About

Paul Hacking is a full professor in the Department of Mathematics and Statistics at the University of Massachusetts Amherst. His research focuses on moduli spaces of algebraic varieties, birational geometry, and mirror symmetry. He earned his PhD from Cambridge University in 2001 and has contributed extensively to the fields of algebraic geometry, including the study of compact moduli spaces of surfaces, exceptional vector bundles, and the birational geometry of cluster algebras. His work involves exploring the structure and classification of algebraic varieties, with significant publications on mirror symmetry, log Calabi-Yau surfaces, and the moduli of surfaces with anti-canonical cycles.

Research topics

• Computer Science
• Mathematics
• Artificial Intelligence
• Algorithm
• Paleontology
• Combinatorics
• Pure mathematics
• Geology
• Crystallography

Selected publications

• Symplectomorphisms of some Weinstein4-manifolds

  Geometry & Topology · 2026-03-16

  preprintOpen access1st authorCorresponding

  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.

  PublisherDOI

• Symplectomorphisms of some Weinstein4-manifolds

  Geometry & Topology · 2026-03-16

  articleOpen access1st authorCorresponding
  PublisherDOI

• Smoothing Gorenstein toric Fano 3-folds

  arXiv (Cornell University) · 2024-12-09

  preprintOpen access

  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.

  PublisherOA PDFDOI

• Homological mirror symmetry for logCalabi–Yau surfaces

  Apollo (University of Cambridge) · 2023-03-16

  articleOpen access1st authorCorresponding

  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→C, where M is a Weinstein four-manifold, such that the directed Fukaya category of w is isomorphic to DbCoh(Y), and the wrapped Fukaya category W(M) is isomorphic to DbCoh(Y∖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.

  PublisherDOI

• Theta functions on varieties with effective anti-canonical class

  Memoirs of the American Mathematical Society · 2022 · 53 citations

  • Artificial Intelligence
  • Computer Science
  • Algorithm

  We show that a large class of maximally degenerating families of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional polarized varieties comes 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -(pseudo-)manifold. Wall structures have previously been constructed inductively for cases with locally rigid singularities [Gross and Siebert, <italic>From real affine geometry to complex geometry</italic> (2011)] and by Gromov-Witten theory for mirrors of log Calabi-Yau surfaces and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Baseline 3"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">K3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surfaces [Gross, Pandharipande and Siebert, <italic>The tropical vertex</italic> ; Gross, Hacking and Keel, <italic>Mirror symmetry for log Calabi-Yau surfaces</italic> (2015); Gross, Hacking, Keel, and Siebert, <italic> Theta functions and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Baseline 3"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">K3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surfaces </italic> (In preparation)]. For trivial wall structures on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -torus we retrieve the classical theta functions. We anticipate that wall structures can be constructed quite generally from maximal degenerations. The construction given here then provides the homogeneous coordinate ring of the mirror degeneration along with a canonical basis. The appearance of a canonical basis of sections for certain degenerations points towards a good compactification of moduli of certain polarized varieties via stable pairs, generalizing the picture for K3 surfaces [Gross, Hacking, Keel, and Siebert, <italic> Theta functions and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Baseline 3"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">K3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surfaces </italic> (In preparation)]. Another possible application apart from mirror symmetry may be to geometric quantization of varieties with effective anti-canonical class.

  PublisherOA PDFDOI

• The Mirror of the Cubic Surface

  Cambridge University Press eBooks · 2022-09-30 · 6 citations

  preprintOpen access

  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.

  PublisherOA PDFDOI

• Homological mirror symmetry for logCalabi–Yau surfaces

  Geometry & Topology · 2022 · 25 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Combinatorics

  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→C, where M is a Weinstein four-manifold, such that the directed Fukaya category of w is isomorphic to DbCoh(Y), and the wrapped Fukaya category W(M) is isomorphic to DbCoh(Y∖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.

  PublisherOA PDFDOI

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

  arXiv (Cornell University) · 2020-08-05 · 3 citations

  preprintOpen access1st authorCorresponding

  We conjecture that any connected component $Q$ of the moduli space of triples $(X,E=E_1+\dots+E_n,Θ)$ 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 $Θ$ is an ample divisor not containing any 0-stratum of $E$, is unirational. More precisely: note that $Q$ has a natural embedding into the Kollár-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.

  PublisherOA PDFDOI

• MIRROR SYMMETRY AND CLUSTER ALGEBRAS

  Proceedings of the International Congress of Mathematicians (ICM 2018) · 2019-05-01 · 9 citations

  article1st authorCorresponding
  PublisherDOI

• Canonical bases for cluster algebras

  Journal of the American Mathematical Society · 2017-09-27 · 371 citations

  articleOpen access

  In an earlier work (Publ. Inst. Hautes Études Sci., 122 (2015), 65–168) 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 tropicalizations 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 (Publ. Inst. Hautes Études Sci., 103 (2006), 1–211). In particular, under suitable hypotheses, for each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the partial compactification of an affine cluster variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> given by allowing some frozen variables to vanish, we obtain canonical bases for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 0 Baseline left-parenthesis upper Y comma script upper O Subscript upper Y Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mi>Y</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^0(Y,\mathcal {O}_Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> extending to a basis of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 0 Baseline left-parenthesis upper U comma script upper O Subscript upper U Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>U</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mi>U</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^0(U,\mathcal {O}_U)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the basic affine space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y comma"> <mml:semantics> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Y,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we obtain a canonical basis of each irreducible representation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L Subscript r"> <mml:semantics> <mml:msub> <mml:mi>SL</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\operatorname {SL}_r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , parameterized by a set which each choice of seed identifies with the 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.

  PublisherOA PDFDOI

Recent grants

• Holomorphic Symplectic Varieties, Mirror Symmetry, and Cluster Algebras

  NSF · $200k · 2016–2020

• Moduli of surfaces, vector bundles, and mirror symmetry

  NSF · $171k · 2012–2016

• Moduli problems in algebraic geometry

  NSF · $110k · 2006–2010

Frequent coauthors

• Seán Keel

  19 shared

• Mark Gross

  12 shared

• Bernd Siebert

  The University of Texas at Austin

  8 shared

• Jenia Tevelev

  6 shared

• Mark D. Gross

  University of Colorado Boulder

  5 shared

• Alessio Corti

  3 shared

• Maxim Kontsevich

  3 shared

• Janós Kollár

  2 shared

Labs

• Paul Hacking's Home PagePI