## Publications

64 results found

Thomas R, Feyzbakhsh S, 2023, Rank r DT theory from rank 0, *Duke Mathematical Journal*, ISSN: 0012-7094

Thomas R, Oh J, 2023, Counting sheaves on Calabi-Yau 4-folds, I, *Duke Mathematical Journal*, Vol: 172, Pages: 1333-1409, ISSN: 0012-7094

Borisov-Joyce constructed a real virtual cycle on compactmoduli spaces of stable sheaves on Calabi-Yau 4-folds, using deriveddifferential geometry.We construct an algebraic virtual cycle. A key step is a localisationof Edidin-Graham’s square root Euler class for SOp2n, Cq bundles tothe zero locus of an isotropic section, or to the support of an isotropiccone.We prove a torus localisation formula, making the invariants computable and extending them to the noncompact case when the fixedlocus is compact.We give a K-theoretic refinement by defining K-theoretic square rootEuler classes and their localised versions.In a sequel we prove our invariants reproduce those of Borisov-Joyce.

Feyzbakhsh S, Thomas R, 2023, Curve counting and S-duality, *Épijournal de Géométrie Algébrique*, Vol: 7, Pages: 1-25, ISSN: 2491-6765

We work on a projective threefold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as P3 or the quintic threefold. We prove certain moduli spaces of 2-dimensional torsion sheaves on X are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in X. When X is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. These latter invariants are predicted to have modular properties which we discuss from the point of view of S-duality and Noether-Lefschetz theory.

Feyzbakhsh S, Thomas RP, 2023, Rank r DT theory from rank 1, *Journal of the American Mathematical Society*, Vol: 36, Pages: 795-826, ISSN: 0894-0347

Fix a Calabi-Yau 3-fold X satisfying the Bogomolov-Gieseker conjecture ofBayer-Macr`ı-Toda, such as the quintic 3-fold.We express Joyce’s generalised DT invariants counting Gieseker semistable sheaves ofany rank r on X in terms of those counting sheaves of rank 1. By the MNOP conjecturethey are therefore determined by the Gromov-Witten invariants of X.

Tanaka Y, Thomas RP, 2020, Vafa-Witten invariants for projective surfaces I: stable case, *Journal of Algebraic Geometry*, Vol: 29, Pages: 603-668, ISSN: 1534-7486

On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a ℂ∗ action with compact fixed locus. Applying virtual localisation we define invariants constant under deformations.When the vanishing theorem of Vafa-Witten holds, the result is the (signed) Euler characteristic of the moduli space of instantons. In general there are other, rational, contributions. Calculations of these on surfaces with positive canonical bundle recover the first terms of modular forms predicted by Vafa and Witten.

Thomas RP, 2020, Equivariant K-theory and refined Vafa-Witten invariants, *Communications in Mathematical Physics*, Vol: 378, Pages: 1451-1500, ISSN: 0010-3616

In [MT2] the Vafa-Witten theory of complex projective surfaces is lifted tooriented $\mathbb C^*$-equivariant cohomology theories. Here we study theK-theoretic refinement. It gives rational functions in $t^{1/2}$ invariantunder $t^{1/2}\leftrightarrow t^{-1/2}$ which specialise to numericalVafa-Witten invariants at $t=1$. On the "instanton branch" the invariants give the virtual$\chi_{-t}^{}$-genus refinement of G\"ottsche-Kool. Applying modularity totheir calculations gives predictions for the contribution of the "monopolebranch". We calculate some cases and find perfect agreement. We also docalculations on K3 surfaces, finding Jacobi forms refining the usual modularforms, proving a conjecture of G\"ottsche-Kool. We determine the K-theoretic virtual classes of degeneracy loci usingEagon-Northcott complexes, and show they calculate refined Vafa-Witteninvariants. Using this Laarakker [Laa] proves universality results for theinvariants.

Gholampour A, Thomas R, 2020, Degeneracy loci, virtual cycles and nested Hilbert schemes II, *Compositio Mathematica*, Vol: 156, Pages: 1623-1663, ISSN: 0010-437X

We express nested Hilbert schemes of points and curves ona smooth projective surface as “virtual resolutions” of degeneracy lociof maps of vector bundles on smooth ambient spaces.We show how to modify the resulting obstruction theories to producethe virtual cycles of Vafa-Witten theory and other sheaf-counting prob-lems. The result is an effective way of calculating invariants (VW, SW,local PT and local DT) via Thom-Porteous-like Chern class formulae.

Feyzbakhsh S, Thomas RP, 2019, An application of wall-crossing to Noether-Lefschetz loci, Publisher: arXiv

Consider a smooth projective 3-fold $X$ satisfying the Bogomolov-Giesekerconjecture of Bayer-Macr\`{i}-Toda (such as $\mathbb P^3$, the quinticthreefold or an abelian threefold). Let $L$ be a line bundle supported on a very positive surface in $X$. If$c_1(L)$ is a primitive cohomology class then we show it has very negativesquare.

Maulik D, Thomas RP, 2019, Sheaf counting on local K3 surfaces, *Pure and Applied Mathematics Quarterly*, Vol: 14, Pages: 419-441, ISSN: 1558-8599

There are two natural ways to count stable pairs or Joyce–Song pairs on X=K3×C; one via weighted Euler characteristic and the other by virtual localisation of the reduced virtual class. Since X is noncompact these need not be the same. We show their generating series are related by an exponential.As applications we prove two conjectures of Toda, and a conjecture of Tanaka–Thomas defining Vafa–Witten invariants in the semistable case.

Gholampour A, Thomas RP, 2019, Degeneracy loci, virtual cycles and nested Hilbert schemes I, *Tunisian Journal of Mathematics*, Vol: 2, Pages: 633-665, ISSN: 2576-7658

Given a map of vector bundles on a smooth variety, consider the deepestdegeneracy locus where its rank is smallest. We show it carries a naturalperfect obstruction theory whose virtual cycle can be calculated by theThom-Porteous formula. We show nested Hilbert schemes of points on surfaces can be expressed asdegeneracy loci. We show how to modify the resulting obstruction theories torecover the virtual cycles of Vafa-Witten and reduced local DT theories. The result computes some Vafa-Witten invariants in terms of Carlsson-Okounkovoperators. This proves and extends a conjecture of Gholampour-Sheshmani-Yau andgeneralises a vanishing result of Carlsson-Okounkov.

Kool M, Thomas RP, 2018, Stable pairs with descendents on local surfaces I: the vertical component, *Pure and Applied Mathematics Quarterly*, Vol: 13, ISSN: 1558-8599

We study the full stable pair theory --- with descendents --- of theCalabi-Yau 3-fold $X=K_S$, where $S$ is a surface with a smooth canonicaldivisor $C$. By both $\mathbb C^*$-localisation and cosection localisation we reduce tostable pairs supported on thickenings of $C$ indexed by partitions. We showthat only strict partitions contribute, and give a complete calculation forlength-1 partitions. The result is a surprisingly simple closed product formulafor these "vertical" thickenings. This gives all contributions for the curve classes $[C]$ and $2[C]$ (andthose which are not an integer multiple of the canonical class). Here theresult verifies, via the descendent-MNOP correspondence, a conjecture ofMaulik-Pandharipande, as well as various results about the Gromov-Witten theoryof $S$ and spin Hurwitz numbers.

Tanaka Y, Thomas RP, 2018, Vafa-Witten invariants for projective surfaces II: semistable case, *Pure and Applied Mathematics Quarterly*, Vol: 13, Pages: 517-562, ISSN: 1558-8599

We propose a definition of Vafa–Witten invariants counting semistable Higgs pairs on a polarised surface. We use virtual localisation applied to Mochizuki/Joyce–Song pairs.For KS≤0we expect our definition coincides with an alternative definition using weighted Euler characteristics. We prove this for degKS<0 here, and it is proved for Sa K3 surface in “Sheaf counting on local K3 surfaces” [D. Maulik and R. P. Thomas, arXiv:1806.02657].For K3 surfaces we calculate the invariants in terms of modular forms which generalise and prove conjectures of Vafa and Witten.

Segal E, Thomas RP, 2018, Quintic threefolds and Fano elevenfolds, *Journal für die reine und angewandte Mathematik*, Vol: 2018, Pages: 245-259, ISSN: 1435-5345

The derived category of coherent sheaves on a general quintic threefold is acentral object in mirror symmetry. We show that it can be embedded into thederived category of a certain Fano elevenfold. Our proof also generates related examples in different dimensions.

Thomas RP, 2018, A K-theoretic Fulton class

Fulton defined classes in the Chow group of a quasi-projective scheme $M$which reduce to its Chern classes when $M$ is smooth. When $M$ has a perfectobstruction theory, Siebert gave a formula for its virtual cycle in terms ofits total Fulton class. We describe K-theory classes on $M$ which reduce to the exterior algebra ofdifferential forms when $M$ is smooth. When $M$ has a perfect obstructiontheory, we give a formula for its K-theoretic virtual structure sheaf in termsof these classes.

Thomas RP, 2018, Notes on homological projective duality, American-Mathematical-Society Summer Research Institute on Algebraic Geometry, Publisher: American Mathematical Society, Pages: 585-609, ISSN: 2324-707X

Jiang Y, Thomas RP, 2016, VIRTUAL SIGNED EULER CHARACTERISTICS, *JOURNAL OF ALGEBRAIC GEOMETRY*, Vol: 26, Pages: 379-397, ISSN: 1056-3911

Roughly speaking, to any space $ M$ with perfect obstruction theory we associate a space $ N$ with symmetric perfect obstruction theory. It is a cone over $ M$ given by the dual of the obstruction sheaf of $ M$ and contains $ M$ as its zero section. It is locally the critical locus of a function.More precisely, in the language of derived algebraic geometry, to any quasi-smooth space $ M$ we associate its $ (\!-\!1)$-shifted cotangent bundle $ N$.By localising from $ N$ to its $ \mathbb{C}^*$-fixed locus $ M$ this gives five notions of a virtual signed Euler characteristic of $ M$:The Ciocan-Fontanine-Kapranov/Fantechi-Göttsche signed virtual Euler characteristic of $ M$ defined using its own obstruction theory,Graber-Pandharipande's virtual Atiyah-Bott localisation of the virtual cycle of $ N$ to $ M$,Behrend's Kai-weighted Euler characteristic localisation of the virtual cycle of $ N$ to $ M$,Kiem-Li's cosection localisation of the virtual cycle of $ N$ to $ M$,$ (-1)^{\textrm {vd}}$ times by the topological Euler characteristic of $ M$.Our main result is that (1)=(2) and (3)=(4)=(5). The first two are deformation invariant while the last three are not.

Pandharipande R, Thomas RP, 2016, Notes on the proof of the KKV conjecture, *Surveys in Differential Geometry*, Vol: 21, Pages: 289-311, ISSN: 2164-4713

The Katz-Klemm-Vafa conjecture expresses the GromovWittentheory of K3 surfaces (and K3-fibred 3-folds in fibre classes)in terms of modular forms. Its recent proof gives the first non-toricgeometry in dimension greater than 1 where Gromov-Witten theory isexactly solved in all genera.We survey the various steps in the proof. The MNOP correspondenceand a new Pairs/Noether-Lefschetz correspondence for K3-fibred3-folds transform the Gromov-Witten problem into a calculation of thefull stable pairs theory of a local K3-fibred 3-fold. The stable pairs calculation is then carried out via degeneration, localisation, vanishing results, and new multiple cover formulae.

Pandharipande R, Thomas RP, 2016, The Katz-Klemm-Vafa conjecture for K3 surfaces, *Forum of Mathematics, Pi*, Vol: 4, ISSN: 2050-5086

We prove the KKV conjecture expressing Gromov–Witten invariants of K3 surfaces in terms of modular forms. Our results apply in every genus and for every curve class. The proof uses the Gromov–Witten/Pairs correspondence for K3-fibered hypersurfaces of dimension 3 to reducethe KKV conjecture to statements about stable pairs on (thickenings of) K3 surfaces. Using degeneration arguments and new multiple cover results for stable pairs, we reduce the KKV conjecture further to the known primitive cases. Our results yield a new proof of the full Yau–Zaslow formula, establish new Gromov–Witten multiple cover formulas, and express the fiberwise Gromov–Witten partition functions of K3-fibered 3-folds in terms of explicit modular forms.

Katz S, Klemm A, Pandharipande R, et al., 2016, On the motivic stable pairs invariants of K3 surfaces, Vol: 315, Pages: 111-146, ISSN: 0743-1643

For a K3 surface S and a class β ∈ Pic(S), we study motivic invariants of stable pairs moduli spaces associated to 3-fold thickenings of S. We conjecture suitable deformation and divisibility invariances for the Betti realization. Our conjectures, together with earlier calculations of Kawai-Yoshioka, imply a full determination of the theory in terms of the Hodge numbers of the Hilbert schemes of points of S. The work may be viewed as the third in a sequence of formulas starting with Yau-Zaslow and Katz-Klemm-Vafa (each recovering the former). Numerical data suggest the motivic invariants are linked to the Mathieu M24 moonshine phenomena. The KKV formula and the Pairs/Noether-Lefschetz correspondence together determine the BPS counts of K3-fibered Calabi-Yau 3-folds in fiber classes in terms of modular forms. We propose a framework for a refined P/NL correspondence for the motivic invariants of K3-fibered CY 3-folds. For the STU model, a complete conjecture is presented.

Thomas RP, 2015, Notes on HPD

Beginning with the theorems of Beilinson and Orlov on derived categories, weshow how these lead naturally to Kuznetsov's beautiful theory of HomologicalProjective Duality. We then survey some examples.

Calabrese JR, Thomas RP, 2015, Derived equivalent Calabi–Yau threefolds from cubic fourfolds, *Mathematische Annalen*, Vol: 365, Pages: 155-172, ISSN: 0025-5831

We describe pretty examples of derived equivalences and autoequivalencesof Calabi-Yau threefolds arising from pencils of cubic fourfolds. The cubic fourfoldsare chosen to be special, so they each have an associated K3 surface. Thus a pencilgives rise to two different Calabi-Yau threefolds: the associated pencil of K3 surfaces,and the baselocus of the original pencil—the intersection of two cubic fourfolds. Theyboth have crepant resolutions which are derived equivalent.

Gholampour A, Sheshmani A, Thomas R, 2014, Counting curves on surfaces in Calabi-Yau 3-folds, *MATHEMATISCHE ANNALEN*, Vol: 360, Pages: 67-78, ISSN: 0025-5831

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- Citations: 10

Addington N, Thomas R, 2014, HODGE THEORY AND DERIVED CATEGORIES OF CUBIC FOURFOLDS, *DUKE MATHEMATICAL JOURNAL*, Vol: 163, Pages: 1885-1927, ISSN: 0012-7094

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- Citations: 73

Kool M, Thomas R, 2014, Reduced classes and curve counting on surfaces II: calculations, *ALGEBRAIC GEOMETRY*, Vol: 1, Pages: 384-399, ISSN: 2313-1691

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- Citations: 13

Kool M, Thomas R, 2014, Reduced classes and curve counting on surfaces I: theory, *ALGEBRAIC GEOMETRY*, Vol: 1, Pages: 334-383, ISSN: 2313-1691

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- Citations: 35

Pandharipande R, Thomas RP, 2014, 13/2 ways of counting curves, Moduli Spaces, Publisher: Cambridge University Press, Pages: 282-333, ISBN: 9781107279544

In the past 20 years, compactifications of the families of curves inalgebraic varieties X have been studied via stable maps, Hilbert schemes,stable pairs, unramified maps, and stable quotients. Each path leads to adifferent enumeration of curves. A common thread is the use of a 2-termdeformation/obstruction theory to define a virtual fundamental class. Therichest geometry occurs when X is a nonsingular projective variety of dimension3. We survey here the 13/2 principal ways to count curves with special attentionto the 3-fold case. The different theories are linked by a web of conjecturalrelationships which we highlight. Our goal is to provide a guide for graduatestudents looking for an elementary route into the subject.

Huybrechts D, Thomas RP, 2014, Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes (vol 346, pg 545, 2010), *MATHEMATISCHE ANNALEN*, Vol: 358, Pages: 561-563, ISSN: 0025-5831

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- Citations: 3

Pandharipande R, Thomas RP, 2014, ALMOST CLOSED 1-FORMS, *GLASGOW MATHEMATICAL JOURNAL*, Vol: 56, Pages: 169-182, ISSN: 0017-0895

Ross J, Thomas R, 2011, WEIGHTED BERGMAN KERNELS ON ORBIFOLDS, *JOURNAL OF DIFFERENTIAL GEOMETRY*, Vol: 88, Pages: 87-107, ISSN: 0022-040X

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- Citations: 11

Ross J, Thomas R, 2011, WEIGHTED PROJECTIVE EMBEDDINGS, STABILITY OF ORBIFOLDS, AND CONSTANT SCALAR CURVATURE KAHLER METRICS, *JOURNAL OF DIFFERENTIAL GEOMETRY*, Vol: 88, Pages: 109-159, ISSN: 0022-040X

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- Citations: 47

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