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Paul Hacking

Publications and source records attributed to Paul Hacking.

At least 19 recordsLinked to original sources
arXiv2025

Homological mirror symmetry for projective K3 surfaces

We prove the homological mirror symmetry conjecture of Kontsevich for K3 surfaces in the following form: The Fukaya category of a projective K3 surface is equivalent to the derived category of coherent sheaves on the mirror, which is a K3 surface of Picard rank $19$ over the field $\mathbb{C}((q))$ of formal Laurent series. This builds on prior work of Seidel, who proved the theorem in the case of the quartic surface, Sheridan, Lekili--Ueda, and Ganatra--Pardon--Shende.

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arXiv2024

Smoothing Gorenstein toric Fano 3-folds

We introduce admissible Minkowski decomposition data (amd) for a 3-dimensional reflexive polytope P. This notion is defined purely in terms of the combinatorics of P. Denoting by X the Gorenstein toric Fano 3-fold whose fan is the spanning fan (a.k.a. face fan) of P, our first result states that amd for P determine a smoothing of X. Our second result amounts to an effective recipe for computing the Betti numbers of the smoothing.

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arXiv2021

Symplectomorphisms of some Weinstein 4-manifolds

Let M be a Weinstein four-manifold mirror to Y\D for (Y,D) a log Calabi--Yau surface; intuitively, this is typically the Milnor fibre of a smoothing of a cusp singularity. We introduce two families of symplectomorphisms of M: Lagrangian translations, which we prove are mirror to tensors with line bundles; and nodal slide recombinations, which we prove are mirror to automorphisms of (Y,D). The proof uses a detailed compatibility between the homological and SYZ view-points on mirror symmetry. Together with spherical twists, these symplectomorphisms are expected to generate all autoequivalences of the wrapped Fukaya category of M which are compactly supported in a categorical sense. A range of applications is given.

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arXiv2020

Secondary fan, theta functions and moduli of Calabi-Yau pairs

We conjecture that any connected component $Q$ of the moduli space of triples $(X,E=E_1+\dots+E_n,\Theta)$ where $X$ is a smooth projective variety, $E$ is a normal crossing anti-canonical divisor with a 0-stratum, every $E_i$ is smooth, and $\Theta$ is an ample divisor not containing any 0-stratum of $E$, is unirational. More precisely: note that $Q$ has a natural embedding into the Koll\'ar-Shepherd-Barron-Alexeev moduli space of stable pairs, we conjecture that the induced compactification admits a finite cover by a complete toric variety. We construct the associated complete toric fan, generalizing the Gelfand-Kapranov-Zelevinski secondary fan for reflexive polytopes. Inspired by mirror symmetry, we speculate a synthetic construction of the universal family over this toric variety, as the Proj of a sheaf of graded algebras with a canonical basis, whose structure constants are given by counts of non-archimedean analytic disks. In the Fano case and under the assumption that the mirror contains a Zariski open torus, we construct the conjectural universal family, generalizing the families of Kapranov-Sturmfels-Zelevinski and Alexeev in the toric case. In the case of del Pezzo surfaces with an anti-canonical cycle of $(-1)$-curves, we prove the full conjecture.

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arXiv2020

Homological mirror symmetry for log Calabi-Yau surfaces

Given a log Calabi-Yau surface $Y$ with maximal boundary $D$ and distinguished complex structure, we explain how to construct a mirror Lefschetz fibration $w: M \to \mathbb{C}$, where $M$ is a Weinstein four-manifold, such that the directed Fukaya category of $w$ is isomorphic to $D^b \text{Coh}(Y)$, and the wrapped Fukaya category $D^b\mathcal{W} (M)$ is isomorphic to $D^b \text{Coh}(Y \backslash D)$. We construct an explicit isomorphism between $M$ and the total space of the almost-toric fibration arising in the work of Gross-Hacking-Keel; when $D$ is negative definite this is expected to be the Milnor fibre of a smoothing of the dual cusp of $D$. We also match our mirror potential $w$ with existing constructions for a range of special cases of $(Y,D)$, notably in work of Auroux-Katzarkov-Orlov and Abouzaid.

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arXiv2019

The mirror of the cubic surface

This paper expands on a remark in the paper "Mirror Symmetry for Log Calabi-Yau Surfaces I" of the first three authors of this paper, explaining fully how various constructions of the authors apply to give the mirror to the cubic surface. We give a full description of the scattering diagram associated to the cubic surface: this is a particularly nice diagram in which rays of every rational slope occur, but they may all be described. The equation of the mirror cubic family is then derived in two ways, first by using broken lines and then by using more recent constructions involving a direct calculation of Gromov-Witten invariants.

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arXiv2016

Theta functions on varieties with effective anti-canonical class

We show that a large class of maximally degenerating families of n-dimensional polarized varieties come with a canonical basis of sections of powers of the ample line bundle. The families considered are obtained by smoothing a reducible union of toric varieties governed by a wall structure on a real n-(pseudo-)manifold. Wall structures have previously been constructed inductively for cases with locally rigid singularities and by Gromov-Witten theory for mirrors of log Calabi-Yau surfaces and K3 surfaces by various combinations of the authors. For trivial wall structures on the n-torus we retrieve the classical theta functions. Possible applications include mirror symmetry, geometric compactifications of moduli of certain polarized varieties via stable pairs and geometric quantization.

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arXiv2014

Canonical bases for cluster algebras

In previous work, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs). Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture. In particular, under suitable hypotheses, for each Y the partial compactification of an affine cluster variety U given by allowing some frozen variables to vanish, we obtain canonical bases for the ring of functions on Y extending to a basis for functions on U. Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell U in the basic affine space Y we obtain a canonical basis of each irreducible representation of SL_r, parameterized by a set which each choice of seed identifies with integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation theoretic considerations. Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.

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arXiv2013

Flipping surfaces

We study semistable extremal threefold neighborhoods following earlier work of Mori, Koll\'ar, and Prokhorov. We classify possible flips and extend Mori's algorithm for computing flips of extremal neighborhoods of type k2A to more general neighborhoods of type k1A. In fact we show that they belong to the same deformation family as k2A, and we explicitly construct the universal family of extremal neighborhoods. This construction follows very closely Mori's division algorithm, which we interpret as a sequence of mutations in the cluster algebra of rank 2 with general coefficients. We identify, in the versal deformation space of a cyclic quotient singularity, the locus of deformations such that the total space admits a (terminal) antiflip. We show that these deformations come from at most two irreducible components of the versal deformation space. As an application, we give an algorithm for computing stable one-parameter degenerations of smooth projective surfaces (under some conditions) and describe several components of the Koll\'ar-Shepherd-Barron boundary of the moduli space of smooth canonically polarized surfaces of geometric genus zero.

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arXiv2013

Birational geometry of cluster algebras

We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the Laurent phenomenon for cluster algebras (of geometric type), extend Speyer's example of an upper cluster algebra which is not finitely generated, and show that the Fock-Goncharov dual basis conjecture is usually false.

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arXiv2012

Moduli of surfaces with an anti-canonical cycle

We prove a global Torelli theorem for pairs (Y,D), where Y is a smooth projective rational surface and D is an effective anti-canonical divisor which is a cycle of rational curves. This Torelli theorem was conjectured by Friedman in 1984. In addition, we construct natural universal families for such pairs.

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arXiv2011

Compact moduli spaces of surfaces of general type

We give an introduction to the compactification of the moduli space of surfaces of general type introduced by Koll\'ar and Shepherd-Barron and generalized to the case of surfaces with a divisor by Alexeev. The construction is an application of Mori's minimal model program for 3-folds. We review the example of the projective plane with a curve of degree d > 3. We explain a connection between the geometry of the boundary of the compactification of the moduli space and the classification of vector bundles on the surface in the case H^{2,0}=H^1=0.

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arXiv2011

Exceptional bundles associated to degenerations of surfaces

In 1981 J. Wahl described smoothings of surface quotient singularities with no vanishing cycles. Given a smoothing of a projective surface X of this type, we construct an associated exceptional vector bundle on the nearby fiber Y in the case H^{2,0}(Y)=H^1(Y)=0. If Y is the projective plane we show that our construction establishes a bijective correspondence between the possible degenerate surfaces X and exceptional bundles on Y modulo dualizing and tensoring by line bundles. If Y is of general type then our construction establishes a connection between components of the boundary of the moduli space of surfaces deformation equivalent to Y and exceptional bundles on Y.

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arXiv2011

Mirror symmetry for log Calabi-Yau surfaces I

We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anti-canonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curves on Y meeting D in a single point. In the case D is contractible, the family gives a smoothing of the dual cusp, and thus a proof of Looijenga's 1981 cusp conjecture.

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arXiv2008

Homology of tropical varieties

Given a closed subvariety of an algebraic torus, the associated tropical variety is a polyhedral fan in the space of 1-parameter subgroups of the torus which describes the behaviour of the subvariety at infinity. We show that the link of the origin has only top rational homology if a genericity condition is satisfied. Our result is obtained using work of Tevelev and Deligne's theory of mixed Hodge structures.

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arXiv2008

The moduli space of curves is rigid

We prove that the moduli stack of stable curves of genus g with n marked points is rigid, i.e., has no infinitesimal deformations. This confirms the first case of a principle proposed by Kapranov. It can also be viewed as a version of Mostow rigidity for the mapping class group.

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arXiv2008

Smoothable del Pezzo surfaces with quotient singularities

We give a complete classification of del Pezzo surfaces with quotient singularities and Picard rank 1 which admit a Q-Gorenstein smoothing. There are 14 infinite families of toric examples. The surfaces in each family correspond to solutions of a Markov-type equation. The remaining surfaces are obtained as deformations of the toric surfaces or belong to a finite list of sporadic surfaces.

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