2019年度年会(於:東京工業大学)

SUMMARY: The finite-dimensional representations of quantum affine algebras have been extensively studied for last three decades originally in connection with the investigation of solutions of the quantum Yang–Baxter equation with spectral parameters. However they have intricate structures and many basic questions are still open. For example, there exists a notion of “character”, called \(q\)-character, but the character formulae for irreducible representations are not known in general.

A quantum affine algebra is specified by its Dynkin type. Recently, non-trivial connections among the representation categories of quantum affine algebras of several different Dynkin types (e.g. \(\mathrm {A}_{2n-1}^{(1)}\) and \(\mathrm {B}_n^{(1)}\), \(\mathrm {D}_{n+1}^{(1)}\) and \(\mathrm {C}_n^{(1)}\)) have been recognized though there are no known explicit algebraic relations on the level of quantum affine algebras themselves. In this talk, I first explain recent developments on the study of such connections. Next, I talk about our results related to this topic, that is, I present ring isomorphisms between “\(t\)-deformed” Grothendieck rings (=quantum Grothendieck rings) associated with the representation categories of quantum affine algebras of type \(\mathrm {A}_{2n-1}^{(1)}\) and \(\mathrm {B}_n^{(1)}\). These isomorphisms imply several new positivity properties of \(t\)-deformed \(q\)-characters of irreducible representations of type \(\mathrm {B}_n^{(1)}\). Moreover, they specialize at \(t = 1\) to the isomorphisms between usual Grothendieck rings which is obtained by Kashiwara, Kim and Oh through other methods. This coincidence gives the affirmative answer to Hernandez’s conjecture in 2002 for type \(\mathrm {B}_n^{(1)}\), which asserts the existence of algorithm to compute the \(q\)-characters of irreducible representations. This talk is based on a joint work with David Hernandez.

SUMMARY: We consider the lattice \(\mathrm {tors} A\) of torsion classes on a finite dimensional algebra. While this lattice is usually infinite, we show that it can still be well understood by studying its Hasse quiver. Moreover, we give some interpretation this Hasse quiver in terms of \(A\)-modules that permits to study algebraic quotients of \(\mathrm {tors} A\), that is quotients of the form \(\mathrm {tors} A \twoheadrightarrow \mathrm {tors} (A/I), \mathcal {T} \mapsto \mathcal {T} \cap \mathrm {mod} (A/I)\) for an ideal \(I\) of \(A\).

As the Hasse quiver of \(\mathrm {tors} A\) contains naturally the exchange graph of support \(\tau \)-tilting modules (as the subset consisting of functorially finite torsion classes), \(\mathrm {tors} A\) can be viewed as a way to extend mutations, even though the behavior at non-functorially finite torsion classes changes drastically, as we will see on several examples, coming from some join work with Aaron Chan.

SUMMARY: \(F\)-singularities are a generic term used to refer to singularities in positive characteristic defined via the Frobenius map. They are conjectured to correspond, via reduction modulo \(p >0\), to singularities in complex birational geometry. I will survey recent developments around this conjecture. In addition, I will explain an application of \(F\)-singularities to birational geometry in positive characteristic. I will also mention some vanishing results on local cohomology to emphasize the different behavior of singularities in characteristic zero and in positive characteristic.

SUMMARY: In arXiv:1002.5047, Feigin and Tipunin introduced \(ADE\) type generalization of triplet \(W\)-algebra by using geometric method. We checked that this VOA has other expected realizations: as intersection of kernels of the narrow screening operators and as some kind of module extension of the corresponding principal \(W\)-algebra. Moreover, we determined the strong generators and proved that the extended part of them are nilpotent in the \(C_2\)-algebra.

SUMMARY: A construction of APN functions using the bent function \(B(x,y)=xy\) is proposed by C. Carlet in 2011. At this time, two families of APN functions using this construction are known, that is, the family of C. Carlet (2011) and the family of Y. Zhou and A. Pott (2013). We propose another family of APN functions with this construction, which are not CCZ equivalent to the former two families on \({\Bbb F}_{2^8}\). We also propose a family of presemifields and determined the middle, left and right nuclei of the associated semifields.

SUMMARY: A new proof of the Aztec diamond theorem is given. The proof is based on a difference equation to which any solution induces a generating or partition function for (domino-)tilings of the Aztec diamonds and a product expression for it. In particular a specific solution is shown which proves the Aztec diamond theorem by Stanley on a multivariate generating function with a nice product expression.

SUMMARY: In this talk, the \(\mathfrak {q}\)-crystal structure of signed unimodal factorizations of reduced words of type \(B\) and that of signed unimodal factorizations of flattened words of type \(D\) are discussed. The relation between signed unimodal factorizations of reduced (flattened) words of type \(B\) (resp. \(D\)) and the type \(B\) (resp. \(D\)) Coxeter–Knuth relation are clarified. The explicit algorithm for odd Kashiwara operators on signed unimodal factorizations of reduced words of type \(B\) is given. This algorithm is also applicable on signed unimodal factorizations of flattened words of type \(D\) without any alterations.

SUMMARY: We give a provisional construction of the Kac–Moody Lie algebra module structure on the hyperbolic restriction of the intersection cohomology complex of the Coulomb branch of a framed quiver gauge theory, as a refinement of the conjectural geometric Satake correspondence for Kac–Moody algebras proposed in an earlier paper with Braverman and Finkelberg. This construction assumes several geometric properties of the Coulomb branch under the torus action. These properties are checked in affine type A, via the identification of the Coulomb branch with a Cherkis bow variety established in a joint work with Takayama.

SUMMARY: Let \(A\) be a symmetric order over a complete discrete valuation ring \(\mathcal {O}\) and \(\kappa \) the residue field of \(\mathcal {O}\). Heller lattices over \(A\) are \(A\)-lattices defined as direct summands of the kernel of the projective cover of indecomposable \(A\otimes \kappa \)-modules as \(A\)-modules.

In \(p\)-modular representation theory, Kawara showed that Heller lattices over group algebras play important roles. Thus, it is natural to study Heller lattices over an arbitrary symmetric order. In this talk, we study Heller lattices over \(\mathcal {O}[x,y]/(x^2,y^2)\). As the main result, we see that the tree classes of stable components containing Heller lattices are \(A_{\infty }\).

SUMMARY: Let \(\Lambda \) be an arbitrary finite dimensional algebra and \(\operatorname {mod} \Lambda \) be the category of finitely generated \(\Lambda \)-modules. In this talk, we show that wide subcategories of \(\operatorname {mod} \Lambda \) associated with \(\tau \)-rigid pairs are semistable. This provides a complement of Ingalls–Thomas-type bijections for finite dimensional algebras.

SUMMARY: In this talk, I deal with the finite-dimensional mesh algebras given by stable translation quivers, which are self-injective. The stable module categories have a structure of triangulated categories coming from syzygies. In order to classify the mesh algebras by stable equivalences, I have determined the Grothendieck groups of the stable module categories as invariants. Combining this result with other invariants, I have proved that there are no non-trivial stable equivalences between such mesh algebras.

SUMMARY: Let \((X,L)\) be a polarized manifold of dimension \(n\). In this talk, we consider the dimension of global sections of adjoint bundle \(K_{X}+mL\). In particular, we study the case where \(n=5\), \(m\geq n+1=6\) and \(h^{0}(L)>0\).

SUMMARY: We answer the following question posed by Arturo Pianzola: describe all twisted forms of the differential Lie algebra \(\mathfrak {sl}_{n}(\mathbb {C}(X))\). Here \(\mathfrak {sl}_{n}(\mathbb {C}(X))\) is given the entry-wise differentiation.

SUMMARY: It was proved by Masuoka and Zubkov that given an affine algebraic supergroup \(G\) and closed sub-supergroup \(H\) over an arbitrary field of characteristic \(\neq 2\), the faisceau \(G \tilde {/} H\) (in the fppf topology) is a superscheme, and is, therefore, the quotient superscheme \(G/H\), which has desirable properties, in fact. We reprove this, by constructing directly the latter superscheme \(G/H\). Our proof describes explicitly the structure sheaf of \(G/H\), and reveals some geometric features of the quotient.

SUMMARY: Let \(k\) be an algebraically closed field of characteristic \(p>0\). Let \(G={\rm SL}_2\) be the special linear group of degree 2 over \(k\) and \(G_r\) the \(r\)-th Frobenius kernel of \(G\). In 1983, Wong gave some generating sets of the Jacobson radical of the hyperalgebra \(\mathcal {U}_r = {\rm Dist}(G_r)\) of \(G_r\) for \(r=1\). Here we report that this result can be generalized to the case for general \(r\), using primitive idempotents of the hyperalgebra \(\mathcal {U}_r\) constructed by the speaker before.

SUMMARY: Let \(B\) be a ring with identity, \(\rho \) an automorphism of \(B\), \(D\) a \(\rho \)-derivation, and \(q\) a central (\(\rho \), \(D\))-constant element in \(B\). By \(B[X;\rho , D]^q\) we denote a \(q\)-skew polynomial ring in which the multiplication is given by \(\alpha X = X\rho (\alpha ) + D(\alpha )\) (\(\forall \alpha \in B\)). In this talk, we shall study a weakly separable polynomial \(f\) in \(B[X;\rho ,D]^q\) of the form \(f=X^m+X^{m-1}a_{m-1} + \cdots Xa_1 +a_0\) (\(m \geq 2\), \(a_i \in B\) (\(0 \leq i \leq m-1\))) , and we shall give a necessary and sufficient condition for a weakly separable polynomial \(f\). In addition, we shall show the difference between the separability and the weak separability in \(B[X;\rho ,D]^q\) under certain conditions.

SUMMARY: For an artin algebra, Brenner showed that how to determine the number of indecomposable direct summands of the middle term of an almost split sequence starting with a simple module. Let \(K\) be an algebraically closed field and \(A=K\Delta _A/I\) a truncated quiver algebra. For a Hochschild extension algebra of \(A\). We give a simple interpretation of a theorem of Brenner by focusing on the number of nonzero cycles in the Hochschild extension algebra.

SUMMARY: It is known that a \(3\)-dimensional quadratic AS-regular algebra is a geometric algebra, however, the converse is not true. In this talk, we give a necessary and sufficient condition that a geometric algebra associated to an elliptic curve \(E\) in \(\mathbb {P}_{k}^{2}\) is a \(3\)-dimensional quadratic AS-regular algebra. Moreover, we show that every \(3\)-dimensional quadratic AS-regular algebra corresponding to an elliptic curve \(E\) in \(\mathbb {P}_{k}^{2}\) is graded Morita equivalent to a \(3\)-dimensional Sklyanin algebra.

SUMMARY: We focus on the structure of the stable category \(\mathsf {\underline {CM}}^{\mathbb Z}(S/(f))\) of graded maximal Cohen–Macaulay module over \(S/(f)\) where \(S\) is a graded (\(\pm 1\))-skew polynomial algebra in \(n\) variables of degree 1, and \(f =x_1^2 + \cdots +x_n^2\). If \(S\) is commutative, then the structure of \(\mathsf {\underline {CM}}^{\mathbb Z}(S/(f))\) is well-known by Knörrer’s periodicity theorem. It will be explained that if \(n\leq 5\), then the structure of \(\mathsf {\underline {CM}}^{\mathbb Z}(S/(f))\) is determined by the number of irreducible components of the point scheme of \(S\) which are isomorphic to \({\mathbb P}^1\).

SUMMARY: We mainly consider Cohen–Macaulay local rings of dimension one. The most typical example of a finite birational extension of them is the endomorphism ring of the maximal ideal. Such a extension have been used to understand representation-theoretic properties of rings. For example, Bass used it to study indecomposable torsion-free modules. The aim of this talk is to study the endomorphism rings of the maximal ideal of Gorenstein local rings. We will give some characterizations of them, and show the relation between self-duality of the maximal ideal and some other properties (Teter’s condition, almost Gorensteiness, etc).

SUMMARY: The Sally module of an ideal is an important tool to interplay between Hilbert coefficients and the properties of the associated graded ring. In this talk we give new insights on the structure of the Sally module. We apply these results characterizing the almost minimal value of the first normal Hilbert coefficient in an analytically unramified Cohen–Macaulay local ring.

SUMMARY: Let \(G\) be a finite simple graph on the vertex set \(V\). Also let \(S=K[V]\) be a polynomial ring over a field \(K\) whose variables are vertices of \(G\). Then we define the edge ideal \(I(G) \subset S\) of \(G\). In the talk, we will show that given integers \(r\) and \(b\) with \(1 \leq b \leq r\), there exists a finite simple connected graph \(G\) such that the regularity of \(S/I(G)\) is equal to \(r\) and the number of extremal Betti numbers of \(S/I(G)\) is equal to \(b\).

SUMMARY: Normal lattice polytopes turn up in many fields of mathematics. It is known that if the Cayley sum of lattice polytopes is normal, then so is their Minkowski sum. In this talk, the Cayley sum of the order polytope of a finite poset and the stable set polytope of a finite simple graph is discussed. We show that the Cayley sum of an order polytope and the stable set polytope of a perfect graph is normal, and hence so is their Minkowski sum. Moreover it turns out that, for an order polytope and the stable set polytope of a graph, the following conditions are equivalent: (i) the Cayley sum is Gorenstein; (ii) the Minkowski sum is Gorenstein; (iii) the graph is perfect.

SUMMARY: Let \(I \subset S= K[x_1, \ldots , x_n]\) be a strongly stable ideal whose generators have degree at most \(d\). It is known that \(I\) admits the alternative polarization b-pol\((I) \subset K[x_{i,j} \mid 1 \le i \le n, 1 \le j \le d]\). This is a very useful tool in the study of strongly stable ideals. We give an easy procedure to construct the irreducible decomposition of \(\operatorname {b-pol}(I)\) from that of \(I\). Furthermore, we describe the Hilbert series of \(H_{\mathfrak m}^i(S/I)\) from the irreducible decomposition of \(I\) via b-pol\((I)\) and the Eliahou–Kervaire formula.

SUMMARY: We give an equivalent condition for a self-dual weight enumerator of genus three to satisfy the Riemann hypothesis. We also observe the truth and falsehood of the Riemann hypothesis for a certain family of invariant polynomials.

SUMMARY: We report a bijection from \(\mathbb {N}^m\) to \(\mathbb {N}\) represented by a polynomial.

SUMMARY: Let \(P\) denote a prime and \(m\) an integer. Positive integers \(a\) and \(A\) are said to be a super perfect number and its partner, if they satisfy \(A=\sigma (a)+m ,\overline {P} \sigma (A)= aP+ P-2+m \overline {P},\) where \(\overline {P}=P-1\) and let \(\sigma (n)\) denote a sum of divisors of \(n\).

Assume that \(P=3,m=-8\). If \(a\) is a prime \(p\) , \(A\) turns out to be \(2q\), \(q\) being a prime. Then both \((q ,p=2q+7) \) are called super twin primes.

Let \(P\) denote a prime and \(m\) an integer. Positive integers \(a\) and \(A\) are said to be a super perfect number and its partner.

SUMMARY: We study the number of integer solutions \((x,y,m)\) of an equation \(f(x,y)=\Pi _K(n)\), where \(f(x,y)\) is a quadratic form with integer coefficients and \(\Pi _K(n)\) is a generalized factorial function over number fields. We show a necessary and sufficient condition for the existence of infinitely many solutions.

SUMMARY: A Heron triangle is a triangle having the property that the lengths of its sides as well as its area are positive integers. In this talk, we show that the exponential Diophantine equation \(~c^{x}+b^{y}=a^{z}~\) concerning Heron triples \(a\),\(~b\),\(~c\) has only the positive integer solution \((x,y,z)=(1,1,2)\) under some conditions. The proof is based on elementary methods and Baker’s method.

SUMMARY: Let \(Z_N(X)\) be the zeta polynomial of the \(N\)th cyclotomic function field of characteristic \(p\). In this talk, we generalize Goss–Bernoulli polynomials, and characterize irreducible components of \(Z_N(X)\) mod \(p\). As an application of our result, for given \(f(X) \in \mathbb {F}_p[X]\), we see that there are infinitely many irreducible polynomial \(N\) such that \(Z_{N}(X)\) mod \(p\) is divided by \(f(X)\).

SUMMARY: Height-one duality is the relations among finite multiple zeta values (FMZVs) derived from the hoffman duality and the reversal relation. Kaneko and Ohno proved an analogue of the height-one duality for multiple zeta-star values and conjectured a kind of duality of multiple zeta-star values for arbitrary heights. This conjecture was proved by Li. On the other hand, (for FMZVs,) Kaneko conjectured a generalization of the height-one duality for arbitrary heights. Moreover, based on the conjecture due to Kaneko and Zagier, it is expected that Kaneko’s conjecture holds also for symmetric multiple zeta values (SMZVs). In this talk, we prove the conjectures for both FMZVs and SMZVs.

SUMMARY: Matsumoto proved that Euler–Zagier double zeta function satisfies a functional equation including confluent hypergeometric functions. After that, Okamoto and Onozuka obtained same type functional equation for Mordell–Tornheim multiple zeta functions, which generalized Matsumoto’s result. On the other hand, we introduced a combinatorial object called 2-colored rooted tree and a multiple zeta function associated with it, which is a common generalization of Euler–Zagier and Mordell–Tornheim multiple zeta functions. In this talk, we will explain that multiple zeta functions associated with certain 2-colored rooted trees satisfy a same type functional equations. This result gives a generalization of Okamoto–Onozuka’s result.

SUMMARY: Cusp forms for the full modular group can be written as linear combination of the Eisenstein series and the double Eisenstein series introduced by Gangl, Kaneko and Zagier. We give an explicit formula for decomposing a Hecke eigenform into double Eisenstein series.

SUMMARY: In this talk, we compute special values of \(L\)-functions of modular forms which are products of the Jacobi theta series or the Borwein theta series. We express \(L\)-values at \(s=1\) of weight 3 theta products in terms of special values of generalized hypergeometric functions.

SUMMARY: We derive an analogue to the Poisson summation formula, in terms of the Laplace transform. This allows us to deduce the functional equation, approximate functional equation, and a fast and absolutely convergent algorithm for the Riemann zeta function, Dirichlet L-functions and the Lerch zeta function, within a unified framework.

SUMMARY: V. Schemmel in 1869 introduced an arithmetic function \(\varphi _m(n)\) which generalizes the Euler’s totient function by introducing a positive integer coefficient \(m\) in the prime factor of the product representation of Euler’s totient function. The Euler’s totient function is a special case when \(m=1\). We extended this definition of \(\varphi _m(n)\) to all integers \(m\) and considered its mean-values with respect to both \(m\) and \(n\).

SUMMARY: We consider zeta functions defined for a family of finite sets. This class of zeta functions includes the Ihara zeta function or other graph zeta functions. In this talk, the conditions which rewrite the exponential expression to the Euler product expression for those zetas.

SUMMARY: We consider zeta functions defined for a family of finite sets. This class of zeta functions includes the Ihara zeta function or other graph zeta functions. In this talk, the conditions which rewrite the Euler product expression to the Hashimoto expression for those zetas.

SUMMARY: The combinatorial zeta function is the zeta function defined for a combinatorial structure, such as a finite graph, discrete dynamical system, a finite group and so on, which has the three wxpressions. We will see that the circulatory of the weight deduces the existence of the three expressions.

SUMMARY: The Ihara expression is an expression of graph zeta functions. Sato obtained the Ihara expression of the second weighted zeta function, which relates to quantum walks via Konno–Sato’s theorem. In this talk, we define the “generalized Sato zeta function” which extends the second weighted zeta function, and we derive its Ihara expression.

SUMMARY: The Nakano–Nishijima–Gell-Mann formula (NNG fomula) is well known as an equation that relates certain quantum numbers of elementary particles to their charge number. This theory is constructed by using the group theory with real number, and introduces the quantum numbers \(I_z\) (isospin), \(S\) (strangeness), etc. phenomenologically. But according to a previous suggestion, in the finite world the relation between quantum numbers and charge numbers is represented by a discrete gauge transformation in a finite field. We rewrite this representation instead of using the NNG formula, and predict relation of quantum numbers of Hadron including Pentaquark. Furthermore, we get discreteness of charge as characteristic of finite field.

SUMMARY: Let \(A_6\) be the alternating group of degree \(6\). We give a negative answer to Noether’s problem for \(N\rtimes A_6\) over \(\mathbb {C}\) where \(N\) is some abelian group.

SUMMARY: We give a stably and retract rational classification of norm one tori of dimension \(p-1\) where \(p\) is a prime number and of dimension up to ten with some minor exceptions.

SUMMARY: We give a stably and retract rational classification of norm one tori of dimension \(n-1\) for \(n=2^e\) \((e\geq 1)\) is a power of \(2\) and \(n=12, 14, 15\). Retract non-rationality of norm one tori for primitive \(G\leq S_{2p}\) where \(p\) is a prime number and for the five Mathieu groups \(M_n\leq S_n\) \((n=11,12,22,23,24)\) is also given.

SUMMARY: In Costa’s paper published in 1977, he asks us whether every retract of \(k^{[n]}\) is also the polynomial ring or not, where \(k\) is a field. In this talk, we give an affirmative answer in the case where \(k\) is a field of characteristic zero and \(n = 3\).

SUMMARY: We improve our previous results on homological shells. Let \(X\) be an arithmetically \(D_2\) closed subscheme of \(P^N(C)\), and \(W\) a homological shell of \(X\). We construct the universal family of homological shells of \(X\) which includes \(W\). We describe the Zariski tangent space at the point \([W]\), the smoothness condition at \([W]\), the differential of the (universal) Koszul graph map at \([W]\) by using cohomological pairing.

SUMMARY: Let \(\mathbf D\) be a triangulated category of coherent sheaves on a smooth projective variety. Then the category \(\mathbf D^{\Delta ^1}\) of morphisms in \(\mathbf D\) is also triangulated. Hence one can assign the space \(\mathrm {Stab} \mathbf D^{\Delta ^1}\) of stability conditions on \(\mathbf D^{\Delta ^1}\) though the non-emptiness of it is not obvious. The aim of this talk is a comparison of the spaces of stability conditions on \(\mathbf D^{\Delta ^1}\) and that on \(\mathbf D\) after the proof of non-emptiness of \(\mathrm {Stab} \mathbf D^{\Delta ^1}\). In particular we discuss the case that \(\mathbf D\) is the derived category of a smooth projective curve.

SUMMARY: We consider the Hilbert scheme \(H\) which parameterizes all closed subschemes of \(\mathbb {P}^n\) with fixed Hilbert polynomial \(P\). If we fix a monomial order \(\prec \) on the polynomial ring \(S\) in \(n+1\) variables, each homogeneous ideal in \(S\) has a unique reduced Gröbner basis with respect to \(\prec \). Using this fact we can decompose the Hilbert scheme \(H\) into the locally closed subschemes of \(H\) called Gröbner schemes. On the other hand, Bialynicki–Birula shows that any smooth projective scheme with a 1-dimensional torus action has a cell decomposition called Bialynicki–Birula decomposition. In this talk, I would like to explain Gröbner schemes and compare two decompositions of the Hilbert scheme \(H\).

SUMMARY: We study the finite F-representation type (abbr. FFRT) property of a two-dimensional normal graded ring R in characteristic p\(>\)0, using notions from the theory of algebraic stacks. Given a graded ring R, we consider an orbifold curve, which is a root stack over the smooth curve C=Proj R, such that R is the section ring associated with a line bundle L on C. The FFRT property of R is then rephrased with respect to the Frobenius push-forwards on the orbifold curve. As a result, we see that if the singularity of R is not log terminal, then R has FFRT only in exceptional cases where the characteristic p divides a weight of the orbifold curve.

SUMMARY: In this talk, we introduce some recent study of the embedded topology of smooth quartics and its bitangent lines via two-graphs and apply it to construct interesting examples for Zariski \(m\)-ples.

SUMMARY: A curve is said to be superspecial if its Jacobian is isomorphic to a product of supersingular elliptic curves. In recent years, the speakers succeeded in enumerating superspecial curves of genus four in small characteristic. This study is the first attempt to obtain an analogous result for genus five. We propose a feasible algorithm to enumerate superspecial curves in the case of trigonal ones of genus five over an arbitrary finite field. We implemented the algorithm over a computer algebra system Magma, and succeeded in enumerating superspecial trigonal curves of genus five over small finite fields.

SUMMARY: I have used the result of Beilinson–Drinfeld some time ago. However, its result was restricted to the chapter 3 of local theory. I would like to continue my trial in the chapter 4 of the global theory and conformal blocks. It is by use of chiral homology of twisted D-modules and the cobordism conjecture of Jacob Lurie. I would like to start from reviewing my computation results of OPEs. Then I will consider the Matsushima obstruction and spin structure by Stiefel–Whitney class and signature. Then I would like to consider the hypothetical \(L_{\infty }\)-algebra structure of string field theory. Noncommutative deformation of Poisson bracket and noncommutative geometry of chiral algebra are the key to define such structures.

SUMMARY: I shall explain my recent study on surfaces with \(c^2 = 9\) and \(\chi =5\) whose canonical classes are divisible by \(3\) in the integral cohomology group, where \(c_1^2\) and \(\chi \) denote the first Chern number of an algebraic surface and the Euler characteristic of the structure sheaf, respectively. The main results are a structure theorem for such surfaces, the unirationality of the moduli space, and a description of the behavior of the canonical map. As a byproduct, we can rule out a certain case mentioned in a paper by Ciliberto–Francia–Mendes Lopes.

SUMMARY: In this talk, we calculate the boundary of movable cones and nef cones of the generalized Kummer 4-fold attached to an abelian surface with Picard number 1.

SUMMARY: We discuss the Doran–Harder–Thompson conjecture, which claims that when a Calabi–Yau manifold \(X\) degenerates to a union of two quasi-Fano manifolds (Tyurin degeneration), a mirror Calabi–Yau manifold of \(X\) can be constructed by gluing the two mirror Landau–Ginzburg models of the quasi-Fano manifolds. We provide a sketch of a proof in the case of elliptic curves and abelian surfaces.

SUMMARY: For a 3-dimensional extremal curve neighborhood (or, extremal curve germ) \((X,C)\) with an extremal curve \(C\) which is not necessary irreducible or reducible, we formulate numerical invariants associated to \(\mathrm {gr}^{n,i}(\mathscr {O},J)\) (S. Mori, 1988) along properties about the normal bundle for such a \((X,C)\) (A. G. Kuznetsov, Y. G. Prokhorov, C. A. Shramov, 2018). In such a process, we use our previous results (the Math. Soc. Japan meeting (Sep. 2018)) on which a line bundle induced from \(\omega _X^{\vee }\) on \(C\) of type (IIA) as a \(l\)-split direct summand is given from the moduli space of certain semi-stable sheaves by a coherent system and Trautmann’s moduli (Le Potier, 1993). As a result, we give a kind of LG-deformation (S. Mori, 1988) for Mukai–Umemura 3-folds,and give an inequality between the associated Chern classes \(c_i\) (\(i \in [1,3]\)) of Miyaoka–Yau type for such a \((X,C)\).