- Divisors on the moduli spaces of stable maps to
varieties and reconstruction -
PDF - J. Reine Angew. Math., 586 (2005),
169 - 205.
We determine generators for the Picard group of the
moduli space of genus 0 stable maps to flag varieties G/P. A precise
description of all relations between the generators is obtained in the
case of type A flags. We also discuss the codimension 2 classes on the
moduli space of stable maps to Grassmannians, determine the corresponding
Chow groups, and study relations between the tautological generators.
- The tautological rings of the moduli spaces of
PDF - Journal of Algebraic Geometry, 15 (2006), 623 - 655.
We define and study the tautological systems of the moduli spaces of
stable maps. We show that the rational cohomology of the moduli space of
genus 0 stable maps to SL-flag varieties is generated by tautological
classes. This result generalizes a well-known theorem of Keel, valid when
the target is a point. As a corollary, we conclude that the Gromov-Witten
invariants capture the entire intersection theory of the moduli space of
stable maps to flag varieties, in genus 0.
classes on the moduli spaces of stable maps
to projective spaces via torus actions -
- Advances in Mathematics, 207 (2006), 661 - 690.
We study the Bialynicki-Birula stratification of
the moduli space of genus 0 stable maps to projective spaces for certain
non-generic torus actions. The Bialynicki-Birula closed cells are
described in terms of the Li-Gathmann spaces of relative stable morphisms.
We show that the decomposition is filterable by explicitly describing an
ordering of the cells. As a corollary, we prove that the rational
cohomology and the rational Chow ring of the moduli spaces of stable maps
are isomorphic, and in fact generated by tautological classes.
- On the intersection theory on the moduli space
two bundles (with Alina Marian) - PDF - Topology, 45
(2006), 531 - 541.
We give an algebro-geometric derivation of the intersection theory
on the moduli space of stable rank 2 bundles of odd degree over a smooth
curve of genus g. We lift the computation computation from the moduli
space to a Quot scheme, where the answer is evaluated via equivariant
- Virtual intersections on the Quot scheme and
Vafa-Intriligator formulas (with Alina Marian) - PDF - Duke Math Journal,
136 (2007), 81 - 113.
We define a virtual fundamental class on the Quot scheme of
arbitrary rank quotients of a fixed bundle over a smooth projective curve.
We use the virtual localization formula to calculate virtual intersection
numbers on Quot. We obtain closed formulas for some intersections of
tautological classes, generalizing the ones written down by Vafa and
Intriligator. Finally, we present an application of our method to the
study of the Pontrjagin ring of the moduli space of stable bundles over a
- Counts of maps to Grassmannians and
intersections on the moduli space of bundles (with Alina Marian)
- PDF - Journal
of Differential Geometry, 76 (2007), 155 - 175.
We show that intersection numbers on the moduli space of stable bundles
of coprime rank and degree over a smooth complex curve can be recovered as
highest degree asymptotics in formulas of Vafa-Intriligator type,
i.e. formulas which capture some of the enumerative geometry of the space
of maps to the Grassmannian. In particular, we evaluate algebraically all
intersection numbers appearing in the Verlinde formula. Our results are in
agreement with previous computations of Witten, Jeffrey-Kirwan and Liu.
Moreover, we prove the vanishing of certain intersections on the Quot
scheme, which can be interpreted as giving equations between counts of
maps to the Grassmannian.
- The level-rank duality for nonabelian theta
functions (with Alina Marian) - PDF - Inventiones
Mathematicae, 168 (2007), 225 - 247.
We prove that the strange duality conjecture of Beauville-Donagi-Tu
holds for all curves, all ranks and degrees. This conjecture relates
non-abelian theta functions, that is sections of the determinant bundle,
on two different moduli spaces; the rank and the level are exchanged in
the process. The conjecture follows from a more general level-rank
which is the nonabelian generalization of the classical Wirtinger
duality on the space of level 2 theta functions.
- A tour of theta dualities on moduli spaces of
sheaves (with Alina Marian) - PDF - Curves and
abelian varieties, 175-202, Contemporary Mathematics, 465, American
Mathematical Society, Providence, Rhode Island (2008).
The purpose of this paper is twofold. First, we survey known
results about theta dualities on the moduli space of bundles on curves
and surfaces. Secondly, we establish new dualities on the moduli
spaces of sheaves over surfaces. Among others, the case of elliptic K3
surfaces is studied in detail; we propose an explicit
approach/reduction of the strange duality conjecture.
- Sheaves on abelian surfaces and Strange Duality
(with Alina Marian) - PDF - Math Annalen,
343 (2009), 1 - 33.
We formulate three versions of a strange duality conjecture for
sections of the theta bundles on different moduli spaces of sheaves on
abelian surfaces. As supporting evidence, we check the equality of
dimensions on dual moduli spaces. This sets the stage for strange duality
for abelian surfaces, and also answers a question raised by
- A note on the Verlinde bundles on elliptic
curves - PDF -
Transactions of the American Math Society, 362 (2010), 3779 - 3799.
The non-abelian theta functions can be organized into Verlinde
vector bundles over the Jacobian. We determine the splitting type of the
Verlinde bundles on elliptic curves in terms of the indecomposable Atiyah
bundles. The argument relies on the analysis of the Heisenberg action on
the symmetric powers of the Schrodinger representation.
- GL Verlinde numbers and Grassmann TQFT
(with Alina Marian) - PDF - Proceedings of the
Lisbon Geometry Summer
Port. Math. 67 (2010), 181 - 210.
We give a brief exposition of the 2d TQFT that captures
the structure of the GL Verlinde numbers,
- The Verlinde bundles and the semihomogeneous
Wirtinger duality - PDF - J.
Reine Angew. Math., 564 (2011), 125 - 180.
We determine the splitting type of the Verlinde vector bundles in
higher genus in terms of simple semihomogeneous factors. In agreement with
strange duality, the simple factors are interchanged by Fourier-Mukai, and
their spaces of sections are naturally dual.
- The moduli space of stable quotients (with
Alina Marian and Rahul Pandharipande) - PDF - Geometry and
Topology, 15 (2011), 1651 - 1706.
A moduli space of stable quotients
of the rank n trivial sheaf on stable curves is introduced.
Over nonsingular curves, the moduli space is
Grothendieck's Quot scheme. Over nodal curves, a
relative construction is made to keep the torsion of
the quotient away from the singularities.
of classical spaces arise naturally: a
nonsingular and irreducible
compactification of the moduli of maps from
genus 1 curves to projective space is obtained.
Localization on the moduli of stable quotients
leads to new relations in the
generalizing Brill-Noether constructions. The moduli space of stable
quotients is proven to carry
a canonical 2-term obstruction theory and
thus a virtual class.
The resulting system of descendent invariants is
proven to equal the Gromov-Witten theory of
the Grassmannian in all genera. Stable quotients
can also be used to study Calabi-Yau geometries.
The conifold is calculated to agree with stable
- Generic strange duality for K3
surfaces (with Alina Marian, and an appendix by Kota Yoshioka) -
Duke Math Journal, 162 (2013), 1463 - 1501.
Strange duality is shown to hold over generic K3 surfaces in a large
number of cases. The isomorphism for elliptic K3 surfaces is obtained
first via Fourier-Mukai techniques. Applications to Brill-Noether theory
for sheaves on K3s are also obtained. The appendix written by K.
Yoshioka discusses the behavior of the moduli spaces under change of
polarization, as needed in the argument.
- Framed sheaves and symmetric obstruction theories,
PDF - Documenta Math, 18 (2013), 323 - 342.
We note that open moduli spaces of sheaves over local Calabi-Yau surface
geometries framed along the divisor at infinity admit symmetric perfect
obstruction theories. We calculate the corresponding Donaldson-Thomas
weighted Euler characteristics as well as the topological Euler
characteristics. Furthermore, for blowup geometries, we discuss the
contributions of exceptional curves.
- On the strange duality conjecture for abelian
surfaces (with Alina Marian) -
PDF - Journal of the European
Math Society, 16 (2014), 1221 - 1251.
We study Le Potier's strange duality conjecture for moduli spaces of
sheaves over generic abelian surfaces. We prove the isomorphism for
abelian surfaces which are products of elliptic curves, when the moduli
spaces consist of sheaves of equal ranks and fiber degree 1. The
birational type of the moduli space of sheaves is also investigated.
Generalizations to arbitrary product elliptic surfaces are given.
- On Verlinde sheves and strange duality over
elliptic Noether-Lefschetz divisors (with
Alina Marian) - PDF - Annales de
l'Institute Fourier, 64 (2014), 2067 - 2086.
We extend results on generic strange duality for K3 surfaces by showing
that the proposed isomorphism holds over an entire Noether-Lefschetz
divisor in the moduli space of quasipolarized K3s. We interpret the
statement globally as an isomorphism of sheaves over this divisor, and
also describe the global construction over the space of polarized K3s.
- The first Chern class of the Verlinde
bundles (with Alina Marian and Rahul Pandharipande) - PDF -
Proceedings of String-Math, Bonn 2012, Proceedings of Symposia in Pure
Mathematics, 87-111 (2015).
A formula for the first Chern class of the Verlinde bundle over the moduli
space of smooth genus g curves is given. A finite-dimensional argument
is presented in rank 2 using geometric symmetries obtained from strange
duality, relative Serre duality, and Wirtinger duality together with the
projective flatness of the Hitchin connection. A derivation using
conformal-block methods is presented in higher rank. An expression for the
first Chern class over the compact moduli space of curves is obtained.
- The Chern character of the Verlinde
bundle over the moduli space of stable curves
(with Alina Marian, Rahul Pandharipande, Aaron Pixton, Dimitri Zvonkine) -
- J. Reine Angew. Math. 732 (2017), 147 - 163
We prove an explicit formula for the total Chern character of the Verlinde
bundle in terms of tautological classes. The Chern characters of the
Verlinde bundles define a semisimple CohFT, related to the fusion algebra
by an element of the Givental group. We determine the element using the
first Chern class of the Verlinde bundle on the moduli space of
pointed smooth curves and the
projective flatness of the Hitchin connection.
- On the strange duality conjecture for abelian
surfaces II (with Barbara Bolognese, Alina Marian, Kota
Yoshioka) - PDF - Journal
of Algebraic Geometry, 475 - 511 (2017)
In the prequel to this paper, two versions of Le Potier's strange duality
conjecture for sheaves over abelian surfaces were studied. A third version
is considered here. In the current setup, the isomorphism involves moduli
spaces of sheaves with fixed determinant and fixed determinant of the
Fourier-Mukai transform on one side, and moduli spaces where both
determinants vary, on the other side. We first establish the isomorphism
in rank one using the representation theory of Heisenberg groups. For
product abelian surfaces, the isomorphism is then shown to hold for
sheaves with fiber degree 1 via Fourier-Mukai techniques. By degeneration
to product geometries, the duality is obtained generically for a large
number of numerical types. Finally, it is shown in great generality that
the Verlinde sheaves encoding the variation of the spaces of theta
functions are locally free over moduli.
- Segre classes and Hilbert schemes of points (with Alina Marian and
Rahul Pandharipande) - PDF - Ann.
Sci. ENS., 239-267 (2017)
We prove a closed formula for the integrals of the top Segre
classes of tautological bundles over the Hilbert schemes of points of a K3
surface X. We derive relations among the Segre classes via equivariant
localization of the virtual fundamental classes of Quot schemes on X. The
resulting recursions are then solved explicitly. The formula proves the
K-trivial case of a conjecture of M. Lehn from 1999.
The relations determining the Segre classes fit into a much wider theory.
By localizing the virtual classes of certain relative Quot schemes on
surfaces, we obtain new systems of relations among tautological classes on
moduli spaces of surfaces and their relative Hilbert schemes of points.
For the moduli of K3 sufaces, we produce relations intertwining the kappa
classes and the Noether-Lefschetz loci. Conjectures are proposed.
- Bundles of generalized theta functions over
abelian surfaces - PDF - accepted
Kyoto J. Math.
We study the bundles of generalized theta functions constructed
moduli spaces of sheaves over abelian surfaces. In degree zero, the
splitting type of
these bundles is expressed in terms of indecomposable
Fourier-Mukai symmetries of the Verlinde bundles are found, consistently
with strange duality. Along the way, a transformation formula for the
theta bundles is derived, extending a theorem of Drezet-Narasimhan from
curves to abelian surfaces.
- On a class of semihomogeneous vector bundles
- PDF -
We study a class of semihomogeneous vector bundles over the
product of an abelian variety and its dual. For abelian surfaces, we
connect these semihomogeneous bundles to the Verlinde bundles of
generalized theta functions constructed from the moduli spaces of sheaves.
- On the combinatorics of Lehn's conjecture (with Alina
Marian and Rahul Pandharipande) - PDF - submitted
Lehn's conjecture is a formula for the top Segre class of the
tautological bundles over the Hilbert scheme of points of surfaces.
Voisin reduced Lehn's conjecture to the vanishing of certain
coefficients of special power series. The first result of this short note
is a proof of the vanishings required by Voisin by residue calculations.
Our second result is an elementary solution of the parallel question for
the top Segre class on the symmetric power of a nonsingular projective
curve C associated to a higher rank vector bundle V on C. Finally, we
propose a complete conjecture for the top Segre class on the Hilbert
scheme of points of S associated to a higher rank vector bundle on S in
the K-trivial case.
- Higher rank Segre integrals over the Hilbert scheme of
points (with Alina
Marian and Rahul Pandharipande) - PDF - preprint
Let S be a nonsingular projective K-trivial surface, and let V be
a vector bundle on S. We calculate the top Segre class of the
tautological bundle over the Hilbert scheme of n points of S associated to
V. In the rank one case, our result specializes to Lehn's previously
- Notes on the moduli space of stable
quotients - PDF
- "Compactifying Moduli Spaces", 69 - 135, Birkhauser, 2016.
This is an expository survey of various results regarding the moduli space
of stable quotients based on lectures for the
Advanced Course "Compactifying Moduli Spaces" at CRM Barcelona, May 2013.
- Tautological relations on stable map spaces - draft
The cohomology of the Kontsevich spaces of genus 0 stable maps
to flag varieties is generated by tautological classes. We study relations
tautological generators. We conjecture that all relations between these
generators are tautological, i.e. they are essentially obtained from the
pullbacks of Keel's
relations on the moduli space of curves, with the aid of the pushforwards
by the natural morphisms. We offer supporting evidence for this conjecture
in low codimensions.
- Divisor calculations over the moduli of K3 surfaces - PDF
We construct curves in the moduli space of K3 surfaces, and prove
several divisorial relations over the moduli space in low