# Main content

Joseph Ayoub: Hodge theory for the conservativity conjecture

Arend Bayer: Bridgeland stability on Kuznetsov components, and applications

I will present a construction of stability conditions on Kuznetsov's semi-orthogonal component of derived categories of Fano threefolds, and on cubic fourfolds, along with applications to Torelli-type questions. This is based on joint work with Lahoz, Macri and Stellari, and some of the applications on joint work in progress also with Nuer and Perry.

Jérémy Blanc: Automorphisms of $$P^1$$-bundles on rational surfaces

In this talk, I will speak about the connected component of automorphisms of $$P^1$$-bundles over rational surfaces. It turns out that this study is reduced to the case of minimal rational surfaces, i.e. Hirzebruch surfaces and the projective plane. Then, one has a large family of decomposable $$P^1$$-bundles (projectivisations of decomposable $$P^1$$-bundles) and some interesting two other countable families, namely Umemura bundles and Schwarzenberger bundles. This work is done using the action of automorphisms of Hirzebruch surfaces on the moduli space of $$P^1$$-bundles on it.  Joint work with Andrea Fanelli and Ronan Terpereau.

Alessandro Chiodo: Néron models and genus-one double ramification cycles via Picard functors

Néron models of Jacobians are naturally described via Picard functors. Over a discrete valuation ring, this can be obtained by Raynaud's theorem via a quotient of the non-separated Picard functor. We can also present a direct approach within the separated functor $$Pic^0$$ of twisted curves. Recently Holmes extended Raynaud's approach on a base scheme of dimension greater than one and was able to provide in this way a universal Néron model over moduli of curves. This construction admits several applications (e.g. the study of limit linear series by Biesel and Holmes). It also allows a new definition of the Double Ramification locus (DR) parametrizing curves equipped with a principal divisor. In collaboration with Holmes, we compute this cycle in genus one and match the formula of Janda, Pandharipande, Pixton and Zvonkine.

Gavril Farkas: The Prym-Green Conjecture on the resolution of paracanonical curves

Daniel Huybrechts: Kuznetsov's K3 category and the global Torelli theorem for cubic fourfolds

Marc Levine: The intrinsic stable normal cone and motivic virtual fundamental classes

We describe how to use motivic stable homotopy theory and the Grothendieck six functor formalism to construct a general theory of the intrinsic normal cone and virtual fundamental classes for perfect obstruction theories. This gives a uniform treatment of these constructions for arbitrary cohomology theories living in the motivic stable homotopy category, including algebraic cycles, algebraic K-theory and algebraic cobordism, as well as the oriented cycles of Barge-Morel-Fasel. This last theory makes possible an enumerative geometry with values in quadratic forms.

Zhiyuan Li: Algebraic cycles on moduli space of polarized hyperkahler manifolds

We define the tautological ring on moduli space of polarized hyperhahler manifolds, which is largely motivated from the work of Marian, Oprea and Pandharipande on moduli space of K3 surfaces. We will talk about the properties of the tautological ring and give some conjectural descriptions such as the tautological conjecture. We will provide some evidence towards these conjectures. This is a joint work of Nicolas Bergeron.

Kieran O'Grady: Moduli of hyperelliptic K3's of degree 4 and VGIT

This is a report  on joint work with Radu Laza. The moduli space of hyperelliptic K3's of degree 4 may be identified with an 18-dimensional locally symmetric variety of Type IV, that we denote by F(18). The generic hyperelliptic K3 of degree 4 is a double cover of $$P^1\times P^1$$, branched over (4,4) smooth curve. There is a birational period map from the GIT quotient of the space of (4,4) divisors on $$P^1\times P^1$$ to the Baily-Borel compactification $$F(18)^{*}$$, which is very far from being regular. Starting from Looijenga's pioneering work on new compactifications defined by arrangements in Type IV BSD's, we have given a very precise conjectural decomposition of the period map into a composition of flips and divisorial contractions. One way of proving our part of our assertions is to do VGIT on the space of (4,4) divisors on $$P^1\times P^1$$.

Georg Oberdieck: Enumerative geometry of elliptic fibrations and modular forms

Led by physics arguments Huang, Katz and Klemm conjectured that curve counting invariants of elliptic Calabi-Yau threefolds form Jacobi forms. On the mathematics side the origin of the modularity is becoming clearer. Part of the modularity arises from sheaf theory. Fourier-Mukai transforms with respect to the Poincare line bundle of the fibration yield modular constraints. The other part has origins in Gromov-Witten theory. I will explain how this philosophy leads to a proof that the generating series of Gromov-Witten invariants of the product of a K3 surface and an elliptic curve (with respect to primitive classes on the K3) is the reciprocal of the Igusa cusp form, a Siegel modular form. The sheaf theory part is joint work with Junliang Shen, the Gromov-Witten part is joint work with Aaron Pixton.

Dragos Oprea: Segre classes and tautological relations over the moduli of K3s

I will discuss the K-trivial case of a conjecture of Lehn regarding the top Segre classes of tautological vector bundles over the Hilbert scheme of points. The method involves the study of the virtual geometry of certain Quot schemes via equivariant localization.

In addition, when the surface varies in moduli, the same method applied to the relative Quot schemes yields relations intertwining the k-classes over the moduli space of K3s and the Noether-Lefschetz loci.

This is based on joint work with Alina Marian and Rahul Pandharipande.

Dan Petersen: Tautological classes with twisted coefficients

Let $$M_g$$, for $$g \geq 2$$, be the moduli space of smooth curves of genus $$g$$. Mumford defined a subring $$R(M_g)$$ of the Chow ring $$CH(M_g)$$ called the tautological ring. I explain how to associate to any irreducible algebraic representation of $$Sp(2g)$$ a relative Chow motive $$V_\lambda$$ over $$M_g$$, and how to define a tautological subgroup $$R(M_g, V_\lambda)$$ inside $$CH(M_g, V_\lambda)$$. Computing $$R(M_g, V_\lambda)$$ for all $$\lambda$$ is equivalent to computing the tautological rings of all fibered powers of the universal curve over $$M_g$$ simultaneously. We are able to completely determine $$R(M_g, V_\lambda)$$ for all $$\lambda$$ when $$g$$ is at most 4. A particular consequence is that the tautological rings of all fibered powers of the universal curve over $$Mg$$ satisfy Poincaré duality in these genera. This was previously known only in genus 2. We also obtain results about conjectural failures of Poincaré duality for $$g$$ $$\geq$$ 5; specifically, we can show that if certain cycles related to modified diagonals on products of very general curves are nonzero in Chow, then Poincaré duality fails in the tautological ring. (Joint with Mehdi Tavakol and Qizheng Yin.)

Ulrike Rieß: Base loci of big and nef line bundles on irreducible symplectic varieties

In this talk, we present  results on base loci of big and nef line bundles on irreducible symplectic varieties, which were motivated by Mayer's remarkable  statements for K3 surfaces. We focus on conditions for the existence of base divisors for big and nef line bundles on certain irreducible symplectic varieties, and sketch the proof.

Furthermore, we will mention that for K3$$^{[2]}$$-type, every big and nef line bundle becomes base point free on a generic deformation.

Stefan Schreieder: Generic vanishing and minimal cohomology classes on abelian fivefolds

We classify generic vanishing subschemes of principally polarised abelian varieties in dimension five, showing that they exist only on Jacobians of curves and intermediate Jacobians of cubic threefolds, and confirming a conjecture of Pareschi and Popa in this case. Our result is implied by a more general statement about subvarieties of minimal cohomology class whose sum is a theta divisor. This is joint work with Casalaina-Martin and Popa.

Qizheng Yin: Gromov-Witten theory, cycles on moduli of K3's, and cycles on hyper-Kähler's

Using virtual class techniques in Gromov-Witten theory, we obtain a decomposition of the small diagonal for the universal family of K3 surfaces, thus generalizing a theorem of Beauville and Voisin. The universal decomposition plays a key role in the proof of the Marian-Oprea-Pandharipande conjecture on the tautological ring of the moduli of K3 surfaces. This is joint work with Rahul Pandharipande.

If time permits, I will also speculate possible uses of Gromov-Witten virtual classes in studying cycles on hyper-Kähler varieties.

Dimitri Zvonkine: Plugging r=0 into the space of r-th roots

Consider a complex curve C endowed with line bundle L whose r-th tensor power is the trivial or the canonical line bundle. The moduli space of pairs (C,L) is a ramified covering of the moduli space $$\overline{\mathcal M}_{g,n}$$ of algebraic curves. It carries several natural cohomology classes whose projections to $$\overline{\mathcal M}_{g,n}$$ turn out to be polynomial in r. We will state several theorems and conjectures that relate the constant term of this polynomial (obtained by plugging r=0) to the Poincaré dual cohomology classes of important geometric loci in $$\overline{\mathcal M}_{g,n}$$. This is joint work with F. Janda, R. Pandharipande, and A. Pixton.

Page URL: https://www.math.ethz.ch/fim/conferences/cycles-and-moduli/talks.html
30.03.2017
© 2017 Eidgenössische Technische Hochschule Zürich