# Beyond Toric Geometry (17w5130)

Arriving in Oaxaca, Mexico Sunday, May 7 and departing Friday May 12, 2017

## Organizers

Nathan Ilten (Simon Fraser University)

Milena Hering (Edinburgh University)

Kalle Karu (University of British Columbia)

## Objectives

There have been a number of important recent developments involving T-varieties, Cox rings, and equivariant cohomology:

1. T-Varieties. Varieties with torus actions by lower-dimensional tori, known as T-varieties, provide a natural generalization of toric varieties. By work of Altmann-Hausen (2006) and Altmann-Hausen-Süß (2008), an $n$-dimensional variety $X$ with an effective action by a $k$-dimensional torus can be encoded by an $n-k$-dimensional quotient variety $Y$ endowed with some additional combinatorial structure. In the toric case, the quotient variety is simply a point and the combinatorial structure is a rational polyhedral fan.

This quasi-combinatorial description has led to a very fruitful study of the geometry of $T$-varieties over the past several years. Many of the classical results concerning toric varieties have been at least partially generalized to this setting: descriptions of divisors and intersection theory (Petersen-Süß 2011); classification of singularities (Liendo-Süß 2013); description of automorphism groups (Arzhantsev-Hausen-Herppich-Liendo 2014); and a characterization of the properties of being Frobenius split (Achinger-Ilten-Süß 2015), just to name a few. These results, and the techniques used, have been applied to a number of other areas, including coding theory (Ilten-Süß 2010), existence of Kähler-Einstein metrics (Süß 2013 and Ilten-Süß 2015), and the study of the affine Cremona group (Liendo 2011).

One particularly well-studied class of T-varieties are equivariant toric vector bundles. They have been described in toric language by Klyachko (1989) using collections of filtered vectored spaces. The recent approach via the machinery of T-varieties has answered questions concerning their Cox rings (see below) and Frobenius-splitting properties. Even more recent work of Di Rocco-Jabbusch-Smith (2014) approaches toric vector bundles via so-called "parliaments" of polytopes, and promises to further illuminate their geometry.

2. Cox rings and Mori Dream Spaces. Cox rings of general algebraic varieties were defined by Hu and Keel (2000), generalizing D. Cox's construction of homogeneous coordinate rings of toric varieties. Cox rings have important applications to birational geometry and the minimal model program. The main problem in the theory of Cox rings is to determine which varieties are Mori Dream Spaces (MDS), which means that they have finitely generated Cox rings. Toric varieties are examples of MDS by the work of Cox (1995).

There has been a great amount of recent work studying Cox rings of varieties closely related to toric varieties. These include Cox rings of toric vector bundles by Gonzalez-Hering-Payne-Süß (2012), complexity one T-varieties by Altmann-Petersen (2012), general varieties with torus action by Hausen-Süß (2010) and many others. Another class of varieties one step away from toric varieties are the moduli spaces $\bar{M}_{0,n}$ of $n$-pointed rational curves. Results supporting the hypothesis that these moduli spaces are MDS include the proof in the case $n=6$ by Castravet (2009), and a conjectural set of hypertree generators by Castravet-Tevelev (2013). On the negative side, we know for large $n$ that the hypertree generators are not enough (Doran-Giansiracusa-Jensen 2014), and most remarkably, that these moduli spaces are not MDS for $n$ large (Castravet-Tevelev 2015).

The work of Castravet-Tevelev has brought back into focus an older proof by Goto-Nishida-Watanabe (1994) that the blowups of certain weighted projective planes at a single point are not MDS. Many more such examples of blowups of weighted projective planes and moduli spaces that are not MDS were found by Gonzalez-Karu (2015).

3. Equivariant cohomology theories. Equivariant intersection cohomology theory of toric varieties has a combinatorial description in terms of sheaves on fans (Barthel-Brasselet-Fiesler-Kaup 1998, Bressler-Lunts 2003). This description has resulted in the proofs of several open conjectures in the combinatorics of polytopes (e.g., Braden-MacPherson (1999), Karu (2004)). More recently similar combinatorial techniques have been used in closely related situations, such as for other varieties with torus action, and for other cohomology theories.

The combinatorial equivariant intersection cohomology of toric varieties is a special case of a more general construction by Goresky-Kottwitz-MacPherson (1998), which applies to other T-varieties, such as Schubert varieties, for which combinatorial description in terms of sheaves on moment graphs was given by Braden-MacPherson (2001). The most remarkable recent result in the intersection cohomology of Schubert varieties is the proof of the Hard Lefschetz theorem and hence non-negativity of the Kazhdan-Lusztig polynomials by Elias-Williamson (2014). Their proof uses the theory of Soergel bimodules instead of sheaves, but the techniques are the same as in the case of toric varieties.

The combinatorics of equivariant (intersection) cohomology for toric varieties has been generalized to other cohomology theories. All these theories can be studied by localization to fixed points and result in combinatorial description as spaces of function on fans. The theories include the operational theories: Chow theory (Payne 2006), K-theory (Anderson-Payne 2015), algebraic cobordism (Krishna-Uma 2013, Gonzalez-Karu 2015) and more general oriented cohomology theories on flag varieties (Harada-Henriques-Holm 2005, Calmes-Zainoulline-Zhong 2013).

This workshop intends to bring together experts in these three areas with several specific objectives in mind. Firstly, the workshop will allow these experts to discuss their techniques and share their recent results with one another. Many of the above-mentioned results are quite new, and have not yet been thoroughly discussed. Secondly, the workshop will provide the opportunity for non-experts in these areas to acquire a working knowledge in this field. To this end, the workshop will include several introductory lectures, and a portion of the participants will be drawn from postdocs and graduate students. Finally, the workshop will provide an environment for old and newly-minted experts alike to tackle open problems in the field, for example, an approach to the classification of Fano T-varieties.

The three proposed areas complement each other and provide ample opportunity for interaction. T-varieties are a natural class of varieties whose Cox rings are more easily understood than in general. In particular, the above-mentioned recent work of Gonzalez-Karu (2015) could be used to construct new examples of T-varieties and toric vector bundles with non-finitely generated Cox rings. Likewise, T-varieties give a natural setting for studying equivariant cohomology theories. On the other hand, the spectrum of the Cox ring of an MDS $X$ is itself a T-variety, with action by the torus whose character lattice is the Picard group of $X$. Furthermore, any cohomology theory on $X$ can be understood as the equivariant cohomology of the universal torsor over $X$, which is an open subvariety of the spectrum of its Cox ring.

A unifying theme among the three areas of this proposal is that all concern varieties endowed with group actions sharing many of the features found in the toric setting. In the equivariant cohomology and the localization formula, the torus usually acts with finitely many fixed points, as is the case for toric varieties. In the case of T-varieties, the simplest non-trivial and best understood case is that where the torus has codimension one, very close to the codimension-zero case of the toric setting. Finally, typical examples of varieties for which Cox rings are computed are quotients of the affine space by a product of the torus and several factors of the additive group $G_a$. This mirrors the situtation for toric varieties, which are quotients of open subsets of affine space by a torus.

It should be noted that this workshop is proposed to run a year after a semester-long program on the broad area of combinatorial algebraic geometry at the Fields Institute in Toronto. This workshop would be an exciting opportunity to build on that program by investigating more deeply the three topics proposed above.

1. T-Varieties. Varieties with torus actions by lower-dimensional tori, known as T-varieties, provide a natural generalization of toric varieties. By work of Altmann-Hausen (2006) and Altmann-Hausen-Süß (2008), an $n$-dimensional variety $X$ with an effective action by a $k$-dimensional torus can be encoded by an $n-k$-dimensional quotient variety $Y$ endowed with some additional combinatorial structure. In the toric case, the quotient variety is simply a point and the combinatorial structure is a rational polyhedral fan.

This quasi-combinatorial description has led to a very fruitful study of the geometry of $T$-varieties over the past several years. Many of the classical results concerning toric varieties have been at least partially generalized to this setting: descriptions of divisors and intersection theory (Petersen-Süß 2011); classification of singularities (Liendo-Süß 2013); description of automorphism groups (Arzhantsev-Hausen-Herppich-Liendo 2014); and a characterization of the properties of being Frobenius split (Achinger-Ilten-Süß 2015), just to name a few. These results, and the techniques used, have been applied to a number of other areas, including coding theory (Ilten-Süß 2010), existence of Kähler-Einstein metrics (Süß 2013 and Ilten-Süß 2015), and the study of the affine Cremona group (Liendo 2011).

One particularly well-studied class of T-varieties are equivariant toric vector bundles. They have been described in toric language by Klyachko (1989) using collections of filtered vectored spaces. The recent approach via the machinery of T-varieties has answered questions concerning their Cox rings (see below) and Frobenius-splitting properties. Even more recent work of Di Rocco-Jabbusch-Smith (2014) approaches toric vector bundles via so-called "parliaments" of polytopes, and promises to further illuminate their geometry.

2. Cox rings and Mori Dream Spaces. Cox rings of general algebraic varieties were defined by Hu and Keel (2000), generalizing D. Cox's construction of homogeneous coordinate rings of toric varieties. Cox rings have important applications to birational geometry and the minimal model program. The main problem in the theory of Cox rings is to determine which varieties are Mori Dream Spaces (MDS), which means that they have finitely generated Cox rings. Toric varieties are examples of MDS by the work of Cox (1995).

There has been a great amount of recent work studying Cox rings of varieties closely related to toric varieties. These include Cox rings of toric vector bundles by Gonzalez-Hering-Payne-Süß (2012), complexity one T-varieties by Altmann-Petersen (2012), general varieties with torus action by Hausen-Süß (2010) and many others. Another class of varieties one step away from toric varieties are the moduli spaces $\bar{M}_{0,n}$ of $n$-pointed rational curves. Results supporting the hypothesis that these moduli spaces are MDS include the proof in the case $n=6$ by Castravet (2009), and a conjectural set of hypertree generators by Castravet-Tevelev (2013). On the negative side, we know for large $n$ that the hypertree generators are not enough (Doran-Giansiracusa-Jensen 2014), and most remarkably, that these moduli spaces are not MDS for $n$ large (Castravet-Tevelev 2015).

The work of Castravet-Tevelev has brought back into focus an older proof by Goto-Nishida-Watanabe (1994) that the blowups of certain weighted projective planes at a single point are not MDS. Many more such examples of blowups of weighted projective planes and moduli spaces that are not MDS were found by Gonzalez-Karu (2015).

3. Equivariant cohomology theories. Equivariant intersection cohomology theory of toric varieties has a combinatorial description in terms of sheaves on fans (Barthel-Brasselet-Fiesler-Kaup 1998, Bressler-Lunts 2003). This description has resulted in the proofs of several open conjectures in the combinatorics of polytopes (e.g., Braden-MacPherson (1999), Karu (2004)). More recently similar combinatorial techniques have been used in closely related situations, such as for other varieties with torus action, and for other cohomology theories.

The combinatorial equivariant intersection cohomology of toric varieties is a special case of a more general construction by Goresky-Kottwitz-MacPherson (1998), which applies to other T-varieties, such as Schubert varieties, for which combinatorial description in terms of sheaves on moment graphs was given by Braden-MacPherson (2001). The most remarkable recent result in the intersection cohomology of Schubert varieties is the proof of the Hard Lefschetz theorem and hence non-negativity of the Kazhdan-Lusztig polynomials by Elias-Williamson (2014). Their proof uses the theory of Soergel bimodules instead of sheaves, but the techniques are the same as in the case of toric varieties.

The combinatorics of equivariant (intersection) cohomology for toric varieties has been generalized to other cohomology theories. All these theories can be studied by localization to fixed points and result in combinatorial description as spaces of function on fans. The theories include the operational theories: Chow theory (Payne 2006), K-theory (Anderson-Payne 2015), algebraic cobordism (Krishna-Uma 2013, Gonzalez-Karu 2015) and more general oriented cohomology theories on flag varieties (Harada-Henriques-Holm 2005, Calmes-Zainoulline-Zhong 2013).

This workshop intends to bring together experts in these three areas with several specific objectives in mind. Firstly, the workshop will allow these experts to discuss their techniques and share their recent results with one another. Many of the above-mentioned results are quite new, and have not yet been thoroughly discussed. Secondly, the workshop will provide the opportunity for non-experts in these areas to acquire a working knowledge in this field. To this end, the workshop will include several introductory lectures, and a portion of the participants will be drawn from postdocs and graduate students. Finally, the workshop will provide an environment for old and newly-minted experts alike to tackle open problems in the field, for example, an approach to the classification of Fano T-varieties.

The three proposed areas complement each other and provide ample opportunity for interaction. T-varieties are a natural class of varieties whose Cox rings are more easily understood than in general. In particular, the above-mentioned recent work of Gonzalez-Karu (2015) could be used to construct new examples of T-varieties and toric vector bundles with non-finitely generated Cox rings. Likewise, T-varieties give a natural setting for studying equivariant cohomology theories. On the other hand, the spectrum of the Cox ring of an MDS $X$ is itself a T-variety, with action by the torus whose character lattice is the Picard group of $X$. Furthermore, any cohomology theory on $X$ can be understood as the equivariant cohomology of the universal torsor over $X$, which is an open subvariety of the spectrum of its Cox ring.

A unifying theme among the three areas of this proposal is that all concern varieties endowed with group actions sharing many of the features found in the toric setting. In the equivariant cohomology and the localization formula, the torus usually acts with finitely many fixed points, as is the case for toric varieties. In the case of T-varieties, the simplest non-trivial and best understood case is that where the torus has codimension one, very close to the codimension-zero case of the toric setting. Finally, typical examples of varieties for which Cox rings are computed are quotients of the affine space by a product of the torus and several factors of the additive group $G_a$. This mirrors the situtation for toric varieties, which are quotients of open subsets of affine space by a torus.

It should be noted that this workshop is proposed to run a year after a semester-long program on the broad area of combinatorial algebraic geometry at the Fields Institute in Toronto. This workshop would be an exciting opportunity to build on that program by investigating more deeply the three topics proposed above.