Abstracts of the Student Seminar
June 24, 2011
Pedro Acosta (University of Michigan) : Orbifold Groupoids
I will present a brief introduction to the notion of orbifold groupoids. Intuitively, orbifolds are
topological spaces that locally look like the quotient of Rn by a finite group. However, thisdefinition of orbifold is not the most appropiate when it comes to defining homotopy groups
and other constructions on a given orbifold. In this talk, I will present an alternative definitionof orbifold using the concept of topological groupoids, and I will show how to define homotopy
groups using this alternative definition.
Roman Budylin (Moscow State University) : Moduli of irreducible representations.
Let G be some group. We want to define some algebraic structure MG on the set of irreducible
G-representations. It is natural to consider families of representations for this purpose. Let S
be some scheme. Then family of representations is a locally free sheaf E with homomorphism ofabstract groups G → AutS (E). We want to find a fine moduli space for such families. It doesn’t
exist as a scheme, but does exist as a stack. I consider examples of finite groups and reductive
groups. In this cases moduli space comes out to be an algebraic stack. Another interestingquestion is functoriality of MG on G. The talk is based on materials from Moscow seminar ofA.N. Parshin.
Emily Clader (University of Michigan) : Localization Computations in GromovWitten Theory.
The technique of localization provides a powerful tool for computing integrals over a scheme
equipped with a torus action by reducing them to integrals over the components of the fixedpoint locus. In particular, the moduli space of stable maps into projective space admits such anaction, and via localization one can exploit this action to compute Gromov-Witten invariants.
We will present the statement of the localization theorem as well as some of its basic applications
in Gromov-Witten theory.
Bashar Dudin (Université Joseph Fourier) : The ELSV model of the Hurwitz stack.
Lando, Ekhedal, Vahenstein and Shapiro defined a cone over M g,n parametrizing in some sense
covers of the projective line with given ramification over a fixed point. We give a modular
construction of this stack-cone, and study its geometry, in particular the relationships to theHurwitz stacks and the stack of stable maps.
Sara Angela Filippini (Università degli Studi dell’Insubria) : A rigid Calabi-Yauthreefold.
Abstract to come.
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Patricio Gallardo (Stony Brook University) : Maximal Semistability on Plane Curves
and Variation of GIT Quotients.
Maximal semistable plane curves are the strictly semistable curves (in the GIT Sense) that
become stable under any perturbation. In this talk, we discuss the combinatorics and singularities
of maximal semistable plane curves. We describe their Newton polytope, their Hilbert-Mumford
numerical function, and algorithms for computing both objects. As an application, we givebounds for the Milnor number of maximal semistable plane curve singularities, and address theproblem of counting their monomial configurations. Finally, we discuss the GIT stability of pairsof curves. This construction depends on a choice of linearization, we discuss an algorithm to findthe critical linearizations (those choices where the stability changes) and the maximal semistableconfigurations in this relative situation.
Alisa Knizel and Alexander Neshitov (St. Petersburg State University) : Algebraicanalogue of Atiyah’s theorem.
In topology there is a well known theorem of Atyiah which states that for a connected Lie group[ ∼G there is an isomorphism R(G)= K0 (BG) where BG is the classifying space of G. In thepresent paper we consider an algebraic analogue of this theorem. In the paper of B.Totaro[Tot] there is a computation of lim K0 (BGi ) for specially selected sequence BGi . However, to←−compute K0 (BG) one needs to prove that lim1 K1 (BGi ) vanishes. For split reductive groups←−we present another aproach and prove that the Borel construction induces a ring isomorphism[R(G)
I(G) = K0 (BG), where BG is an étale classifying space introduced by Voevodsky and Morelin [VM]. Our approach makes possible to compute Ki (BG), which we expect to provide in anext paper.
References[VM] V. Voevodsky, F. Morel A1 -homotopy theory of schemes. Publications Mathématiques del’IHES(90)[Tot] The Chow Ring of a Classifying Space. Proc. Symp. Pure Math. (K-Theory, 1997)
Uros Kuzman (University of Ljubljana) : On the theory of J-holomorphic discs.
Let (M, J) be a smooth manifold with a smooth almost complex structure J. In general theexistence of complex submanifolds is a problem of an overdetermined PDE system, and therefore,generically there are no solutions. A special case are complex submanifolds of complex dimension
1, usualy called pseudo-holomorphic curves. We present the basic notion of pseudo-holomorphic
discs and the fundamental theorem of Nijenhuis-Woolf on their existence. The proof is basedon the implicit function theorem giving a parametrization of small J-holomorphic discs by holomorphic ones. We also give a brief discussion on the latest results in case of big discs where thelinearized Cauchy-Riemann operator turns out to be Fredholm.
Alessio Lo Giudice (SISSA) : A compactification of the Moduli space of Principal
Higgs bundles over singular curves.
A principal Higgs bundle (P, φ) over a singular curve X is a pair consisting of a principal bundle
P and a morphism φ : X → AdP ⊗ Ω1X . We construct the moduli space of Principal Higgs
G-bundles over a nodal curve X using the theory of decorated vector bundles, more precisely
given a faithful representation ρ : G → GLr (C) of G, we consider pairs (E, f ) where E is a vector
bundle of rank r over X and f : Eρ → L is a morphism of bundles, being L a line bundle and2

Eρ a vector bundle depending on the Higgs field φ. Moreover, we show that this moduli spacecoincides with the space of framed bundles for suitable representations.
Gunnar Magnusson (Université Joseph Fourier) : Canonical metrics and modulispaces.
Every complex curve carries a canonical Kähler metric. We will use these metrics to construct theWeil-Petersson metric on the moduli space of curves. If time permits, we will see how propertiesof the canonical metrics translate into geometric statements about the moduli space of curves.
Clélia Pech (Université Joseph Fourier) : Quantum cohomology of the odd symplectic Grassmannian of lines.
Odd symplectic Grassmannians are a generalization of symplectic Grassmannians to odd-dimensional
spaces. We will compute the classical and quantum cohomology of the odd symplectic Grassmannian of lines. Although these varieties are not homogeneous, we obtain Pieri and Giambelli
formulas that are very similar to the symplectic case. We notice that their quantum cohomology
is semi-simple, which enables us to check Dubrovin’s conjecture for this case.
Flavia Poma (SISSA) : Gromov-Witten invariants in positive and mixed characteristic.
I will describe how to define Gromov-Witten invariants for smooth projective schemes overany field and, more generally, over a regular scheme, focusing on the construction of a virtual
fundamental class. I will show that they satisfy the fundamental axioms and some more properties(e.g. WDVV equation, reconstruction theorem). If there is time, I will present a result on thecomparison of invariants in different characteristics for smooth projective schemes defined in
mixed characteristic.
Alexandra Popa (Stony Brook University) : Two-point Gromov-Witten formulas for
symplectic toric manifolds.
In 2007 A. Zinger computed certain two-point genus zero hyperplane integrals in Gromov-Witten
theory. In 2010 A. Zinger and the speaker extended these results to projective complete intersections. Then, the speaker extended most of these results to symplectic toric manifolds. I will
explain ideas behind the proof of these formulas in the projective setting and how they extend
to the toric case.
Nathan Priddis (University of Michigan) : FJRW ring and Mirror symmetry.
Several years ago Fan, Jarvis and Ruan developed a theory that plays an important role inMirror Symmetry. It is also being used currently to describe a Landau-Ginzburg/Calabi Yaucorrespondence. In this theory one constructs a state space with a quantum multiplication
structure, which yields a Frobenius algebra. I will briefly outline the construction of this ring,with some examples. I will also explain some results relating to mirror symmetry of LandauGinzburg models.
Dustin Ross (Colorado State University) : Open Gromov-Witten Theory and theCrepant Resolution Conjecture.
The crepant resolution conjecture relates the GW theory of an orbifold to the GW theory of a
crepant resolution. We suspect that the CRC for toric Calaby-Yau threefolds can be approached3locally via “open” GW theory. In this talk I will lay out the foundations for open GW theoryof CY threefolds and describe the recent success of the local approach for the specific orbifold[O(−1) ⊕ O(−1)/Z2 ]. This is joint work with my advisor Renzo Cavalieri.
Pablo Solis (Berkeley) : G-bundles on Curves and Loop Groups.
Recently Frenkel, Teleman, and Tolland have constructed a geometric completion of the stackof maps from stable marked curves to the quotient stack [pt/GL(1)]. This completion is used todefine gauge-theoretic Gromov-Witten invariants. I’d like to discuss this construction and givean alternative construction using the representation theory of the loop group LGL(1). If timeallows I’ll present a preliminary results on how to construct a similar completion with GL(1)replaced by a semisimple group G, again using the representation theory of the loop group LG.
Loek Spitz (University of Amsterdam) : An ELSV type formula relating completed
Hurwitz numbers to the moduli space of spin curves.
Hurwitz numbers count the number of covers of P1 with given ramification over two special
points, and simple ramification elsewhere. It is well known that these numbers are related inseveral ways to intersection theory on moduli spaces of curves and Gromov-Witten theory. In
this talk I will define a type of Hurwitz number where the ramification over the non-special points
is given by completed r + 1-cycles. These completed Hurwitz numbers are conjecturally related
to the moduli space of r-spin curves by an analog of the ELSV formula ([Zvo]). I will explain
this formula and give you an idea of a proof of a special case, based on joint work with Sergey
Shadrin and Dimitri Zvonkine ([SSZ]).
References[Zvo] D. Zvonkine, A preliminary text on the r-ELSV formula, preprint 2006.
[SSZ] S. Shadrin, L. Spitz, D. Zvonkine, On Double Hurwitz numbers with completed cycles,
arXiv:1103.3120.
Zhiyu Tian (Stony Brook University) : Symplectic geometry and rationally connected varieties.
A smooth projective variety is called rationally connected if there is a rational curve through any
two given points. In the 90’s, Kollar conjectured that rational connectedness is preserved undersymplectic deformation. I will sketch a proof of the conjecture for 3-folds and explain an idea of
Kollar which reduces the conjecture to the study of the symplectic analogues of two properties ofrationally connected varieties, the symplectic MRC quotient and the symplectic Graber-HarrisStarr theorem. I will also discuss in more detail the symplectic Graber-Harris-Starr theorem.
Amos Turchet (Università degli Studi di Udine) : Algebraic Hyperbolicity and Vojta
conjectures in Cubic surfaces.
The problem of determining whether there exist infinitely many integral points on varieties is an
important open problem in Arithmetic Geometry. For algebraic surfaces the problem focuses on
the density of integer points with respect to Zariski topology. The celebrated Vojta’s conjecture
asserts that for a surface of log-general type the integral points are not Zariski dense. This still
unproved conjecture admits two analogous in complex analysis and in algebraic function field
theory. In this latter case it predicts that given a smooth curve C and a finite subset S ⊆ C,
there should exist a bound for the degree (in a suitable projective embedding) for the imagesof non-constant morphisms C \ S → X , where X is again an algebraic variety of log-general

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type. In the arithmetic and function field case, starting from their proof of Siegel’s theorem
via Schmidt’s Subspace Theorem, Pietro Corvaja and Umberto Zannier solved several particularcases of Vojta’s conjecture in the last years.
In this talk we present some recent results, namely for the complement of the union of aconic and two lines in the projective plane and for a smooth cubic surface minus two completelyreducible hyperplane sections. In this latter case we describe our personal work proving that ingeneral the only families of curves where infinite integral points lay are the 21 lines that remainon the surface.
Michel van Garrel (California Institute of Technology) : Higher rank DonaldsonThomas invariants.
Donaldson-Thomas invariants of smooth complex three-folds are defined either via integrationof a virtual fundamental class, or as weighted Euler characteristic. The former, earlier version
works for moduli spaces of ideal sheaves (rank 1). While the latter applies to moduli spaces ofsheaves of any rank, it assumes that the variety is Calabi-Yau. I will discuss elements of these
constructions and talk about an attempt to generalize the first one to moduli spaces of higher
rank sheaves for a class of varieties that are not Calabi-Yau.

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