We prove the ascending chain condition for minimal log discrepancies on an arbitrary fixed threefold.
We construct relatively bounded toroidal and toric models of relatively bounded fibrations over curves.
In this paper, we give an explicit result on the Mordell conjecture over function field of characteristic zero. Based on the idea of Dimitrov–Gao–Habegger, we refine Vojta’s proof and use Looper–Silverman–Wilms’ explicit bound on small points to get our main theorem.
Assuming natural variational realization conjectures, we give uniform bounds for the obstruction to the integral Tate conjecture in 1-dimensional families of algebraic varieties over an infinite finitely generated field.
Given a smooth projective variety X and a smooth nef divisor D, we identify genus zero relative Gromov-Witten invariants of (X, D) with (n + 1) relative markings with genus zero orbifold Gromov-Witten invariants of multi-root stacks over the I1-bundle P & ratio;= I(OX (-D) circle plus OX ) with n orbifold markings. This is a generalization of the local-relative correspondence beyond maximal contacts. Repeating this process, we identify genus zero relative Gromov-Witten invariants of ambient insertions with genus zero absolute Gromov-Witten invariants of toric bundles. We also present how this correspondence can be used to compute genus zero two-point relative Gromov-Witten invariants.
Polystability of (twisted) Stokes representations (i.e. wild monodromy representations) will be characterised, in terms of the corresponding differential Galois group (generalising the Zariski closure of the monodromy group in the tame case). This extends some results of Richardson. Further, the intrinsic approach to such results will be established, in terms of reductions of Stokes local systems.
We give a complete description of the behavior of the volume function at the boundary of the pseudoeffective cone of certain Calabi-Yau complete intersections known as Wehler N-folds. We find that the volume function exhibits a pathological behavior when N>=3, we obtain examples of a pseudoeffective R-divisor D for which the volume of D+sA, with s small and A ample, oscillates between two powers of s, and we deduce the sharp regularity of this function answering a question of Lazarsfeld. We also show that h^0(X,[mD]+A) displays a similar oscillatory behavior as m increases, showing that several notions of numerical dimensions of D do not agree and disproving a conjecture of Fujino. We accomplish this by relating the behavior of the volume function along a segment to the visits of a corresponding hyperbolic geodesics to the cusps of a hyperbolic manifold.
Given a smooth genus two curve $C$, the moduli space SU$_C(3)$ of rank three semi-stable vector bundles on $C$ with trivial determinant is a double cover in $\mathbb{P}^8$ branched over a sextic hypersurface, whose projective dual is the famous Coble cubic, the unique cubic hypersurface that is singular along the Jacobian of $C$. In this paper we continue our exploration of the connections of such moduli spaces with the representation theory of $GL_9$, initiated in \cite{GSW} and pursued in \cite{GS, sam-rains1, sam-rains2, bmt}. Starting from a general trivector $v$ in $\wedge^3\mathbb{C}^9$, we construct a Fano manifold $D_{Z_{10}}(v)$ in $G(3,9)$ as a so-called orbital degeneracy locus, and we prove that it defines a family of Hecke lines in SU$_C(3)$. We deduce that $D_{Z_{10}}(v)$ is isomorphic to the odd moduli space SU$_C(3, \mathcal{O}_C(c))$ of rank three stable vector bundles on $C$ with fixed effective determinant of degree one. We deduce that the intersection of $D_{Z_{10}}(v)$ with a general translate of $G(3,7)$ in $G(3,9)$ is a K3 surface of genus $19$.
Using pagoda flop transitions between smooth projective threefolds, a relation is derived between the Euler numbers of moduli spaces of stable pairs which are scheme-theoretically supported on a fixed singular space curve and Euler numbers of Flag Hilbert schemes associated to a plane curve singularity. When the space curve singularity is locally complete intersection, one obtains a relation between the latter and Euler numbers of Hilbert schemes of the space curve singularity. It is also shown that this relation yields explicit results for a class of torus-invariant locally complete intersection singularities. These results spark a series of open questions in low-dimensional topology, combinatorics, and representation theory.
We develop an equivariant version of the non-archimedean Arakelov theory of [BGS95] in the case of toric varieties. We define the equivariant analogues of the non-archimedean differential forms and currents appearing in loc. cit. and relate them to piecewise polynomial functions on the polyhedral complexes defining the toric models. In particular, we give combinatorial characterizations of the Green currents associated to invariant cycles and combinatorial descriptions of the arithmetic Chow groups.
We discover a simple construction of a four-dimensional family of smooth surfaces of general type with $p_g(S)=q(S)=0$, $K^2_S=3$ with cyclic fundamental group $C_{14}$. We use a degeneration of the surfaces in this family to find (complicated) explicit equations of six new pairs of fake projective planes. Our methods for finding new fake projective planes involve nontrivial computer calculations which we hope will be applicable in other settings.
We study a pair consisting of a smooth variety over a field of positive characteristic and a multi-ideal with a real exponent. We prove the finiteness of the set of minimal log discrepancies for a fixed exponent if the dimension is less than or equal to three. We also prove that the set of log canonical thresholds (lct for short) of ideals on a smooth variety in positive characteristic is contained in the set of lct’s of ideals on a smooth variety over C {\mathbb {C}} , assuming the dimension is less than or equal to three. Under the same dimension assumption, it follows that the accumulation points of lct are rational. Our proofs also show the same statements for the higher dimensional case if all such pairs admit log resolutions by a composite of blow-ups by smooth centers.
In this paper, we establish a structure theorem and prove an isomorphism theorem for cohomology groups of pseudo-effective line bundles over holomorphically convex manifolds, which generalizes the results of Takegoshi, Demailly-Peternell-Schneider, Meng-Zhou, and Wu. As applications, we first give an answer to a question proposed by Matsumura, and establish an injectivity theorem for purely log terminal pairs generalized to pseudo-effective line bundles with transcendental singularities, and then we obtain a Kollár-Nadel-Ohsawa type vanishing theorem which extends the results of Matsumura, Fujino, Meng-Zhou, and others.
Let f : A -> B be a family of abelian varieties over a compact Riemann surface B and fix an effective horizontal divisor D subset of A. We study (S, D)integral sections sigma of the family A where S subset of B is arbitrary. These sections sigma are algebraic and satisfy the geometric condition f(sigma(B) boolean AND D) subset of S. Developing the work of Parshin, we establish new quantitative results concerning the finiteness and the polynomial growth of large unions of (S, D)-integral sections where S can vary and is required to be finite only in a thin analytic open subset of B. Such results are out of the range of purely algebraic methods and imply new evidence and interesting phenomena to the Geometric Lang-Vojta Conjecture.
The authors provide correct proofs of Theorems 7.4 and 7.6 of Brian Lehmann and Sho Tanimoto in “Rational curves on prime Fano threefolds of index 1” [J. Algebraic Geom. 30 (2021), 151–188].
We study a pair consisting of a smooth variety over a field of positive characteristic and a multi-ideal with a real exponent. We prove the finiteness of the set of minimal log discrepancies for a fixed exponent if the dimension is less than or equal to three. We also prove that the set of log canonical thresholds (lct for short) of ideals on a smooth variety in positive characteristic is contained in the set of lct's of ideals on a smooth variety over C, assuming the dimension is less than or equal to three. Under the same dimension assumption, it follows that the accumulation points of log canonical thresholds are rational. Our proofs also show the same statements for the higher dimensional case if all such pairs admit log resolutions by a composite of blow-ups by smooth centers.
Suppose that E \mathcal {E} is a vector bundle on a smooth projective variety X X . Given a family of curves C C on X X , we study how the Harder-Narasimhan filtration of E | C \mathcal {E}|_{C} changes as we vary C C in our family. Heuristically we expect that the locus where the slopes in the Harder-Narasimhan filtration jump by μ \mu should have codimension which depends linearly on μ \mu . We identify the geometric properties which determine whether or not this expected behavior holds. We then apply our results to study rank 2 2 bundles on P 2 \mathbb {P}^{2} and to study singular loci of moduli spaces of curves.
The purpose of this paper is to prove a basic $p$-adic comparison theorem for smooth rigid analytic and dagger varieties over the algebraic closure $C$ of a $p$-adic field: $p$-adic pro-\'etale cohomology, in a stable range, can be expressed as a filtered Frobenius eigenspace of de Rham cohomology (over $\bf{B}^+_{\rm dR}$). The key computation is the passage from absolute crystalline cohomology to Hyodo-Kato cohomology and the construction of the related Hyodo-Kato isomorphism. We also "geometrize" our comparison theorem by turning $p$-adic pro-\'etale and syntomic cohomologies into sheaves on the category ${\rm Perf}_C$ of perfectoid spaces over $C$ (this geometrization will be crucial in our proof of the $C_{\rm st}$-conjecture in the sequel to this paper).
Let G G be a finite group, X X be a smooth complex projective variety with a faithful G G -action, and Y Y be a resolution of singularities of X / G X/G . Larsen and Lunts asked whether [ X / G ] − [ Y ] [X/G]-[Y] is divisible by [ A 1 ] [\mathbb {A}^1] in the Grothendieck ring of varieties. We show that the answer is negative if B G BG is not stably rational and affirmative if G G is abelian. The case when X = Z n X=Z^n for some smooth projective variety Z Z and G = S n G=S_n acts by permutation of the factors is of particular interest. We make progress on it by showing that [ Z n / S n ] − [ Z ⟨ n ⟩ / S n ] [Z^n/S_n]-[Z\langle n\rangle / S_n] is divisible by [ A 1 ] [\mathbb {A}^1] , where Z ⟨ n ⟩ Z\langle n\rangle is Ulyanov’s polydiagonal compactification of the n n th configuration space of Z Z .
