## Ohio State University Algebraic Geometry Seminar## Year 2014-2015Time: Tuesdays 3-4pmLocation: MW 154 |
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(Dave Anderson): Associated to a sequence of simple roots for a reductive group G, there is a certain tower of P^1-bundles called a Bott-Samelson variety. These varieties are basic tools in geometry and representation theory, mainly because of their combinatorial structure and the fact that well-chosen sequences produce desingularizations of Schubert varieties. In this talk I'll describe the geometry of these spaces, focusing on the non-trivial but tractable problem of computing their cones of effective divisors.

(Maksym Fedorchuk): We develop a new combinatorial approach to proving base point freeness of divisors on $\overline{M}_{0,n}$. On one hand, this leads to a simple sufficient criterion for an S_n-symmetric divisor to be semiample. On the other hand, we construct infinite series of semiample divisors associated to cyclic quadratic forms satisfying certain positivity properties..

(Nick Addington): Autoequivalences of the derived category of a variety can be seen as "hidden symmetries" of the variety. Many examples come from birational geometry: twists of one kind or another around subvarieties that can be contracted or flopped. A few years ago I introduced a rather different autoequivalence for the Hilbert scheme of n points on a K3 surface, built from the universal ideal sheaf. In joint work with C. Meachan and W. Donovan, we can show that the same construction works for certain moduli spaces of torsion sheaves on a K3 surface. This geometry is much richer in this new example, the autoequivalence factors as a product of equivalences associated to abelian fibrations and Mukai flops. Along the way we produce pairs of compact hyperkaehler varieties of every (even) dimension that are derived-equivalent but not birational.

(Matthew Ballard): We discuss how the derived category of a smooth algebraic stack of finite type changes as one removes certain types of closed substacks. As an application, we show how wall-crossing in moduli of stable sheaves and Bridgeland stable objects yields semi-orthogonal decompositions relating their derived categories.

(Yunfeng Jiang): I will talk about the joint work recently with Richard Thomas. Basically speaking, for any scheme or DM stack M with a perfect obstruction theory, we associate to a space N the symmetric obstruction theory in the sense of Behrend. The space N is a cone over M and contains M as its zero section. We construct two invariants on M using the symmetric obstruction theory on N, one is Behrend's weighted Euler characterisitcs of N weighted by Behrend function and localized to M, the other is Kiem-Li's cosection localization invariant of the virtual cycle of N to M. We prove that these two invariants are the same.

(Dawei Chen): The cone of effective divisors contains important information about the birational geometry of a variety. In this talk I will give an introduction to this subject, with a focus on the case when the ambient variety is the moduli space of curves. If time permits, I will also talk about effective cycles of higher codimension.

(Zhengyu Zong): Based on the work of Eynard-Orantin and Marino, the remodeling conjecture was proposed in the papers of Bouchard-Klemm-Marino-Pasquetti in 2007 and 2008. The remodeling conjecture can be viewed as an all genus mirror symmetry for toric Calabi-Yau 3-orbifolds. It relates the higher genus open Gromov-Witten potential of a toric Calabi-Yau 3-orbifold to the higher genus B-model potential which is obtained by applying the topological recursion on the mirror curve. In this talk, I will explain the proof of the remodeling conjecture for general toric Calabi-Yau 3-orbifolds. This work is joint with Bohan Fang and Chiu-Chu Melissa Liu.

(Giulio Codogni): We show that being a general fibre of a Mori fibre space is a rather restrictive condition for a Fano variety. In order to detect this property, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with rational singularities to be realised as a fibre of a Mori fibre space, which turn into a characterisation in the rigid case. As an application, we apply our criteria to solve the classically known smooth two-dimensional case, give an almost exhaustive answer for smooth threefolds and flag varieties and a further characterisation on the polytope in the Gorenstein toric case. An interesting connection with K-semistability is also investigated. Joint work with A. Fanelli, R. Svaldi and L. Tasin.

(Deepam Patel): In his fundamental paper on the projective line minus three points, Deligne constructed certain extensions of mixed Tate motives arising from the fundamental group of the projective line minus three points. Since then, motivic structures on homotopy groups have been studied by many authors. In this talk, we will show how to construct a motivic structure on the (nilpotent completion of) *n*-th homotopy group of *P**n* minus *n*+2 hyperplanes in general position.

(Tony Pantev): I will explain how Lagrangian foliations in (shifted) symplectic geometry give rise to global potentials. I will also give natural constructions of isotropic foliations on moduli spaces and will discuss the associated potentials. I will give applications to the moduli of representations of fundamental groups and to non-abelian Hodge theory. This is based on joint works with Calaque, Katzarkov, Toen, Vaquie, and Vezzosi.

(Angela Gibney): I will talk about joint work with Belkale and Mukhopadhyay, in which we study aspects of vector bundles of conformal blocks on the moduli space of curves, using the quantum cohomology of Grassmannians.

(Mihai Fulger): It is classical to study the geometry of a projective variety through positive cones of numerical classes of divisors or curves. The Mori cone in particular plays an important role in the classification of projective algebric varieties. A number of pathological examples have shifted attention from the higher (co)dimensional case. They show that the analogous definitions do not lead to the same positivity properties. To correct the negative outlook, I look at stronger positivity conditions. A sample result is that the pseudoeffective cone of numerical k-dimensional cycle classes is pointed for all k. The proof works in all characteristics, and without restrictions on singularities. This is in joint work with Brian Lehmann.

(Jason Miller): Spherical varieties are a generalization of certain varieties equipped with a well behaved group action such as toric and flag varieties. For a spherical G-variety X, one can use methods from Newton-Okounkov theory to encode information about X and its G-orbit closures via faces on an associated polytope. I will discuss recent work examining an extension of the above correspondence to all the Borel orbit closures for another special class of spherical varieties, the wonderful group compactifications. This correspondence enjoys similar properties as in the case of G-orbits.

(Artan Sheshmani): I will talk about joint work with Amin Gholampour and Richard Thomas on proving the S-duality conjecture regarding modularity of DT invariants of sheaves with 2 diemnsional support in an ambient CY threefold. One of the crucial ingredients needed for our analysis is the relative Hilbert scheme of points on a surface. More precisely, together with Gholampour we have proven that the generating series, associated to the Hibert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson for absolute Hilbert schemes. We extend their constructions to the relative setting, and using localization and degeneration techniques, express the intersection numbers of the relative Hilbert scheme in terms of tangent bundle of the surface with logarithmic zeros and derive a nice formula as a modular form. I will then show how this leads to the proof of a well known conjecture in string theory, called S-duality modularity conjecture.

(Amin Gholampour): We study the geometry of moduli space of rank 2 torsion free sheaves on toric 3-folds. As an outcome, we develop a vertex theory for the rank 2 Donaldson-Thomas invariants. These invariants are expressed combinatorially in terms of a new labelled box configuration. We compare this to the vertex theories for the rank 1 Donaldson-Thomas theory of Maulik-Nekrasov-Okounkov-Pandharipande as well as the stable pair theory of Pandharipande-Thomas.

(Yefeng Shen): The occurrence of modular forms and quasi-modular forms in Gromov-Witten theory is an interesting phenomenon. I will present the following work joint with Jie Zhou. We show that the WDVV equations for elliptic orbifolds are equivalent to the Ramanujan identities for some modular groups. We then apply this to prove the genus zero Gromov-Witten correlation functions for all elliptic orbifolds are quasi-modular forms. Combining with the tautological relations on the moduli space of pointed curves, we also obtain the modularity for all genera. This generalizes an earlier result of Milanov-Ruan and solves a modularity conjecture for the Gromov-Witten theory of the elliptic orbifold curve with four -orbifold points.

(Edwin León Cardenal): Take a local field \(K\) (for instance \(\mathbb{R}\), \(\mathbb{C}\) or \(\mathbb{Q}_p\);) and let \(f(x)\in K[x]=K[x_1,\ldots,x_n]\) be a polynomial function or an analytic function over an open set \(U\) in \(K^n\). If \(\Phi\) is a Bruhat-Schwartz function on \(K^n\) and \(s\) in a complex number with \(Re(s)>0\), then the local zeta function attached to \((f,\Phi)\) is
\[
Z_{\Phi}(s,{f})=\int\limits_{K^{n}\smallsetminus f^{-1}(0)}\Phi(x)\ |f(x)|_K^{s}\ |dx|,
\]
where \(|dx|\) is the Haar measure on \(K^{n}\). This type of local zeta functions have been studies extensively since their introduction in the 60's by Gelfand and Weil.
In this talk we describe the basic properties of \(Z_{\Phi}(s,{f})\) and some interesting problems about its behavior. We will describe their generalization to the case of analytic mappings, which are vectors of analytic functions of the form \(\boldsymbol{f}=(f_{1},\ldots,f_{l}):U\rightarrow K^{l}\). The talk is based on the works [Veys W. & Zúñiga-Galindo W.A., *Zeta functions associated with polynomial mappings, log-principalization of ideals, and Newton polyhedra.* Trans. Amer. Math. Soc. 360 (2008), 2205-2227] and [León-Cardenal E., Veys W. & Zúñiga-Galindo W. A., *Poles of Archimedean zeta functions for analytic mappings.* J. Lond. Math. Soc. (2) 87 (2013), no. 1, 1-21.]

(Matthew Satriano): No prior knowledge of stacks will be assumed for this talk. We will discuss the theory of toric stacks, which like toric varieties, is a class of stacks which is particularly amenable to computation. Using the theory developed, we will show that it is impossible to construct a degree map for 0-cycles on even the nicest of Artin stacks. We will discuss our result within the broader context of Grothendieck-Riemann-Roch for stacks. This is based on joint work with Dan Edidin and Anton Geraschenko.

(Wilberd van der Kallen): Let k be a field. Let G be a reductive algebraic k-group acting algebraically on a finitely generated k-algebra A. Then the cohomological finite generation property (CFG) holds: the cohomology algebra \(H^*(G,A)\) is a finitely generated k-algebra. This result fits into a long story, going from the First Fundamental Theorem of invariant theory to strict polynomial bifunctors and their cohomology. We will sample this story. The slides can be found at http://www.staff.science.uu.nl/~kalle101/Ohio2015.pdf

**Ohio State University Algebraic Geometry Seminar-Year 2013-2014**

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