Abstract In this paper we provide applications of general results of Baldi-Klingler-Ullmo and Khelifa-Urbanik on the geometry of the Hodge locus associated to an integral polarized variation of Hodge structures to the case of Noether-Lefschetz loci for families of smooth surfaces. In particular, we consider the family of smooth surfaces in the linear system of a sufficiently ample line bundle on a smooth projective threefold Y , in case Y is a Fano or a Calabi-Yau threefold, and we prove results on the different behaviour of the union of the general, respectively exceptional, components of its Noether-Lefschetz locus.

The main aim of this paper is to construct a complex analytic family of symmetric projective K3 surfaces through a compactifiable deformation family of complete quasi-projective varieties from $\operatorname{CP}^2 \#9\overline{\operatorname{CP}}^2$. Firstly, for an elliptic curve $C_0$ embedded in $\operatorname{CP}^2$, let $S \cong \operatorname{CP}^2 \#9\overline{\operatorname{CP}}^2$ be the blow up of $\operatorname{CP}^2$ at nine points on the image of $C_0$ and $C$ be the strict transform of the image. Then if the normal bundle satisfies the Diophantine condition, a tubular neighborhood of the elliptic curve $C$ can be identified through a toroidal group. Fixing the Diophantine condition, a smooth compactifiable deformation of $S\backslash C$ over a 9-dimensional complex manifold is constructed. Moreover, with an ample line bundle fixed on $S$, complete Kähler metrics can be constructed on the quasi-projective variety $S\backslash C$. So complete Kähler metrics are constructed on each quasi-projective variety fiber of the smooth compactifiable deformation family. Then a complex analytic family of symmetric projective K3 surfaces over a 10-dimensional complex manifold is constructed through the smooth compactifiable deformation family of complete quasi-projective varieties and an analogous deformation family.

Abstract Let X be an algebraic surface with $$\mathcal {L}$$ an ample line bundle on X. Let $$\Gamma (X, \mathcal {L})$$ be the geometric monodromy group associated to a family of nonsingular curves in X, that are zero loci of sections of $$\mathcal {L}$$ . We provide obstructions to $$\Gamma (X, \mathcal {L})$$ being finite index in the mapping class group. We also show that for any $$k \ge 0$$ , the image of monodromy is finite index in appropriate subgroups of the quotient of the mapping class group by the $$k^{th}$$ term of the Johnson filtration, assuming that $$\mathcal {L}$$ is sufficiently ample. This enables us to construct several subgroups of the mapping class group with unusual properties, in some cases providing the first examples of subgroups with those properties.

Let M be a moduli space of stable vector bundles of rank r and determinant ξ on a compact Riemann surface X. Fix a semistable holomorphic vector bundle F on X such that χ(E ⊗ F) = 0 for E∈M. Then any E∈M with H0(X, E ⊗ F) = 0 = H1(X, E ⊗ F) has a natural holomorphic projective connection. The moduli space of pairs (E, ∇), where E∈M and ∇ is a holomorphic projective connection on E, is an algebraic T*M–torsor on M. We identify this T*M–torsor on M with the T*M–torsor given by the sheaf of connections on an ample line bundle over M.

Abstract We give a criterion for slope-stability of the syzygy bundle of a globally generated ample line bundle on a smooth projective variety of Picard number $1$ in terms of Hilbert polynomial. As applications, we prove the stability of syzygy bundles on many varieties, such as smooth Fano or Calabi–Yau complete intersections, hyperkähler varieties of Picard number 1, abelian varieties of Picard number $1$, rational homogeneous varieties of Picard number 1, weak Calabi–Yau varieties of Picard number $1$ of dimension $\leq 4$, and Fano varieties of Picard number $1$ of dimension $\leq 5$. Also, we prove the stability of syzygy bundles on all hyperkähler varieties.

We show that if $f\colon X \to T$ is a surjective morphism between smooth projective varieties over an algebraically closed field $k$ of characteristic $p>0$ with geometrically integral and non-uniruled generic fiber, then $K_{X/T}$ is pseudo-effective. The proof is based on covering $X$ with rational curves, which gives a contradiction as soon as both the base and the generic fiber are not uniruled. However, we assume only that the generic fiber is not uniruled. Hence, the hardest part of the proof is to show that there is a finite smooth non-uniruled cover of the base for which we show the following: If $T$ is a smooth projective variety over $k$ and $\mathcal{A}$ is an ample enough line bundle, then a cyclic cover of degree $p \nmid d$ given by a general element of $\left|\mathcal{A}^d\right|$ is not uniruled. For this we show the following cohomological uniruledness condition, which might be of independent interest: A smooth projective variety $T$ of dimenion $n$ is not uniruled whenever the dimension of the semi-stable part of $H^n(T, \mathcal{O}_T)$ is greater than that of $H^{n-1}(T, \mathcal{O}_T)$. Additionally, we also show singular versions of all the above statements.Comment: Comments are more than welcome

A sequence of point configurations on a compact complex manifold is asymptotically Fekete if it is close to maximizing a sequence of Vandermonde determinants. These Vandermonde determinants are defined by tensor powers of a Hermitian ample line bundle and the point configurations in the sequence possess good sampling properties with respect to sections of the line bundle. In this paper, given a collection of Hermitian ample line bundles, we address the question of existence of a sequence of point configurations which is asymptotically Fekete with respect to each of the line bundles. We give a conjectural necessary and sufficient condition and prove this in the toric case.

Abstract Let X be a smooth projective variety over a complete discretely valued field of mixed characteristic. We solve non-Archimedean Monge–Ampère equations on X assuming resolution and embedded resolution of singularities. We follow the variational approach of Boucksom, Favre, and Jonsson proving the continuity of the plurisubharmonic envelope of a continuous metric on an ample line bundle on X. We replace the use of multiplier ideals in equicharacteristic zero by the use of perturbation friendly test ideals introduced by Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron, and Witaszek building upon previous constructions by Hacon, Lamarche, and Schwede.

Abstract Let be integers and let denote the Hirzebruch surface with invariant . We compute the Seshadri constants of an ample line bundle at an arbitrary point of the ‐point blow‐up of when and at a very general point when or . We also discuss several conjectures on linear systems of curves on the blow‐up of at very general points.

Abstract We study when the Picard group of smooth surfaces of degree $d\geq 5$ in $\mathbb{P}^{3}$ acquires extra classes. In particular we show that the so-called exceptional components of the Noether–Lefschetz locus are not Zariski dense. This answers a 1991 question of C. Voisin. We also obtain similar results for the Noether–Lefschetz locus for suitable $(Y,L)$, where $Y$ is a smooth projective three-fold and $L$ a very ample line bundle. Both results are applications of the Zilber–Pink viewpoint recently developed by the authors for arbitrary (polarized, integral) variations of Hodge structures.

Abstract Let $X$ be a variety with a stratification ${\mathcal{S}}$ into smooth locally closed subvarieties such that $X$ is locally a product along each stratum (e.g., a symplectic singularity). We prove that assigning to each open subset $U \subset X$ the set of isomorphism classes of locally projective crepant resolutions of $U$ defines an ${\mathcal{S}}$-constructible sheaf of sets. Thus, for each stratum $S$ and basepoint $s \in S$, the fundamental group acts on the set of germs of projective crepant resolutions at $s$, leaving invariant the germs extending to the entire stratum. Global locally projective crepant resolutions correspond to compatible such choices for all strata. For example, if the local projective crepant resolutions are unique, they automatically glue uniquely. We give criteria for a locally projective crepant resolution $\rho : \tilde X \to X$ to be globally projective. We show that the sheafification of the presheaf $U \mapsto \operatorname{Pic}(\rho ^{-1}(U)/U)$ of relative Picard classes is also constructible. The resolution is globally projective only if there exist local relatively ample bundles whose classes glue to a global section of this sheaf. The obstruction to lifting this section to a global ample line bundle is encoded by a gerbe on the singularity $X$. We show the gerbes are automatically trivial if $X$ is a symplectic quotient singularity. Our main results hold in the more general setting of partial crepant resolutions, that need not have smooth source. We apply the theory to symmetric powers and Hilbert schemes of surfaces with du Val singularities, finite quotients of tori, multiplicative, and Nakajima quiver varieties, as well as to canonical three-fold singularities.

We prove that any commutative group scheme over an arbitrary base scheme of finite type over a field with connected fibers and admitting a relatively ample line bundle is polarizable in the sense of Ng\^o. This extends the applicability of Ng\^o's support theorem to new cases, for example to Lagrangian fibrations with integral fibers and has consequences to the construction of algebraic classes.Comment: comments welcome

Line bundles on the moduli space of Lie algebroid connections over a curve

Let X be a smooth curve of genus g which is a multiple covering ϕ : X → Y of a smooth curve Y. We define an invariant Δ ϕ of the map ϕ , which is the maximum among the integers m such that for a base point free line bundle L on X with deg L ≤ g − 1 the inequality Cliff ( L ) ≤ m implies that L is composed with ϕ . Our main result is that a very ample special line bundle L on X with deg L > 3 g − 3 2 is normally generated when Cliff ( L ) ≤ Δ ϕ under additional numerical conditions.

Degree of the 3-secant variety

Abstract A notable example due to Heier, Lu, Wong, and Zheng shows that there exist compact complex Kähler manifolds with ample canonical line bundle such that the holomorphic sectional curvature is negative semi‐definite and vanishes along high‐dimensional linear subspaces in every tangent space. The main result of this note is an upper bound for the dimensions of these subspaces. Due to the holomorphic sectional curvature being a real‐valued bihomogeneous polynomial of bidegree (2,2) on every tangent space, the proof is based on making a connection with the work of D'Angelo on complex subvarieties of real algebraic varieties and the decomposition of polynomials into differences of squares. Our bound involves an invariant that we call the holomorphic sectional curvature square decomposition length, and our arguments work as long as the holomorphic sectional curvature is semi‐definite, be it negative or positive.

Let X ⊂ P r X \subset \mathbb {P}^r be a linearly normal variety defined by a very ample line bundle L L on a projective variety X X . Recently it is shown by Kangjin Han, Wanseok Lee, Hyunsuk Moon, and Euisung Park [Compos. Math. 157 (2021), pp. 2001–2025] that there are many cases where ( X , L ) (X,L) satisfies property Q R ( 3 ) \mathsf {QR} (3) in the sense that the homogeneous ideal I ( X , L ) I(X,L) of X X is generated by quadratic polynomials of rank 3 3 . The locus Φ 3 ( X , L ) \Phi _3 (X,L) of rank 3 3 quadratic equations of X X in P ( I ( X , L ) 2 ) \mathbb {P}\left ( I(X,L)_2 \right ) is a projective algebraic set, and property Q R ( 3 ) \mathsf {QR} (3) of ( X , L ) (X,L) is equivalent to that Φ 3 ( X ) \Phi _3 (X) is nondegenerate in P ( I ( X ) 2 ) \mathbb {P}\left ( I(X)_2 \right ) . In this paper we study geometric structures of Φ 3 ( X , L ) \Phi _3 (X,L) such as its minimal irreducible decomposition. Let Σ ( X , L ) = { ( A , B ) ∣ A , B ∈ P i c ( X ) , L = A 2 ⊗ B , h 0 ( X , A ) ≥ 2 , h 0 ( X , B ) ≥ 1 } . \begin{equation*} \Sigma (X,L) \!=\! \{ (A,B) \mid A,B \!\in \! {Pic}(X),~L \!=\! A^2 \otimes B,~h^0 (X,A) \!\geq \! 2,~h^0 (X,B) \!\geq \! 1 \}. \end{equation*} We first construct a projective subvariety W ( A , B ) ⊂ Φ 3 ( X , L ) W(A,B) \subset \Phi _3 (X,L) for each ( A , B ) (A,B) in Σ ( X , L ) \Sigma (X,L) . Then we prove that the equality Φ 3 ( X , L ) = ⋃ ( A , B ) ∈ Σ ( X , L ) W ( A , B ) \begin{equation*} \Phi _3 (X,L) ~=~ \bigcup _{(A,B) \in \Sigma (X,L)} W(A,B) \end{equation*} holds when X X is locally factorial. Thus this is an irreducible decomposition of Φ 3 ( X , L ) \Phi _3 (X,L) when P i c ( X ) {Pic} (X) is finitely generated and hence Σ ( X , L ) \Sigma (X,L) is a finite set. Also we find a condition that the above irreducible decomposition is minimal. For example, it is a minimal irreducible decomposition of Φ 3 ( X , L ) \Phi _3 (X,L) if P i c ( X ) {Pic}(X) is generated by a very ample line bundle.

Abstract The Chebyshev potential of a Hermitian metric on an ample line bundle over a projective variety, introduced by Witt Nyström, is a convex function defined on the Okounkov body. It is a generalization of the symplectic potential of a torus‐invariant Kähler potential on a toric variety, introduced by Guillemin, that is a convex function on the Delzant polytope. A folklore conjecture asserts that a curve of Chebyshev potentials associated to a subgeodesic in the space of positively curved Hermitian metrics is linear in the time variable if and only if the subgeodesic is a geodesic in the Mabuchi metric. This is classically true in the special toric setting, and in general Witt Nyström established the sufficiency. The main obstacle in the conjecture is that it is difficult to compute Chebyshev potentials, that are currently only known on the Riemann sphere and toric varieties. The goal of this article is to disprove this conjecture. To that end we characterize the geodesics consisting of Fubini–Study metrics for which the conjecture is true on the hyperplane bundle of the projective space. The proof involves explicitly solving the Monge–Ampère equation describing geodesics on the subspace of Fubini–Study metrics and computing their Chebyshev potentials.

We study syzygies of Kummer varieties proving that their behavior is half of the abelian varieties case. Namely, an m m th power of an ample line bundle on a Kummer variety satisfies the Green-Lazarsfeld property ( N p ) (N_p) , if m > p + 2 2 m > \frac {p+2}{2} .

We solve the Ein–Lazarsfeld–Mustopa conjecture for the blow up of a projective space along a linear subspace. More precisely, let X be the blow up of {mathbb {P}}^n at a linear subspace and let L be any ample line bundle on X. We show that the syzygy bundle M_{L} defined as the kernel of the evalution map H^{0}(X,L)otimes {mathcal {O}}_{X}rightarrow L is L-stable. In the last part of this note we focus on the rigidness of M_{L} to study the local shape of the moduli space around the point [M_{L}].