Week 1 abstracts

Valery Alexeev: Concrete functorial compactifications of moduli of K3 surfaces

Typically in compactification problems there are two approaches: functorial, by adding some "stable" objects on the boundary, and combinatorial, e.g. toroidal compactifications. Sometimes they happily meet: classically for curves but also for abelian varieties where the moduli of stable pairs (X,B) matches the toroidal compactification for the 2nd Voronoi fan.

In this work, joint with Alan Thompson, we investigate whether two approaches match for moduli spaces of polarized K3 surfaces of low degree. In particular, in degree 2 we match a specific toroidal compactification to a moduli space of stable pairs (X,B) for a particular choice of divisor B. 

I will also briefly discuss the work of Adrian Brunyate who investigated this question in the case of elliptic K3 surfaces.

Carolina Araujo: Foliations with positive tangent sheaf

In recent years, techniques from higher dimensional algebraic geometry, specially from the minimal model program, have been successfully applied to the study of global properties of holomorphic foliations. In this talk I will discuss foliations with positive tangent sheaf, with focus on those with ample anti-canonical class. I will present some results and techniques developed in a series of joint works with Stéphane Druel.

Paul Aspinwall: Mirror Symmetry and Extremal Transitions in the Toric World

We review the mathematical machinery of the gauged linear sigma-model which lies at the heart of most examples of mirror symmetry. We propose a more general picture going beyond the reflexive case of Batyrev and Borisov. This is inspired by looking at extremal
transitions between models. We also emphasize the importance of non-geometric phases.

Benjamin Bakker: Bounding torsion in geometric families of abelian varieties

A celebrated theorem of Mazur asserts that only finitely many groups occur as the torsion part of the group of rational points of an elliptic curve over Q; it is conjectured that the same is true for abelian varieties over a number field K, though very little progress has been made in proving it.  The natural geometric analog where K is replaced by the function field of a complex curve---known as the geometric torsion conjecture---is equivalent to the nonexistence of low genus curves in congruence covers of the moduli space of abelian varieties. In joint work with J. Tsimerman, we prove the conjecture for abelian varieties with real multiplication.  We'll discuss a general method for bounding the genus of curves in locally symmetric varieties using hyperbolic geometry to bound Seshadri constants, as well as a new technique for computing the Kodaira dimension of these varieties in the rank one case.

Sébastien Boucksom: K-stability, growth of functionals and singularities of pairs

K-stability is an algebro-geometric property of a polarized variety, which can be viewed as a limit version of the more classical Geometric Invariant Theory notions. In this talk, I will review the relation of K-stability to Kähler geometry, where algebro-geometric invariants are interpreted as slopes at infinity of certain natural functionals, and to singularities of pairs, in the sense of the Minimal Model Program. 

Morgan Brown: Homotopy Equivalence of Berkovich spaces via Birational Geometry

Let $K$ be a field with valuation $v$. This valuation induces a norm on the field $K$, and for any quasiprojective variety $X$ over $K$, the norm gives a topology on $X$ analogous to the Euclidean topology on a complex variety. Berkovich spaces are a natural setting for studying analysis in this topology. There are many connections between the study of Berkovich spaces and other areas of mathematics, including number theory, tropical geometry, and, more recently, birational geometry. I will present how techniques from the minimal model program can be used to show that if $X$ and $Y$ are smooth projective varieties over $\mathbb{C}((t))$, and $f:X\to Y$ has rationally connected fibers, then the induced map on the Berkovich spaces is a homotopy equivalence. This is joint work with Tyler Foster.

Frédéric Campana: Pseudoeffectivity properties of orbifold cotangent bundles

Abstract: If (X,D) is a smooth and complex projective log-canonical pair with pseudo-effective canonical bundle, we show that the determinant of any quotient Q of any tensor power of its cotangent bundle is pseudo-effective. This strengthens our former result CP13, where such det(Q)'s were shown to be generically nef. The proof is entirely similar, the main improvement being a strengthening of the Bogomolov-McQuillan algebraicity criterion for foliations. This new result permits to obtain very elementarily (and without using BCHM) the corollaries of CP13, and even new results previously not accessible even using BCHM. This is joint work (in development) with M. Paun.

Serge Cantat: Groups of birational transformations

I shall describe methods from p-adic dynamics and geometric group 
theory that may be used to study groups of birational transformations of
projective varieties.

Junyan Cao: Kodaira dimension of algebraic fiber spaces over abelian varieties

Let f be a fibration between two projective manifolds. The Iitaka conjecture states that the Kodaira dimension of total space is not less than the sum of the Kodaira dimension of the generic fiber and that of the base manifold.  We prove a log-version (with klt pair) of the conjecture, under the assumption that the base is an abelian variety . The proof relies mainly on the positivity properties of direct images of relative pluri-canonical bundles. This is a joint work with Mihai Pãun.

Sebastian Casalaina-Martin: On descending cohomology geometrically

In this talk I will present some joint work with Jeff Achter and Charles Vial concerning the problem of determining when the cohomology of a smooth projective variety over the rational numbers can be modeled by an abelian variety. The primary motivation is a problem posed by Barry Mazur. We provide an answer to Mazur's question in two situations. First, we show that the odd cohomology groups can be modeled by the cohomology of an abelian variety over the rationals provided the geometric coniveau is maximal. This provides an answer to Mazur’s question for all uniruled threefolds, for instance, and provides a complete answer for all threefolds assuming the generalized Hodge, generalized Tate, or Bloch conjectures. Second, we show how a result of Beauville provides further information about this abelian variety in the case of a fibration in quadrics.

Paolo Cascini: Birational geometry and singularities in positive characteristic. 

Many of the results in the Minimal Model Program over complex projective varieties depend on Kodaira's vanishing theorem and its generalizations over singular varieties. Because of the failure of these tools over an algebraically closed field of positive characteristic, it is not known whether these results generalize to this case.  The main tool available for the study of varieties over a field of positive characteristic is the Frobenius morphism. In particular, it is crucial to understand singularities from this point of view. 
I will survey some recent progress in this direction.

Ana-Maria Castravet: Mori Dream Spaces

Mori Dream Spaces form an ideal class of algebraic varieties for which, as the name suggests, the Minimal Model Program works very well. Introduced by Hu and Keel, Mori Dream Spaces can be algebraically characterized as varieties whose total coordinate ring is finitely generated. Prime examples are toric varieties, or more generally, log Fano varieties. Among the examples, there are many moduli spaces, as often variation of GIT gives new birational models. However, there does not exist a satisfactory structure theory for Mori Dream Spaces. In this talk, I will discuss known results and open questions, with special attention payed to the birational geometry of moduli spaces of stable rational curves.  

Fabrizio Catanese: Projective K(\pi,1) spaces and applications to moduli problems

An interesting  theme of research   is the study of projective varieties Z which are K(H,1)'s , i.e. classifying
spaces BH for some discrete group H. For  such varieties Z, a bold conjecture is that  also their Galois conjugates Z^s are classifying spaces BH' for some discrete group H' . The well known examples are: curves and abelian varieties, the latter being exactly the projective K(\pi,1) spaces where the group \pi is abelian. Interesting examples are the Bagnera-De Franchis varieties and Generalized Hyperelliptic varieties, quotients A/G of an Abelian variety A by a finite group G. Hypersurfaces in BdF varieties are special cases of the notion of Inoue type varieties, whose moduli spaces have been investigated in our joint work with I. Bauer and D. Frapporti (especially for the classification of algebraic surfaces with low invariants). Inoue type varieties are defined as  the quotient X = W/G of an ample divisor W in a  projective varieties Z which is a  K(H,1), by the free action of a finite group G.
In order to obtain general results on the moduli spaces of ITV, implying that if X' is homotopically equivalent to an ITV X,  also X'  is an Inoue type variety, we need to extend the definition to multiple Inoue type varieties. This is done via a theorem, recently proven in joint work  with Yongnam Lee, giving an explicit characterization of  deformations to embeddings as smooth hypersurfaces.

Time permitting, I shall also discuss a  remarkable series of algebraic surfaces, counterexamples to Fujita's question on VHS (in joint work with Michael Dettweiler).

THM. If one has a  Kaehler family fibred over a curve B, then the direct image V of the relative dualizing sheaf is the direct sum of an ample vector bundle A and of a unitary flat vector bundle W. V does not need to be semiample, equivalently, the bundle W can have infinite monodromy.

The examples are given by surfaces S which  are abelian coverings  with group  (\ZZ/n)^2 of the Del Pezzo surface Z of degree 5, branched on a union of lines which forms a bianticanonical divisor.

The Albanese map a of S  is a semistable fibration onto a curve B of genus b,  with  fibres of genus g = (n-1)/2, and where g=2b. a has only 3 singular fibres, the union of two smooth curves of genus b.

The simplest case is for n=5, where  S is a ball quotient.  Do all  these surfaces have negative curvature, are they projective K(\pi,1)'s?

Jungkai Chen: Geography of threefolds of general type

The classification of algebraic varieties leads to the study of Fano-type variety, Calabi-Yau varieties and varieties of general type. For varieties of general type, main questions concern the universal lower bound of volume, effective pluricanonical maps, and relations between various birational invariants. These types of questions are considered to be well-understood for surfaces. However, not so much is known even in dimension three. In this talk, we will survey the recent progress of explicit studies of threefolds of general type and their application to various geographic problems.  

Aldo Conca: Multigraded ideals with a radical gin

I will report on a project with Emanuela De Negri and Elisa Gorla related to the study of multigraded ideals with a radical generic initial ideal. Our main new result is that if a multigraded ideal has a radical  multigraded generic initial ideal then the same is true for every multigraded hyperplane section and for every multigraded projection. Connection to universal Gr\"obner bases for determinantal ideals, Koszul algebras associated to subspaces configurations  and to ideals associated to the multiview varieties of Aholt, Sturmfels and Thomas will be discussed.

Tommaso de Fernex: Birational geometry of projective hypersurfaces

I will review old and new results on the birational geometry of hypersurfaces in projective space. The main focus will be on the birational rigidity of Fano hypersurfaces of index one, a problem tracing back to works of Iskovskikh and Manin, and Fano and Segre before them, which has been completely solved in recent years for smooth hypersurfaces. Time permitting, I will also discuss some new results on the birational rigidity of singular hypersurfaces.

Simon Donaldson: Stability of algebraic varieties and Kahler geometry

We will begin by reviewing background in Geometric Invariant Theory, the Kempf-Ness metrical criterion for stability and  the Kobayashi-Hitchin correspondence for vector bundles.  Then we will explain the notion of K-stability for varieties and the formal picture relating this to Kahler geometry. These ideas  will be illustrated by the case of toric manifolds.  We will then outline  the proof (with Chen and Sun) of Yau's conjecture for Kahler-Einstein metrics on Fano manifolds and particularly the interaction between algebraic geometry and Riemannian convergence theory. As time allows, we will describe more recent work (with Sun) on the algebro-geometric meaning of Riemannian "tangent cones" and mention other developments (by a number of different groups)  concerning moduli spaces of Fano manifolds.

Dan Edidin: Strong regular embeddings and the geometry of hypertoric stacks

We explain how the notion of “strong regular embeddings” can be used to compare the geometry of a stack to that of a regularly embedded substack. This theory can be applied to understand the relationship between singular hypertoric varieties and singular Lawrence toric varieties. While this talk is about stacks, the motivating ideas come from simple observations about invariant rings for actions of finite groups.

David Eisenbud: Higher Matrix Factorizations for Complete Intersections: An introduction and an application

Higher matrix factorizations for complete intersections, introduced in my joint work with Irena Peeva, generalize the matrix factorizations of an element in a commutative ring---the case of a hypersurface. They give insight into the properties of minimal free resolutions that generalize the eventual periodicity of resolutions over a hyper surface. I'll explain this structure, and an application of it discovered in joint work with Peeva and Frank-Olaf Schreyer.

Gavril Farkas: The Green-Lazarsfeld secant conjecture

Generalizing the well-known Green Conjecture on syzygies of canonical curves, Green and Lazarsfeld formulated in 1986 the Secant Conjecture predicting that a line bundle L of sufficiently high degree on a curve has a non-linear p-syzygy if and only if L fails to be (p+1)-very ample. I will discuss a solution to this conjecture (in various degrees of generality), including a complete affirmative answer in the extremal case. This is joint work with Michael Kemeny.

Osamu Fujino: On semi-log canonical pairs 

The notion of semi-log canonical singularities was introduced by Kollár and Shepherd-Barron in order to investigate compactifications of moduli spaces for surfaces of general type. In this talk, I will explain various properties of semi-log canonical pairs. The semipositivity theorems for semi-log canonical pairs, which come from graded polarizable admissible variations of mixed Hodge structure, play important roles for moduli problems. I have completed Kollár's projectivity criterion for the moduli spaces of higher-dimensional stable varieties. 

Taro Fujisawa: Limits of Hodge structures in several variables

I would like to talk about the results in my recent preprint "Limits of Hodge structures in several variables II" (arxiv:1506.02271). On the relative log de Rham cohomology groups of a proper Kaehler semistable morphism, I constructed mixed Hodge structures in my  previous paper. In the talk, I will present the two results: First, the weight filtrations above coincide with the monodromy weight filtrations. Second, the mixed Hodge structures above form admissible variations of mixed Hodge structure.

Patrick Graf: The jumping coefficients of non-Q-Gorenstein multiplier ideals

De Fernex and Hacon associated a multiplier ideal sheaf to a pair (X, a^c) consisting of a normal variety and a closed subscheme, which generalizes the usual notion where the canonical divisor K_X is assumed to be Q-Cartier. I will discuss some of the properties of the jumping numbers associated to these multiplier ideals.
The set of jumping numbers of a pair is unbounded, countable and satisfies a certain periodicity property. Furthermore, the jumping numbers form a discrete set of real numbers if the locus where K_X fails to be Q-Cartier is zero-dimensional. In particular, discreteness holds whenever X is a threefold with rational singularities.

Daniel Greb: Movable curves & semistable sheaves

I will discuss several geometric and moduli-theoretic motivations for generalising the notion of "semistable sheaf" from the classical case where stability is measured with respect to ample divisors to the general setup where degrees are computed with respect to movable curve classes. Afterwards, I will present recent results obtained in joint projects with Kebekus-Peternell and Toma: First, I will explain that most of the basic properties one expects from semistability still hold true. Then, I will address the question of boundedness of the family of sheaves that are semistable with respect to a movable curve class, as well as the relation between natural chamber structures on the ample and the movable cone. Finally, as an application of these fundamental results, I will sketch the construction of a higher-dimensional analogue of the Donaldson-Uhlenbeck compactification.

Paul Hacking: Theta functions for K3 surfaces

Andrei Tyurin conjectured the existence of a canonical basis of global sections for an ample line bundle on a K3 surface, analogous to theta functions for abelian varieties. I'll describe joint work with Gross, Keel, and Siebert which uses ideas from mirror symmetry to prove the conjecture for K3 surfaces near a cusp of the moduli space.

Christopher Hacon and James McKernan: Birational geometry and moduli spaces of varieties of general type

The moduli space of stable curves is one of the most heavily studied varieties in algebraic geometry.  Its construction uses geometric invariant theory.  Unfortunately this does not seem to work in higher dimensions.  

In the last three decades there has been considerable progress towards constructing analogous moduli spaces for higher dimensional varieties using techniques from the minimal model program.  We review recent progress which leads to a complete solution of the corresponding boundedness problem in all dimensions.

Gordon Heier: Holomorphic sectional curvature and the structure of projective Kaehler manifolds

I will present recent results on the consequences of semi-definite holomorphic sectional curvature for the geometric structure of projective Kaehler manifolds. In the semi-positive case, the focus is on the existence of rational curves. In the semi-negative case, the focus is on semi-positivity of the canonical line bundle and splitting theorems. This talk covers various joint works with S. Lu, B. Wong and F. Zheng.

Andreas Höring: MMP for compact Kähler threefolds

Compact Kähler manifolds are a natural generalization of complex projective manifolds and many statements from the minimal model program are expected to hold in this more general setting. Yet while the MMP for projective manifolds has made tremendous progress for more than thirty years, the proofs typically do not work for higher-dimensional Kähler manifolds. In this talk I will explain the main challenges for establishing the Kähler MMP and how to resolve these problems for threefolds.

Mattias Jonsson: Degenerations of Calabi-Yau manifolds and Berkovich spaces

Kontsevich and Soibelman have given a conjectural description of the Gromov-Hausdorff limit of a maximally degenerate family of polarized Calabi-Yau manifolds in terms of the Berkovich space attached to the degeneration. Motivated by this, Mustata, Nicaise and Xu recently studied the essential skeleton of this Berkovich space, which is a natural realization of the dual complex of a minimal model of the degeneration.

I will present joint work with Sebastien Boucksom, in which we show that the volume form induced by a holomorphic form of top degree on a fiber converges, in a suitable sense, to an explicit Lebesgue type measure on the essential skeleton.

Stefan Kebekus: Higgs sheaves on singular spaces and the Miyaoka-Yau Inequality for minimal varieties of general type

We establish the Miyaoka-Yau inequality for the tangent sheaf of any minimal, complex, projective variety X of general type, with only klt singularities. In the case of equality, we prove that the canonical model of X has only quotient singularities and is uniformized by the unit ball. Joint with Greb, Peternell, Taji.

Sándor Kovács: Projectivity of the moduli space of stable log-varieties

This is a report on joint work with Zsolt Patakfalvi. We prove a strengthening of Kollár's Ampleness Lemma and use it to prove that any proper coarse moduli space of stable log-varieties of general type is projective. We also confirm the Iitaka-Viehweg conjecture on the subadditivity of log-Kodaira dimension for fiber spaces whose general fiber is of log general type.

Martí Lahoz: Rational cohomology tori

Complex tori can be topologically characterized among compact Kähler manifolds by their integral cohomology ring. I will discuss the structure of compact Kähler manifolds whose rational cohomology ring is isomorphic to the rational cohomology ring of a torus and give some examples. This is joint work with Olivier Debarre and Zhi Jiang.

Adrian Langer: Higgs sheaves in positive characteristic

I would like to talk about non-abelian Hodge theory in positive characteristic and its applications to study Chern classes of logarithmic Higgs sheaves. This has various applications, e.g. to logarithmic version of Bogomolov-Miyaoka-Yau inequality in positive characteristic and to p-adic uniformization of curves (this last result was obtained by Lan-Sheng-Yang-Zuo).

Radu Laza: Birational geometry of the moduli space of hyperelliptic quartic K3s

The study of compactions of the moduli space of K3 surfaces is a problem of great interest. For low degree cases, E. Looijenga has constructed a framework that provides a comparison between the two naturally available compactifications in this case: GIT and Baily-Borel. In this talk, I will discuss an enrichment of this picture, essentially a continuous interpolation between the GIT and BB models.   While the discussion will be mostly concerned with the case of hyperelliptic quartic K3 surfaces, we expect such an interpolation to hold quite generally. This is inspired and quite analogous to the Hassett-Keel program that studies the birational geometry of the moduli space \bar M_g of curves. 

This is a report on joint work with K. O’Grady.

Robert Lazarsfeld: Syzygies of algebraic curves of large degree

In the mid 1980's, Mark Green and I conjectured that one could read off the gonality of an algebraic curve C from the syzygies among the equations defining any one sufficiently positive embedding of C. About a year ago, Lawrence Ein and I noticed that a small variant of the ideas used by Voisin in her work on canonical curves leads to a very quick proof of this gonality conjecture. More recently, Ein and Yang and I found a partial generalization of the main vanishing involved to smooth projective varieties of all dimensions. I will discuss this circle of ideas. In the unlikely event that time allows, I will explain how these results fit into the larger project of trying to understand the asymptotic behavior of the syzygies of a projective variety as the positivity of the embedding line bundle increases.

Anton Leykin: Effective Noetherianity up to symmetry

Given a polynomial ring in infinitely many variables with an action of a large group or monoid, we consider invariant ideals that are finitely generated up to symmetry. Some rings can be shown to be equivariantly Noetherian, but in general not all ideals are finitely generated by orbits of finitely many elements. Once finite generation is proven, a natural question arises: how to find generators? 

I will overview our results that establish Noetherianity for infinite-dimensional toric varieties, highlight the gap between non-effective and effective theory, and talk about the work in progress on equivariant Groebner bases. 

(Based on joint works with Draisma, Eggermont, Hillar, Kahle, and Krone.)

John Lesieutre: Constraints on threefolds admitting positive entropy automorphisms

There numerous examples of smooth, projective surfaces which admit automorphisms of positive entropy. However, relatively few examples are known for algebraic varieties in higher dimensions. I will give some constraints on the geometry of threefolds which can admit such automorphisms. For example, I will show that if one starts with a smooth threefold with no positive entropy automorphisms, and performs a sequence of blow-ups, any automorphism of the resulting threefold must be imprimitive. I'll also mention a related example of a non-uniruled, terminal threefold with infinitely many K_X-negative rays on the cone of curves.

Anatoly Libgober: Sections of Pfaffians

This is report on recent joint work with L.Borisov on calculation of stringy Hodge numbers of linear sections of Pfaffians. The results involve a study of singularities of Pfaffians, their resolutions, and identities for q-hypergeometric functions. They provide indirect confirmation of conjectures on double mirrors and some of A.Kuznetsov conjectures on homological projective duality.

Daniel Lowengrub: A Cancellation Theorem for Segre Classes

The Riemann Singularity theorem is a classical theorem relating two important objects associated to smooth curves. It says that the multiplicity of a point on the theta divisor of the curve is equal to the dimension of the fiber of the Abel Jacobi map from the Hilbert scheme of points to the Jacobian. Sebastian Casalania-Martin and Jesse Kass proved an analog of this for nodal curves and conjectured what the formula should be for general planar curves. In this talk, we will prove a theorem about Segre classes which will allow us to generalize Fulton's proof of the Riemann Singularity theorem to arbitrary planar curves, and thus obtain the conjectured formula.

Martin Möller: The volume of the moduli space of flat surfaces

The Hodge bundle over the moduli space of curves is statified according to the type of zeros of the differential form. These strata have a natural volume form, invariant under the Teichmueller geodesic flow.

We compute the volumes of these moduli spaces in a way that allows to deduce large genus asympotitcs. We will present applications of this to Lyapunov exponents, to Siegel-Veech constants and to slopes of curves in the moduli space of curves.

Mircea Mustata: On the divisors computing minimal log discrepancies

I will give an introduction to an invariant of singularities--the minimal log discrepancy. Motivated by Shokurov's ACC Conjecture for minimal log discrepancies, we discuss a boundedness question for the divisors computing the minimal log discrepancy on a fixed germ. This is joint work in progress with Yusuke Nakamura.

Yoshinori Namikawa: A finiteness theorem for symplectic singularities

The notion of a symplectic singularity was first introduced by Beauville. An affine symplectic singularity with a good C^*-action is called a conical symplectic variety. Many interesting varieties are conical symplectic varieties. For example, among them are nilpotent orbit closures of a complex semisimple Lie algebbra, Slodowy slices to such orbits, Nakajima quiver varieties, hypertoric varieties and so on.

In this talk we discuss how many such varieties exist. If we fix the dimension of conical symplectic varieties X and the maximal weight N of the minimal homogeneous generators of the coordinate ring R of X, then there are only finitely many such X up to isomorphism.

We first relate a conical symplectic variety with a log Fano klt pair, which has a contact structure. Then the boundedness result for log Fano klt pairs with fixed Cartier index assures that the family of conical symplectic varieties of a fixed dimension and with a fixed maximal weight, forms a bounded family. Finally we prove the rigidity of conical symplectic varieties by using Poisson deformations.

Kieran O'Grady: EPW sextics

Abstract: EPW sextics are special hypersurfaces, 
which come with a double cover which is a deformation of 
the Hilbert  square of a K3 surface.  We will explain why they
form an interesting family of varieties.

Angela Ortega: The Prym map of degree-7 cyclic coverings 

One can associate to a finite morphism between smooth projective curves its Prym variety, which is an abelian variety carrying a natural polarization (not necessarily principal). The corresponding  Prym map is the map between the moduli space of coverings (for a given degree and genera)  and  the moduli space of abelian varieties with some fixed polarization type. By dimension reasons, only in very few cases one can expect the Prym map to be generically finite over its image. The most famous Prym map where this occurs is that from the moduli space of étale double coverings over a genus 6-curve mapping dominantly onto the moduli of principally polarized abelian varieties of dimension 5, whose fibers  carry the structure of the 27 lines of a smooth cubic surface.

In this talk we will explain another instance of this situation: the case of  étale cyclic coverings of degree 7 over a genus 2-curve.  We show that the Prym map is generically finite over a special subvariety of the moduli space of 6-dimensional abelian varieties with polarization type (1,1,1,1,1,7).  By extending the map to a proper map on a partial compactification of the space of coverings and performing a local analysis we compute that the degree of this Prym map is 10. 

This a joint work with Herbert Lange. 

Karol Palka: The geometry of rational cuspidal curves in the complex projective plane

We will discuss our recent proofs (together with M. Koras) of the following results concerning rational cuspidal curves contained in the complex projective plane:
(1) (the Coolidge-Nagata conjecture): every such curve is Cremona equivalent to a line, (2) Its singular locus consists of at most four singular points. The method is based on using the two-dimensional logMMP.

Mihai Păun: Metric properties of direct images of twisted relative canonical bundles

Abstract: We will present a recent joint work with S. Takayama. Our main results can be seen as quantitative counterparts of the classical positivity properties of relative canonical bundles and their direct images.

Jorge Vitorio Pereira: Adjoint dimension of foliations

The classification of foliated surfaces by Brunella, McQuillan and Mendes carries many similarities with  Enriques-Kodaira classification of surfaces but also many important differences. I will discuss an alternative classification scheme where the role of differential forms along the leaves is replaced by differential forms along the leaves with values in fractional powers of the conormal bundle of the foliation. In this alternative setup  one obtains a classification of foliated surfaces closer to the usual Enriques-Kodaira classification.  If time permits, I will show how to apply this alternative classification to describe the Zariski closure of the set foliations which admit rational first integral of bounded genus in families of foliated surfaces. Joint work with Roberto Svaldi.

Mihnea Popa: Positivity for Hodge modules and geometric applications

M. Saito's theory of Hodge modules provides a powerful generalization of classical Hodge theory that is beginning to find basic applications to birational geometry. One of the main reasons for this is that generalizations of Hodge bundles and de Rham complexes arising in this context satisfy analogues of well-known vanishing and positivity theorems. I will review a number of such results that have been obtained recently, and how they can be applied to deduce new statements regarding generic vanishing, holomorphic one-forms, or families of varieties of general type. Much of this work is joint with C. Schnell. 

Bangere Purnaprajna: Fundamental groups and Shafarevich conjecture on holomorphic convexity

In this article we prove new results on fundamental groups for some classes of fibered algebraic surfaces with a finite group of automorphisms. The methods actually compute the fundamental groups of fibered surfaces under study. The corollaries include an affirmative answer to Shafarevich conjecture on holomorphic convexity, Nori's insightful question on fundamental groups and free abeliansess of second homotopy groups for these surfaces. We also prove a theorem, that bounds the multiplicity of  the multiple fibers of a fibration for any algebraic surface with a finite group of automorphisms G. If X/G is a P^1 fibration, we show that multiplicty actually divides |G|. This theorem on multiplicity, which is of independent interest, plays a important role in our theorems. Our earlier results in this topic have appeared in Math Annalen (2009) and Inventiones Mathematicae (2011) 

Claudiu Raicu: Characters of equivariant D-modules on spaces of matrices

I will describe the characters of the simple GL-equivariant D-modules on a complex vector space of matrices (general, symmetric, or skew-symmetric) and explain how this information can be used to compute local cohomology modules, as well as to (dis)prove some cases of a conjecture of Levasseur.

Sönke Rollenske: Gorenstein stable surfaces with K^2=1

Stable surfaces are the two-dimensional analogs of stable curves, that is, they are the singular surfaces occurring in a compactification of the Giesecker moduli space of surfaces of general type. We combine a generalisation of classical techniques using the canonical map with Koll\'ars point of view that  a non-normal stable surface is a log-canonical pair with suitable glueing data to study Gorenstein stable surfaces with $K_X^2=1$ and worse than canonical singularities.

This is joint work, partly in progress, with M. Franciosi and R. Pardini.

Julius Ross: Variation of Gieseker Moduli Spaces via Quiver GIT

There are various notions of stability for sheaves (e.g. Mumford stability, Geiseker semistability) that are important in the production of projective moduli spaces.  These notions usually depend non-trivially on a choice of ample line bundle, or Kahler class and different choices give rise to different, but related, moduli spaces. 

In this talk I will discuss joint work with Daniel Greb and Matei Toma in which we introduce a notion of Gieseker-stability that depends on several polarisations.  We use this to study the change in the moduli space of Giesker semistable sheaves on manifolds giving new results in dimensions at least three, and to study the notion of Gieseker-semistability for sheaves taken with respect to an irrational Kahler class.

Justin Sawon: Lagrangian fibrations

In holomorphic symplectic geometry, `Lagrangian fibrations' are fibrations of holomorphic symplectic manifolds by complex tori such that the holomorphic symplectic form vanishes identically on the fibres. These are higher-dimensional analogues of elliptic fibrations on K3 surfaces. In this talk we will survey Lagrangian fibrations, discussing some existence and classification results.

Frank-Olaf Schreyer:  Matrix factorizations and models of curves in P^4

For curves in P^3 the Hilbert-Burch theorem applied with vector bundles gives methods
for constructions of curves. In the talk I will explain, how matrix factorizations 
may serve similarly to construct a flag of a curve and a hyperfurface in P^4.

I apply illustrate this technique for curve of genus g=15 to 20.

Karl Schwede: On the moduli part of the F-different

Suppose X is a variety with a Frobenius splitting, compatibly splitting a subvariety W.  How can we describe the induced splitting on W (as a divisor)?  What is the induced splitting on W as the characteristic varies?

These questions are closely tied to questions in birational algebraic geometry.  Indeed, given a log canonical center W of some log canonical pair (X, D), one expects/obtains a divisor D_W on W such that K_W + D_W is linearly equivalent to the restriction of K_X + D to W.  This divisor D_W has two parts, the "fixed" divisorial part and the moduli part which is not fixed.  From the point of view of F-singularities, the moduli part appears to be fixed too -- this just corresponds to the induced splitting on W.  We will explore the meaning of this fixed D_W both in terms of a fibration coming from a resolution of X and as we vary the characteristic.

This is work in progress with Omprakosh Das.

Gregory G. Smith:  Nonnegativity certificates on real projective varieties

How can one use sums of squares to characterize nonnegative polynomials?  In this talk, we will review the general methods for certifying that a homogeneous polynomial is nonnegative on a real projective subvariety.  We will discuss an analogue of Bertini's Theorem in convex algebraic geometry and present new optimal degree bounds for certificates on real projective curves.  This talk is based on joint work with Grigoriy Blekherman and Mauricio Velasco.

Andrew Snowden: Connections between commutative algebra and representations of categories

In recent years, there has been a surge of interest in the representation theory of categories, with applications in algebraic geometry, topology, and elsewhere. In cases of interest, the
representations behave very similarly to modules over a commutative ring, and notions from commutative algebra (such as noetherianity, local cohomology, Grobner bases, etc) have proved useful. I will outline some of the motivations and applications of the subject, and
then go into more detail about the connections with commutative algebra.

Frank Sottile: Galois groups of Schubert problems

In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem.  Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems.  While difficult to determine in general, several methods have been developed recently to partially determine Galois groups in the Schubert calculus.

I will describe this background and discuss a project that seeks to determine Galois groups of all Schubert problems of moderate size, investigating millions of problems.  This project combines theoretical results with supercomputers employing several overlapping methods, including combinatorics, symbolic computation, and numerical homotopy continuation.  It is driving the development of new algorithms and software, and a partial classification is emerging from this study.

Jason Starr: Spaces of Rational Curves on Fano Manifolds

I will quickly review how theorems on existence and density of rational points on Fano manifolds defined over non-algebraically closed fields (e.g., function fields of curves and surfaces) lead to questions about the geometry of the compactified spaces of rational curves on the Fano manifold, the spaces of stable maps. Then I will explain recent results on the geometry of these spaces, including recent joint work with Zhiyu Tian determining the Picard groups, ample cones and (regular) contractions of the spaces of stable maps on (most) Fano hypersurfaces.

Mike Stillman: Applications of computational algebraic geometry to vacuum
moduli spaces of supersymmetric models in physics

Given a supersymmetric potential function, such as one for the MSSM (minimal super-symmetric extension of the standard model of particle physics), there is a naturally associated affine algebraic variety, which is (essentially) the moduli space of possible vacua of the theory.  In this talk we describe the structure of some of these moduli spaces, including the Electro-weak sector of the MSSM, which were obtained with the help of computational algebraic geometry and Macaulay2.  We consider theories where one allows for more than 3 generations of particles.  Since nature seems to have chosen 3 generations, theories for which this number of generations is forced would be ideal.  Although it does not appear that 3 generations is forced here, we see how the geometry varies with different numbers of generations of particles.

We will assume no knowledge of physics during this talk.  We will briefly describe the physics needed, and then we will describe the algebraic geometric and computational methods and results which allow the structure to become apparent.

This talk is based on joint work with the following authors, some of which appears in the recent paper arxiv.org/1408.6841. Joint work with:

Yang-Hui He (Oxford and City University London, UK)
Vishnu Jejjala (University of the Witwatersrand, Johannesburg, South Africa)
Cyril Matti (City University, London, UK)
Brent Nelson (Northeastern University, USA)

Hendrik Suess: Torus equivariant K-stability in complexity one

Using the combinatorial description of Fano varieties with torus actions of complexity one we give an effective criterion to test equivariant K-stability for such Fano varieties. By recent results of Datar and Szekelyhidi this leads to new examples of Kaehler-Einstein Fano threefolds and non-toric non-trivial Kaehler-Ricci solitons in dimension 3. This is joint work with Nathan Ilten.

Song Sun: Singularities of Kahler-Einstein metrics and stability

We will first review the fact that under appropriate assumptions Gromov-Hausdorff limits of polarized Kahler-Einstein manifolds are naturally endowed with the structure of a normal projective variety. Then we will move on to discuss some recent results,  which give an algebro-geometric description of the metric tangent cone at a singular point. This is related to a possible notion of ``stability” for an algebraic singularity, and we will explain a conjectural picture.

This talk is based on joint work with Simon Donaldson.

Gabor Szekelyhidi: The equivariant Yau-Tian-Donaldson conjecture

I will discuss an equivariant version of the Yau-Tian-Donaldson conjecture on the existence of Kahler-Einstein metrics, strengthening the result of Chen-Donaldson-Sun. This gives new examples of Kahler-Einstein manifolds, and it can also be applied to the existence problem for Kahler-Ricci solitons. It is joint work with Ved Datar.

Tomasz Szemberg: Sylvester-Gallai and beyond

Configurations of lines are a classical subject of study in various branches of mathematics: algebraic geometry, combinatorics, topology. Recently they played a prominent role in a number of results in algebraic geometry and commutative algebra concerning for example containment relations between ordinary and symbolic powers of homogeneous ideals, the birational invariance of the bounded negativity property on surfaces and related estimates on Harbourne constants. Some results available so far for lines have been extended to configurations of curves of higher degree. I will give an account of what was done in the last couple of years and discuss some promising future research directions.

Behrouz Taji: On a conjecture of Shafarevich and Campana

In 1962 Shafarevich conjectured that a smooth family of curves of genus $g\geq 2$ over non-hyperbolic algebraic curves, namely $\mathbb{C}$, $\mathbb{C}^*$, $\mathbb{P}^1$ and elliptic curve $E$, is isotrivial. More generally, it was conjectured that any smooth family of canonically-polarized manifolds over these curves has no (algebraic) variation. We prove a higher dimensional generalization of this conjecture by Campana (itself a generalization of Viehweg's hyperbolicity conjecture) via the results of Birkar, Cascini, Hacon and McKernan on the existence of minimal models of general type and Campana-Paun's generic orbifold-semipositivity result.

Shunsuke Takagi: Frobenius action on local cohomology and the Hodge filtration

M. Mustata and V. Srinivas recently introduced the so-called weak ordinarity conjecture, which states that if X is an n-dimensional smooth projective variety over a field of characteristic zero, then there exists a dense set of reductions X_p to positive characteristic such that the action of Frobenius on H^n(X_p, O_{X_p}) is bijective. In this talk, I will discuss the relationship of this conjecture and F-singularities. In particular, I will show that a variant of the weak ordinarity conjecture gives a Hodge theoretic interpretation of F-nilpotent isolated singularities. This leads to a characterization of low-dimensional F-nilpotent isolated singularities in terms of divisor class groups and Brauer groups. This talk is based on joint work with B. Bhatt, K. Schwede and with V. Srinivas.

Sofia Tirabassi: Deformations of minimal cohomology classes

We show that there is a one-to-one correspondence between the infinitesimal deformations of a non-hyperelliptic curve of genus g and those of the Brill-Noether loci W_d (C). We apply this to the study of a celbrate cojecture of Debarre concerning ppav admitting a d-dimensional subscheme representing the minimal cohomology class.

Valentino Tosatti: Nakamaye's theorem on complex manifolds

A well-known result of Nakamaye states that the augmented base locus of a
nef and big line bundle on a smooth projective variety over the complex
numbers equals its null locus. I will discuss an extension of this theorem
to all nef and big real (1,1) classes on compact complex manifolds, which
also gives an analytic proof of Nakamaye's original result. I will also
mention some striking consequences of this theorem, and some recent
further developments. This is joint work with Tristan Collins.

Frédéric Touzet: Compact leaves of codimension one holomorphic foliations

This talk is concerned with the study of codimension one
foliations on projective/Kähler manifolds having a compact leaf
(free of singularities). The embedding of this hypersurface in the
ambient manifold can be investigated mainly through Ueda theory (order
of flatness of the normal bundle) and through the holonomy
representation (dynamics of the foliation in the transverse direction).
We address in particular the following problems: existence of foliation
having as a leaf a given hypersurface with topologically torsion normal
bundle, study of foliations having a compact leaf whose holonomy is
abelian (resp. solvable) and factorization results. This is a joint work
with B. Claudon, F.Loray and J.V. Pereira.

Nikolaos Tziolas: Automorphisms of canonically polarized surfaces in positive characteristic

Let X be a smooth canonically polarized surface defined over an algebraically closed field of characteristic p>0. In this talk I will present some results about the geometry of X in the case when the automorphism scheme Aut(X) of X is not smooth, or equivalently X has nontrivial global vector fields. This is a situation that appears only in positive characteristic and is intimately related to the structure of the moduli stack of canonically polarized surfaces in positive characteristic. One of the results that will be presented in this talk is that smooth canonically polarized surfaces with non smooth automorphism scheme and “small” invariants are algebraically simply connected and uniruled.

Dror Varolin: A survey of L^2 Extension and its applications in analytic and algebraic geometry

Since the L^2 extension theorem was proved by Ohsawa and Takegoshi in 1987, there has been a huge amount of work on both further L^2 Extension results and their applications.  I will give a somewhat biased survey of the methods and results, and discuss a number of applications in analytic and algebraic geometry.

Claire Voisin: Stable birational invariants and the Lüroth problem

Kei-ichi Watanabe: p_g-ideals and core of integrally closed ideals in normal surface singularities

Let (A.m) be a 2-dimensional normal local ring and I be an integrally closed m-primary ideal of A. Let f : X --> Spec(A) be a log resolution of I so that IO_X=O_X(-Z).  We say I is a p_g-ideal if h^1'O_X(-Z))= p_g(A). We show that p_g ideals have very nice properties. For example, the Rees algebra of I is normal and Cohen-Macaulay iff I is a p_g ideal. We compute the core of such ideals and give the formula for integrally closed ideals of rational singularities.  We also give a characterization of rational singularities using the core of integrally closed ideals.  

Jörg Winkelmann: On h-principle on specialness

This is joint work with F. Campana.

A complex space X is said to satisfy the h-principle if, for every Stein manifold S and every continuous map g:S\to X there is a holomorphic map f:S\to X homotopic to g. This notion, which was introduced by Gromov, is known to be satisfied for ``elliptic'' complex manifolds, a class which contains in particular every homogeneous complex manifold.

We investigate this property for compact complex projective manifolds, in particular exploring relationships with algebro-geometric properties. The main result is that the h-principle may hold only if X is ``special''. In particular this implies that a variety of general type never satisfies the h-principle.

We also prove that the h-principle is not satisfied for any compact complex manifold which is hyperbolic in the sense of Kobayashi.

David Witt Nyström: Growth conditions associated to ample (or big) line bundles

I will discuss a new construction which associates to any ample (or big) line bundle on a projective manifold a growth condition on the tangent space of any given point. The growth condition can be seen to encode such classical invariants as the volume and the Seshadri constant. It is inspired by toric geometry, and in fact in the toric case the growth condition is "equivalent" to the moment polytope. As in the toric case the growth condition says a lot about the Kähler geometry of the manifold. I will present a theorem about Kähler embeddings of large balls, which generalizes the connection between Seshadri constants and Gromov width established by McDuff and Polterovich.

Chenyang Xu: Dual complex of singular pairs

Abstract: We will discuss the recent progress of understanding the dual complex D(D) of a singular pair (X,D), which is the combinatorial data characterizing how the components of the divisor D intersect each other. In particular, we will focus on the recent joint work with Kollár on studying the fundamental group of D(D) when (X,D) is a log Calabi-Yau pair. This also has direct application to the degeneration of Calabi-Yau manifolds.