'/C) of the induced covering f:C'{:}{=}f^{-1}(C)longrightarrow C: Prym(C′/C)∼∏d|n,d≠1Pd(C′/C).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {{\\,\\mathrm{Prym}\\,}}(C'/C)\\sim \\prod _{d|n,\\ d\\ne 1}{\\mathcal {P}}_{d}(C'/C). \\end{aligned}$$\\end{document}We show that for the very general member [C]in |{mathcal {H}}| the isogeny component {mathcal {P}}_{d}(C'/C) is mu _{d}-simple with {{,mathrm{End},}}_{mu _{d}}({mathcal {P}}_{d}(C'/C))cong {mathbb {Z}}[zeta _{d}]. In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map {mathcal {P}}_{d}(C'/C)subset {{,mathrm{Jac},}}(C')longrightarrow {{,mathrm{Alb},}}(S').

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.

Central extensions and the Riemann-Roch theorem on algebraic surfaces

We study canonical central extensions of the general linear group over the ring of adeles on a smooth projective algebraic surface $X$ by means of the group of integers. Via these central extensions and the adelic transition matrices of a rank $n$ locally free sheaf of $\mathcal{O}_X$-modules we obtain a local (adelic) decomposition for the difference of Euler characteristics of this sheaf and the sheaf $\mathcal{O}_X^n$. Two distinct calculations of this difference lead to the Riemann-Roch theorem on $X$ (without Noether's formula). Bibliography: 21 titles.

Slope inequalities and a Miyaoka–Yau type inequality

For a minimal smooth projective surface S of general type over a field of characteristic p>0 , we prove that K^2_S\le 32\chi(\mathcal{O}_S). Moreover, if 18\chi(\mathcal{O}_S)<K^2_S\le 32\chi(\mathcal{O}_S) , the Albanese morphism of S must induce a genus 2 fibration. A classification of surfaces with K^2_S=32\chi(\mathcal{O}_S) is also given. The inequality also implies \chi(\mathcal{O}_S)>0 , which answers completely a question of Shepherd-Barron.

Naive 𝔸1-Homotopies on Ruled Surfaces

Abstract We explicitly describe the $\mathbb{A}^1$-chain homotopy classes of morphisms from a smooth henselian local scheme into a smooth projective surface, which is birationally ruled over a curve of genus $> 0$. We consequently determine the sheaf of naive $\mathbb{A}^1$-connected components of such a surface and show that it does not agree with the sheaf of its genuine $\mathbb{A}^1$-connected components when the surface is not a minimal model. However, the sections of the sheaves of both naive and genuine $\mathbb{A}^1$-connected components over schemes of dimension $\leq 1$ agree. As a consequence, we show that the Morel–Voevodsky singular construction on a smooth projective surface, which is birationally ruled over a curve of genus $> 0$, is not $\mathbb{A}^1$-local if the surface is not a minimal model.

On the BMY inequality on surfaces

In this paper, we are concerned with the relation between the ordinarity of surfaces of general type and the failure of the BMY inequality in positive characteristic. We consider semistable fibrations where S is a smooth projective surface and C is a smooth projective curve. Using the exact sequence relating the locally exact differential forms on S, C, and S/C, we prove an inequality relating and c 2 for ordinary surfaces which admit generically ordinary semistable fibrations. This inequality differs from the BMY inequality by a correcting term which vanishes if the fibration is ordinary.

Stable maps to Looijenga pairs: orbifold examples

In arXiv:2011.08830 we established a series of correspondences relating five enumerative theories of log Calabi-Yau surfaces, i.e. pairs $(Y,D)$ with $Y$ a smooth projective complex surface and $D=D_1+\dots +D_l$ an anticanonical divisor on $Y$ with each $D_i$ smooth and nef. In this paper we explore the generalisation to $Y$ being a smooth Deligne-Mumford stack with projective coarse moduli space of dimension 2, and $D_i$ nef $\mathbb{Q}$-Cartier divisors. We consider in particular three infinite families of orbifold log Calabi-Yau surfaces, and for each of them we provide closed form solutions of the maximal contact log Gromov-Witten theory of the pair $(Y,D)$, the local Gromov-Witten theory of the total space of $\bigoplus_i \mathcal{O}_Y(-D_i)$, and the open Gromov-Witten theory of toric orbi-branes in a Calabi-Yau 3-orbifold associated to $(Y,D)$. We also consider new examples of BPS integral structures underlying these invariants, and relate them to the Donaldson-Thomas theory of a symmetric quiver specified by $(Y,D)$, and to a class of open/closed BPS invariants.

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 variety that cannot be dominated by one that lifts

We prove a strong version of a theorem of Siu and Beauville on morphisms to higher genus curves, and we use it to show that if a variety X in characteristic p lifts to characteristic 0, then any morphism X→C to a curve of genus g≥2 can be lifted along. We use this to construct, for every prime p, a smooth projective surface X over F¯p that cannot be rationally dominated by a smooth proper variety Y that lifts to characteristic 0.

Bounding cohomology on a smooth projective surface with Picard number 2

The following conjecture arose out of discussions between B. Harbourne, J. Roé, C. Cilberto and R. Miranda: for a smooth projective surface X there exists a positive constant cX such that for every prime divisor C on X. When the Picard number we prove that if either the Kodaira dimension and X has a negative curve or X has two negative curves, then this conjecture holds for X.

Convergence of the mirror to a rational elliptic surface

The construction introduced by Gross, Hacking and Keel allows one to construct a formal mirror family to a pair $(S,D)$ where $S$ is a smooth rational projective surface and $D$ a certain type of Weil divisor supporting an ample or anti-ample class. In that paper they proved two convergence results. Firstly that if the intersection matrix of $D$ is not negative semi-definite then the family they construct lifts to an algebraic family. Secondly they prove that if the intersection matrix is negative definite then their construction lifts along certain analytic strata on the base, and then over a formal neighbourhood of this. In the original version of that paper they claimed that if the intersection matrix were negative semi-definite then family in fact extends over an analytic neighbourhood of the origin but gave an incorrect proof. In this paper we correct this error. We explain how the general Gross-Siebert program can be used to reduce construction of the mirror to such a surface to calculating certain relative Gromov-Witten invariants. We then relate these invariants to the invariants of a new space where we can find explicit formulae for the invariants. From this we deduce analytic convergence of the mirror family, at least when the original surface has an $I_4$ fibre.

Formality conjecture for minimal surfaces of Kodaira dimension 0

Let $\mathcal {F}$ be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the differential graded (DG) Lie algebra $R\operatorname {Hom}(\mathcal {F},\mathcal {F})$ of derived endomorphisms of $\mathcal {F}$ is formal. The proof is based on the study of equivariant $L_{\infty }$ minimal models of DG Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for K3 surfaces) on previous results on the same subject.

Counting Rational Curves on K3 Surfaces With Finite Group Actions

Abstract Göttsche gave a formula for the dimension of the cohomology of Hilbert schemes of points on a smooth projective surface $S$. When $S$ admits an action by a finite group $G$, we describe the action of $G$ on the Hodge structure. In the case that $S$ is a K3 surface, each element of $G$ gives a trace on $\sum _{n=0}^{\infty }\sum _{i=0}^{\infty }(-1)^{i}H^{i}(S^{[n]},\mathbb{C})q^{n}$. When $G$ acts faithfully and symplectically on $S$, the resulting generating function is of the form $q/f(q)$, where $f(q)$ is a cusp form. We relate the Hodge structure of Hilbert schemes of points to the Hodge structure of the compactified Jacobian of the tautological family of curves over an integral linear system on a K3 surface as $G$-representations. Finally, we give a sufficient condition for a $G$-orbit of curves with nodal singularities not to contribute to the representation.

On automorphisms and the cone conjecture for Enriques surfaces in odd characteristic

We prove that, for an Enriques surface in odd characteristic, the automorphism group is finitely generated and it acts on the effective nef cone with a rational polyhedral fundamental domain. We also construct a smooth projective surface in odd characteristic which is birational to an Enriques surface and whose automorphism group is discrete but not finitely generated.

Int-amplified Endomorphisms on Normal Projective Surfaces

We investigate int-amplified endomorphisms on normal projective surfaces. We prove that the output of the equivariant MMP is either a Q-abelian surface, a (equivariant) quasi-étale quotient of a smooth projective surface, a Mori dream space, or a projective cone of an elliptic curve.