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Let V be a smooth quasi-projective complex surface such that the first three logarithmic plurigenera
$\overline P_1(V)$
,
$\overline P_2(V)$
and
$\overline P_3(V)$
are equal to 1 and the logarithmic irregularity
$\overline q(V)$
is equal to
$2$
. We prove that the quasi-Albanese morphism
$a_V\colon V\to A(V)$
is birational and there exists a finite set S such that
$a_V$
is proper over
$A(V)\setminus S$
, thus giving a sharp effective version of a classical result of Iitaka [12].
The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let X be a nonsingular irreducible complex surface, and let E be a vector bundle of rank n on X. We use the m-elementary transformation of E at a point
$x \in X$
to show that there exists an embedding from the Grassmannian variety
$\mathbb{G}(E_x,m)$
into the moduli space of torsion-free sheaves
$\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$
which induces an injective morphism from
$X \times M_{X,H}(n;\,c_1,c_2)$
to
$Hilb_{\, \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$
.
We introduce a conjecture on Virasoro constraints for the moduli space of stable sheaves on a smooth projective surface. These generalise the Virasoro constraints on the Hilbert scheme of a surface found by Moreira and Moreira, Oblomkov, Okounkov and Pandharipande. We verify the conjecture in many nontrivial cases by using a combinatorial description of equivariant sheaves found by Klyachko.
We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-Kähler manifold using the extended Mukai lattice. This enables us to define a Mukai vector for certain objects in the derived category taking values inside the extended Mukai lattice which is functorial for derived equivalences. As applications, we obtain a structure theorem for derived equivalences between hyper-Kähler manifolds as well as an integral lattice associated to the derived category of hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of a K3 surface mimicking the surface case.
The most fundamental example of mirror symmetry compares the Fermat hypersurfaces in
$\mathbb {P}^n$
and
$\mathbb {P}^n/G$
, where G is a finite group that acts on
$\mathbb {P}^n$
and preserves the Fermat hypersurface. We generalize this to hypersurfaces in Grassmannians, where the picture is richer and more complex. There is a finite group G that acts on the Grassmannian
$\operatorname {{\mathrm {Gr}}}(n,r)$
and preserves an appropriate Calabi–Yau hypersurface. We establish how mirror symmetry, toric degenerations, blow-ups and variation of GIT relate the Calabi–Yau hypersurfaces inside
$\operatorname {{\mathrm {Gr}}}(n,r)$
and
$\operatorname {{\mathrm {Gr}}}(n,r)/G$
. This allows us to describe a compactification of the Eguchi–Hori–Xiong mirror to the Grassmannian, inside a blow-up of the quotient of the Grassmannian by G.
We prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In generic cases, this gives a bijection between the set ${\mathcal Enr}(X)$ of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank $20,$ we prove that the fibers of ${\mathcal Enr}(X)\to \mathrm {{Br}}(X)[2]$ above the nonzero points have the same cardinality.
For a nonconstant elliptic surface over $\mathbb {P}^1$ defined over $\mathbb {Q}$, it is a result of Silverman [‘Heights and the specialization map for families of abelian varieties’, J. reine angew. Math.342 (1983), 197–211] that the Mordell–Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is nonisotrivial, one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann [‘Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. I’, Math. Nachr.49 (1971), 107–123] and Setzer [‘Elliptic curves of prime conductor’, J. Lond. Math. Soc. (2)10 (1975), 367–378]. In this note, we show that the Legendre elliptic surface has the desired property, conditional on the existence of infinitely many Mersenne primes.
We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups:
${\mathfrak{A}}_5$
,
${\text{PSL}}_2(\textbf{F}_7)$
,
${\mathfrak{A}}_6$
,
${\text{SL}}_2(\textbf{F}_8)$
,
${\mathfrak{A}}_7$
,
${\text{PSp}}_4(\textbf{F}_3)$
,
${\text{SL}}_2(\textbf{F}_{7})$
,
$2.{\mathfrak{A}}_5$
,
$2.{\mathfrak{A}}_6$
,
$3.{\mathfrak{A}}_6$
or
$6.{\mathfrak{A}}_6$
. All of these groups with a possible exception of
$2.{\mathfrak{A}}_6$
and
$6.{\mathfrak{A}}_6$
indeed act on some rationally connected threefolds.
In this paper, we prove a Clifford type inequality for the curve $X_{2,2,2,4}$, which is the intersection of a quartic and three general quadratics in $\mathbb {P}^5$. We thus prove a stronger Bogomolov–Gieseker inequality for characters of stable vector bundles and stable objects on Calabi–Yau complete intersection $X_{2,4}$. Applying the scheme proposed by Bayer, Bertram, Macrì, Stellari and Toda, we can construct an open subset of Bridgeland stability conditions on $X_{2,4}$.
In this note, we prove the semiampleness conjecture for Kawamata log terminal Calabi–Yau (CY) surface pairs over an excellent base ring. As applications, we deduce that generalized abundance and Serrano’s conjecture hold for surfaces. Finally, we study the semiampleness conjecture for CY threefolds over a mixed characteristic DVR.
In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type
$\mathbf {A}_1+\mathbf {A}_3$
and prove an analogue of Manin’s conjecture for integral points with respect to its singularities and its lines.
We study lc pairs polarized by a nef and log big divisor. After proving the minimal model theory for projective lc pairs polarized by a nef and log big divisor, we prove the effectivity of the Iitaka fibrations and some boundedness results for dlt pairs polarized by a nef and log big divisor.
Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either
$X_{\overline {K}}$
has infinitely many rational curves or X has infinitely many unirational specialisations.
Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K.
In characteristic
$0$
, symplectic automorphisms of K3 surfaces (i.e., automorphisms preserving the global
$2$
-form) and non-symplectic ones behave differently. In this paper, we consider the actions of the group schemes
$\mu _{n}$
on K3 surfaces (possibly with rational double point [RDP] singularities) in characteristic p, where n may be divisible by p. We introduce the notion of symplecticness of such actions, and we show that symplectic
$\mu _{n}$
-actions have similar properties, such as possible orders, fixed loci, and quotients, to symplectic automorphisms of order n in characteristic
$0$
. We also study local
$\mu _n$
-actions on RDPs.
This paper gives a description of the full space of Bridgeland stability conditions on the bounded derived category of a contraction algebra associated to a
$3$
-fold flop. The main result is that the stability manifold is the universal cover of a naturally associated hyperplane arrangement, which is known to be simplicial and in special cases is an ADE root system. There are four main corollaries: (1) a short proof of the faithfulness of pure braid group actions in both algebraic and geometric settings, the first that avoid normal forms; (2) a classification of tilting complexes in the derived category of a contraction algebra; (3) contractibility of the stability space associated to the flop; and (4) a new proof of the
$K(\unicode{x3c0} \,,1)$
-theorem in various finite settings, which includes ADE braid groups.
We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type
$\frac {1}{4}(1,1)$
. The moduli space of such log-Enriques surfaces with k singular points is a modular variety of orthogonal type associated with a unimodular lattice of signature
$(2,10-k)$
. We prove that the invariant, viewed as a function of the modular variety, is given by the Petersson norm of an explicit Borcherds product. We note that this torsion invariant is essentially the BCOV invariant in the complex dimension
$2$
. As a consequence, the BCOV invariant in this case is not a birational invariant, unlike the Calabi-Yau case.
The mathematical physicists Bershadsky–Cecotti–Ooguri–Vafa (BCOV) proposed, in a seminal article from 1994, a conjecture extending genus zero mirror symmetry to higher genera. With a view towards a refined formulation of the Grothendieck–Riemann–Roch theorem, we offer a mathematical description of the BCOV conjecture at genus one. As an application of the arithmetic Riemann–Roch theorem of Gillet–Soulé and our previous results on the BCOV invariant, we establish this conjecture for Calabi–Yau hypersurfaces in projective spaces. Our contribution takes place on the B-side, and together with the work of Zinger on the A-side, it provides the first complete examples of the mirror symmetry program in higher dimensions. The case of quintic threefolds was studied by Fang–Lu–Yoshikawa. Our approach also lends itself to arithmetic considerations of the BCOV invariant, and we study a Chowla–Selberg type theorem expressing it in terms of special
$\Gamma $
-values for certain Calabi–Yau manifolds with complex multiplication.
O’Grady’s generalised Franchetta conjecture (GFC) is concerned with codimension 2 algebraic cycles on universal polarised K3 surfaces. In [4], this conjecture has been studied in the Betti cohomology groups. Following a suggestion of Voisin, we investigate this problem in the Deligne-Beilinson (DB) cohomology groups. In this paper, we develop the theory of Deligne-Beilinson cohomology groups on (smooth) Deligne-Mumford stacks. Using the automorphic cohomology group and Noether-Lefschetz theory, we compute the 4th DB-cohomology group of universal oriented polarised K3 surfaces with at worst an
$A_1$
-singularity and show that GFC for such family holds in DB-cohomology. In particular, this confirms O’Grady’s original conjecture in DB cohomology.