MAGADAN VOLUME on the 5Th of December 2015, Seventeen

Fedor Bogomolov, Ivan Cheltsov, Tatiana Kozlovskaya, and Jihun Park

On the 5th of December 2015, seventeen mathematicians arrived in Magadan, a small city in the Russian Far East, located on the Sea of Okhotsk between Nagaev and Gertner bays. Ten mathematicians flew in from Moscow: Elena Bunkova, Sergey Galkin, Sergey Gorchinskiy, Igor Netay, Yuri Prokhorov, Vic- tor Przyjalkowski, Marat Rovinsky, Constantin Shramov, Evgeny Smirnov and Andrey Trepalin. Andrey Mironov came from Novosibisrk, and Alexei Savva- teev from Irkutsk. The remaining five traveled from outside of Russia: Fedor Bogomolov came from New York in the US, Ivan Cheltsov from Edinburgh in the UK, Kento Fujita from Kyoto in Japan, Jihun Park and Dmitrijs Sakovics from Pohang in Korea. They all came to Magadan to participate in a conference. Two local mathematicians, Tatiana Kozlovskaya and Olga Starikova, joined them in Magadan. This volume carries 21 papers related to the Magadan Conference that was held at North–East State University in Magadan for the period 6–12 December 2015. Why Magadan? The idea of organizing a conference there came from the song “I am going to Magadan” by Vasya Oblomov. Ivan Cheltsov’s kids (Vanya, Fedya and Katya) often listened to this song during the family trip to Bormes- les-Mimosas in August 2014. This song is not the only song about Magadan and songs about the city are somewhat surprisingly numerous. Nearly everyone in Russia knows that Magadan is the capital of Kolyma (these words are from another song “Port Vanino”). Unfortunately, the image of Magadan conjured up by songs, books and movies is usually somewhat less than enticing, to say the least. A typical example is the well-known dialogue in the comedy The Diamond Arm by Leonid Gaidai: — Come visit us in Kolyma! — No, I think it’s better you visit us instead. Most Russians hearing the words “Magadan” or “Kolyma” would probably tend to immediately think of the land of eternal frost, labor camps and the Gulag. Not exactly your first choice if you were looking for a holiday destination. (Kento Fujita, one of the participants of the conference, visited Russia for the first time in his life. His first town in Russia was Magadan!) However, contemporary Magadan has managed to move on somewhat from the past, although the locals have not in any way been subsumed by a collective amnesia and forgotten the uncomfortable events of its history. In case anyone might, however, the Mask of Sorrow, a monument standing high on a hill above Magadan, serves as a reminder of the thousands of people who perished there. Magadan was founded in 1930 as a port of entry to Soviet arctic gold mines and also as the headquarters of the network of the Gulag camps dispersed

i MAGADAN VOLUME throughout the huge Kolyma region. Facing south it also played a role as a resort town for the Kolyma which runs down from frozen mountains towards the Arctic Ocean. From 1932 to 1953, the city was operated by the Dalstroy organization that mined gold throughout the Kolyma region using extensively the work of prisoners living in the concentration camps. This company also built the famous Kolyma highway, a 2000 km road that connects Magadan to Yakutsk, the capital of Russian diamond production. This is the only road that links Magadan to the rest of the world. However, most people arrive either by air or by sea (no railways go to Magadan). Because of this, locals often feel like they live on an island and refer to other parts of Russia as the mainland. Despite its remote location, harsh climate (the daily mean temperature in December is −15◦C), small size (less than 100,000 people) and turbulent history, Magadan is a vibrant modern city. In fact, the hardships tend to bring people together and locals are almost always ready to help a stranger. Interestingly, many people who thought they had left Magadan forever returned later, drawn back by Magadan’s nature and the openness of its people. The economy is mostly focused on the gold mining and machine-building industries. The city hosts the headquarters of all local gold mining companies, but the gold is not mined or refined there. Because of this, Magadan is a middle class city. It has a number of cultural institutions including a a very rich anthropological museum, a somewhat unique kind of geological museum, and an extremely informative museum of the Gulag history. It also has a big library, several modern cinemas and a famous drama theater. In addition to this, Magadan is a home to North–East State University, which has around three thousand students. The university provides excellent education for locals, and has all possible facilities for an international conference. We are very grateful to the university for hosting our Magadan Conference. Nineteen participants is not very many for a scientific meeting. In fact, we expected more people to join the conference. Unfortunately, Transaero, the only airline that operated the direct Moscow to Magadan flight, went into liquidation on the 1st of October 2015. Because of this many Europeans and Americans canceled their participation. Despite this, the Magadan Conference was a successful event. Thirty mathematicians (those who came to Magadan and those who planned to come but failed to do so) contributed 21 papers to this volume. Briefly their contribution are as follows.

In the paper Endomorphisms of projective bundles over a certain class of varieties, Ekaterina Amerik and Alexandra Kuznetsova study endomorphisms of projectivizations of vector bundles on simply-connected projective varieties. To be precise, let X be a simply-connected projective variety such that the first cohomology groups of all line bundles on X are zero, and let E be a vector bundle over X. Then a power of any endomorphism of P(E) must map fibers of the natural projection P(E) → X back to fibers. This is not very hard to show. In their paper, Amerik and Kuznetsova proved that if E admits an endomorphism which is of degree greater than 1 on the fibers, then E splits into a direct sum of line bundles.

ii F. BOGOMOLOV, I. CHELTSOV, T. KOZLOVSKAYA, AND J. PARK

In the paper Equivariant matrix factorizations and Hamiltonian reduction, Sergey Arkhipov and Tina Kanstrup established an equivalence of two categories related to the moment map ∗ ∗ µ: TX → g . Here X is a smooth scheme with an action of an algebraic group G, and g is the Lie algebra of the group G. The first category is the derived category of G-equivariant coherent sheaves on the derived fiber µ−1(0), and the second ∗ category is the derived category of G-equivariant matrix factorizations on TX ×g with potential given by µ. To describe the paper Cylinders in del Pezzo surfaces with Du Val singularities by Grigory Belousov, let us fix a del Pezzo surface Sd with Du Val singularities such that K2 = d. For 3 d 9, the anticanonical divisor is very Sd 6 6 ample and the anticanonical linear system embeds Sd into the projective space d 3 P . In particular, for d = 3, the surface S3 is a cubic surface in P . Meanwhile, for d = 2, the surface Sd is a hypersurface in the weighted projective space P(1, 1, 1, 2) defined by a quasi-homogeneous polynomial of degree 4. Similarly, for d = 1, the surface Sd is a hypersurface P(1, 1, 2, 3) of degree 6.

Definition. For an ample Q-divisor H on the surface Sd, an H-polar cylinder in Sd is an open subset U ⊂ Sd such that • the subset U is isomorphic to Z × A1 for some affine curve Z, • there exists an effective Q-divisor D on the surface Sd such that U = S \ Supp(D)

and D ∼Q −KSd . The group of Kishimoto, Prokhorov, and Zaidenberg and the group of Cheltsov,

Park, and Won have studied the existence of (−KSd )-polar cylinders in the surface Sd. In the case when Sd is smooth, they showed that Sd has a (−KSd )- polar cylinder if and only if d > 4. In the singular case, Cheltsov, Park and Won proved the following.

Theorem. Suppose that the surface Sd is singular. Then it does not contain

(−KSd )-polar cylinders if and only if • either d = 1 and Sd allows only singular points of types A1, A2, A3 or D4, • or d = 2 and Sd allows only singular points of type A1.

Moreover, they pointed out that the existence of (−KSd )-polar cylinders is closely related to the existence of non-log canonical tigers on the surfaces Sd. The latter notion was introduced by McKernan and Keel. Let us recall that a tiger on Sd is an eﬀective Q-divisor D such that

D ∼Q −KSd , and the singularities of the log pair (Sd,D) are not Kawamata log terminal. If in addition, the singularities of the log pair (Sd,D) are not log canonical, then D is said to be non-log canonical tiger. The surface Sd contains a non-log canonical tiger if and only if α(Sd) < 1, where α(Sd) is the α-invariant of Tian of the surface Sd. The number α(Sd) is iii MAGADAN VOLUME computed in all possible cases, and in most of them we have α(Sd) < 1. For 5 example, if d > 2, then α(Sd) 6 6 . Thus, most del Pezzo surfaces with Du Val singularities contain non-log canonical tigers. However, many such tigers are just divisors in the anticanonical linear system | − KSd |, or eﬀective divisors whose support contains tigers in | − KSd |. In the paper Cylinders in del Pezzo surfaces with Du Val singularities, Belousov proved the following.

Theorem. The surface Sd contains a (−KSd )-polar cylinder if and only if it contains a non-log canonical tiger D such that Supp(D) does not contain a a non-log canonical tiger in | − KSd |.

The paper On contraction of algebraic points by Fedor Bogomolov and Jin Qian addresses the following very natural problem. Let Σ be a subset of the 1 natural numbers and let S1 and S2 be two subsets of points in ¯ . Then S1 PQ 1 1 can be Σ-contracted to S2 if there is a rational map f : ¯ → ¯ such that the PQ PQ image of S1 under f and all branch points of f are contained in S2 with all local ramification indices of f belonging to the set Σ. By Belyi’s theorem, any 1 finite subset S1 ⊂ ¯ can be -contracted to the set {0, 1, ∞} if we omit the PQ N restriction on ramification indices. The maps with natural restrictions on indices constructed in the paper provide the existence of curves whose unramified coverings dominate by meromorphic maps broad classes of curves. The program of finding such curves was formulated in the previous work of Bogomolov and Tschinkel. In the paper Birational rigidity is not an open property, Ivan Cheltsov and Mikhail Grinenko constructed a counterexample to a conjecture of Alessio Corti. To explain it, let us recall that a Mori fibre space is a surjective morphism π : X → S such that

• the variety X has terminal and Q-factorial singularities, • the inequality dim(S) < dim(X) holds and π∗(OX ) = OS, • the divisor −KX is relatively ample for π, • the equality rk Pic(X) = rk Pic(X) + 1 holds. Moreover, the Mori ﬁbre space π : X → S is said to be birationally rigid if, 0 0 0 0 given any birational map ξ : X 99K X to another Mori ﬁbre space π : X → S , there exists a commutative diagram

ρ ξ X / X / X0

π π0 σ S / S0 for some birational maps ρ and σ such that the composition map ξ ◦ ρ induces an isomorphism of the generic ﬁbers of the Mori ﬁbre spaces π and π0. In the paper Birational rigidity is not an open property, Cheltsov and Grinenko constructed an explicit three-dimensional example that shows that the following conjecture fails in general. iv F. BOGOMOLOV, I. CHELTSOV, T. KOZLOVSKAYA, AND J. PARK

Conjecture (Corti). For any scheme T , and a ﬂat family of Mori ﬁbre spaces parametrised by T X / S

T the set of all t ∈ T such that the corresponding fibre Xt → St is birationally rigid is open in T . In the paper Okounkov bodies and Zariski decompositions on surfaces, Sung Rak Choi, Jinhyung Park and Joonyeong Won investigate the close relationship between Okounkov bodies and Zariski decompositions of pseudoeffective divisors on smooth projective surfaces. In particular, they completely determine the limiting Okounkov bodies on such surfaces, and give applications to Nakayama constants and Seshadri constants. Moreover, they study how the shapes of Okounkov bodies change as they vary the divisors in the big cone and how the behaviors of their shapes decomposes the cone. In the paper On three-dimensional semi-terminal singularities, Kento Fujita classifies three-dimensional non-normal semi-terminal singularities. The notion of terminal singularities is very important in the minimal model program. For the two-dimensional case, the notion of terminal singularities is equivalent to the notion of smoothness. Three-dimensional terminal singularities are understood very well thanks to the work of Mori, Kollárand Shepherd-Barron. On the other hand, non-normal varieties are also important for algebraic geometry. In his earlier paper, Fujita introduced the notion of semi-terminal singularities which is a natural generalization of terminal singularities in non-normal case. In this paper, he classifies all of the non-normal three-dimensional semi-terminal singularities. In the paper Integral Chow motives of threefolds with K-motives of unit type, Sergey Gorchinskiy proves that if a smooth projective algebraic variety of dimension less than or equal to three has a unit type integral K-motive, then its integral Chow motive is of Lefschetz type. As a consequence, Gorchinskiy deduces the following result: if a smooth projective variety of dimension less than or equal to three admits a full exceptional collection, then its the integral Chow motive is of Lefschetz type. In the paper Heegaard splittings of branched cyclic coverings of connected sums of lens spaces, Tatiana Kozlovskaya investigates the relations between two descriptions of closed orientable 3-manifolds: as branched coverings and as Heegaard splittings. She presents an explicit relation for a class of 3-manifolds which are branched cyclic coverings of connected sums of lens spaces, where the branching set is an axis of a hyperelliptic involution of a Heegaard surface. In the paper On commuting ordinary diffrential operators with polynomial coefficients corresponding to spectral curves of genus two, Andrey Mironov and Valentina Davletshina study the natural action of the group of automorphisms of the first Weyl algebra on commuting ordinary differential operators with polynomial coefficients. They prove that for fixed generic spectral curve of genus two the set of orbits is infinite.

v MAGADAN VOLUME

The paper Sharygin triangles by Igor Netay and Alexei Savvateev is about triangles in R2 such that the triangle formed by the intersection points of bisec- tors with opposite sides is isosceles. If the original triangle is not itself isosceles, then it is called a Sharygin triangle. It turns out that Sharygin triangles are parametrized by an open subset (in the Euclidean topology) of an elliptic curve. The authors give many nice examples of Sharygin triangles, and prove that there are infinitely many non-similar integer Sharygin triangles. In the paper Second Chern numbers of vector bundles and higher adeles, De- nis Osipov gives a construction of the second Chern number of a vector bundle over a smooth projective surface defined over a perfect field k by means of adelic transition matrices for the vector bundle. His construction does not use algebraic K-theory and depends on the canonical Z-torsor of a locally linearly compact k-vector space. One of the key statements which he uses in the construction is the splitting of certain central extensions of groups constructed using Parshin-Beilinson adeles on the algebraic surface. He proves also an analog of this statement for an arithmetic surface and adeles on it. In the paper A simple proof of the non-rationality of a general quartic double solid, Yuri Prokhorov gives a very short proof of the following nice result.

Theorem. Let X be the double cover of P3 branched in the quartic surface 3 3 3 3 x1x2 + x2x3 + x3x4 + x4x1 = 0. Then X is not rational. This result together with a standard degeneration technique gives a new proof of the following well-known result.

Corollary. A double cover of P3 branched in a very general quartic surface is not rational. In the paper Laurent phenomenon for Landau-Ginzburg models of complete intersections in Grassmannians of planes, Victor Przyjalkowski and Constantin Shramov provide a particular algorithm for constructing birational isomor- phisms of Landau-Ginzburg models for smooth Fano complete intersections in Grassmannians of planes suggested by Batyrev, Ciocan-Fontanine, Kim, and van Straten with complex tori. In particular, superpotentials of the Landau- Ginzburg models after birational transformations become complex tori. The authors also show that the periods for the Laurent polynomials coincide with Givental integrals, which means that these polynomials are weak Landau-Ginz- burg models. An alternative algorithm is given in Prince’s paper in this volume, which we described above. The paper Birationally rigid complete intersections of codimension two by Alexander Pukhlikov and Daniel Evans studies birational geometry of higher- dimensional complete intersections of codimension two. The purpose is to obtain explicit estimates for the codimension of the complement of the set of birationally rigid varieties in a natural parameter space. To be precise, the authors prove the following explicit result. Fix an integer M > 13, and ﬁx two integers d2 > d1 > 2 such that d1 + d2 = M + 2. Consider the space

P = Pd1,M+3 × Pd2,M+3 vi F. BOGOMOLOV, I. CHELTSOV, T. KOZLOVSKAYA, AND J. PARK of pairs of homogeneous polynomials (f1(x0, . . . , xM+2), f2(x0, . . . , xM+2)) of degrees d1 and d2, respectively. This is the natural parameter space for Fano complete intersections of the type (d1, d2) in projective space. Denote by the symbol V (f1, f2) the set of common zeros of f1 and f2.

Theorem. There exists a Zariski open subset Preg ⊂ P such that (M − 9)(M − 10) codim P\P ⊂ P − 1 reg > 2 and for every pair (f1, f2) ∈ Preg

(1) the closed set V (f1, f2) is irreducible, reduced and of codimension 2 in M+2 P with singular locus of codimension at least 7 in V (f1, f2), so that V (f1, f2) is a factorial projective algebraic variety, (2) the singularities of V (f1, f2) are terminal, so that V (f1, f2) is a primi- tive Fano variety of index 1 and dimension M, (3) the Fano variety V (f1, f2) is birationally rigid, (4) one has Bir V f1, f2 = Aut V f1, f2 .

As a standard immediate consequence of this theorem, we obtain the non- rationality of regular complete intersections described above. In the paper Hilbert’s Theorem 90 for non-compact groups, Marat Rovinsky presents three examples of sites (or pairs (K,G) consisting of a field K and a group of its automorphisms G) such that the simple ‘coherent’ sheaves (or certain irreducible K-semilinear representations of the group G) admit a simple description. In the first example, the group G is precompact. In the second example, the field K is a field of rational functions and G permutes the variables. In the third example, the field K is a universal domain over field of characteristic zero and G its automorphism group. The paper The Bogomolov-Prokhorov invariant of surfaces as equivariant cohomology by Evgeny Shinder gives a conceptual proof of the following theorem, which is a generalization of a result of Bogomolov and Prokhorov: for a complex smooth projective surface M with an action of a finite cyclic group G satisfying a certain technical condition there is an isomorphism between the Bogomolov-Prokhorov invariant

1 2 H G, H M, Z and the first cohomology of the divisors fixed by the action. The proof relies on G-equivariant cohomology groups and the Gysin maps between cohomology of the fixed points of G and the equivariant cohomology of M. In the paper Singularities of divisors on flag varieties via Hwang’s product theorem, Evgeny Smirnov gives a new conceptual proof of the following result by Pasquier:

Theorem. Let X be a generalized ﬂag variety, and let D be an eﬀective Q- divisor on X that is stable with respect to a Borel subgroup. Then the pair (X,D) is Kawamata log terminal if and only if dDe = 0.

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The paper Minimal del Pezzo surfaces of degree 2 over finite fields by Andrey Trepalin studies del Pezzo surfaces of degree 2 over the finite field Fq. Namely, let S2 be a del Pezzo surface of degree 2 over Fq, and denote by S2 the same surface considered over the algebraic closure Fq of the field Fq. Suppose S2 is minimal over Fq, i.e. its Picard group over Fq is generated by the anticanonical divisor −KS2 . Denote by Γ the image of the Galois group Gal(Fq/Fq) in the group Aut(Pic(S2)). Then Γ is a cyclic subgroup of the Weyl group W (E7). Note that there are 60 conjugacy classes of cyclic subgroups in W (E7), but only 18 of them correspond to minimal del Pezzo surfaces. In his paper, Trepalin studies which possibilities of these subgroups for minimal del Pezzo surfaces of degree 2 can be achieved for a given q. In the paper The Jordan constant for Cremona group of rank 2, Egor Yasin- sky proves the following result: Theorem. The Jordan constant of the group Bir( 2) is 7200. Pk Here k is an arbitrary algebraically closed field of characteristic 0. Yasinsky also proves that the constant equals 120 when k is the field of real or rational numbers. To explain what Yasinsky’s result really means, let us recall Definition. A group G is said to be Jordan if there exists a positive integer m such that every finite subgroup H ⊂ G contains a normal abelian subgroup A/H of index at most m. The minimal such m is called the the Jordan constant of the group G. Informally, this means that all finite subgroups of G are “almost” abelian. The name Jordan comes from the classical theorem of Camille Jordan: the group GL ( ) is Jordan for every n. The group Bir( 2) is known to be Jordan. n k Pk This was proven by Serre. In fact, very recent results by Birkar, Prokhorov and Shramov imply that the group Bir( n) is Jordan for all n. Pk We should not really close this foreword without including a story about the family of the owner of the restaurant where the conference dinner was served. We happened to have a conference dinner at a Korean–Russian restaurant in Magadan. The restaurant was run by a Korean-Russian couple. To be precise, they are called Koryo-saram ( Kor saram) in Russian. This Russian phrase is borrowed from Korean and it literally means a person from Koryo (Korea). Their ancestors are Koreans who settled in the Maritime Province (Primorsky Krai in Russian) in the 19th century when the borders were neither clear nor under full control. In 1937 the Soviet government forcibly relocated approximately 170,000 Koreans from the Maritime Province to Kazakhstan and Uzbekistan, 6000 km away, for political reasons. The Korean deportees faced many hardships in Central Asia. Since most of them were rice farmers and fish- ers, they had difficulty in adapting to the arid climate of the new settlement. It was reported that about 40,000 deported Koreans died between 1937 and 1938 for this reason. However they overcame these hardships and recovered their original standard of living after a short period. Nowadays about a half million Koryo-saram reside in Russia and the former Soviet Union countries. The grandparents and the parents of the owner of the restaurant endured and overcame their lives of hardship. The husband’s grandfather was originally from

viii F. BOGOMOLOV, I. CHELTSOV, T. KOZLOVSKAYA, AND J. PARK

Seoul. His family migrated to the Maritime Province in the late 19th century and was deported to Kazakhstan. After a one-century odyssey, eventually the family of the owner of the restaurant settled down in Magadan, far above the Maritime Province. To our surprise, 400 or so Koryo-saram live in Magadan, according to the owner.

On our visit to the Mask of Sorrow at night, this huge monument of com- memoration reminded us of a poem of Brecht: Hörteich diese Freunde von mir sagen: “Die Stärkeren überleben.” Und ich haßte mich. We sincerely thank those in the past who sacrificed themselves so that we in the present have the chance to be who we are.

Fedor Bogomolov Courant Institute of Mathematical Sciences New York University 251 Mercer Street, New York, NY 10012, USA and Laboratory of Algebraic Geometry National Research University Higher School of Economics 6 Usacheva street, Moscow, 119048, Russia E-mail address: [email protected]

Ivan Cheltsov School of Mathematics The University of Edinburgh James Clerk Maxwell Building The King’s Buildings Mayfield Road, Edinburgh EH9 3JZ, UK and Laboratory of Algebraic Geometry National Research University Higher School of Economics 6 Usacheva street, Moscow, 119048, Russia E-mail address: [email protected]

Tatiana Kozlovskaya Magadan Institute of Economy of Saint Petersburg University of Management Technologies and Economics 27 Novaya street Magadan, 685000, Russia E-mail address: konus [email protected]

Jihun Park Center for Geometry and Physics Institute for Basic Science (IBS) 77 Cheongam-ro, Nam-gu, Pohang, Gyeongbuk, 37673, Korea and Department of Mathematics POSTECH 77 Cheongam-ro, Nam-gu, Pohang, Gyeongbuk, 37673, Korea E-mail address: [email protected]

ix Magadan Conference December 6∼12, 2015