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Given a compact K\"ahler manifold $X$ it is interesting to ask whether it admits a constant scalar curvature K\"ahler (cscK) metric. In this short note we show that there always exist cscK metrics on compact K\"ahler manifolds with nef canonical bundle, thus on all smooth minimal models, and also on the blowup of any such manifold. This confirms an expectation of Jian-Shi-Song \cite{JianShiSong} and extends their main result from $K_X$ semi-ample to $K_X$ nef, with a direct proof that does not appeal to the Abundance conjecture. As a byproduct we obtain that the connected component $\mathrm{Aut}_0(X)$ of a compact K\"ahler manifold with $K_X$ nef is either trivial or a complex torus., Comment: 9 pages, new result added (Corollary 2). Accepted in Annali Della Scuola Normale Superiore Di Pisa - Classe Di Scienze

We classify all minimal models X of dimension n, Kodaira dimension n - 1 {n-1} and with vanishing Chern number c 1 n - 2 ⁢ c 2 ⁢ ( X ) = 0 {c_{1}^{n-2}c_{2}(X)=0} . This solves a problem of Kollár. Completing previous work of Kollár and Grassi, we also show that there is a universal constant ϵ > 0 {\epsilon>0} such that any minimal threefold satisfies either c 1 ⁢ c 2 = 0 {c_{1}c_{2}=0} or - c 1 ⁢ c 2 > ϵ {-c_{1}c_{2}>\epsilon} . This settles completely a conjecture of Kollár.

There are two parallelohedra in dimension 2 and five in dimension 3. We study smallest polyhedral quotients of the tilings defined by them on tori. For 2-tori the situation is well understood, and the three minimal models are well known. Here we investigate the five 3-dimensional cases with the result: Each of the five parallelohedra admits a minimal (and weakly neighborly) model, moreover in three cases this is unique up to natural equivalence. Altogether there are 14 inequivalent minimal models.

For a map $$f:X\rightarrow Y$$ f : X → Y , there is the relative model $$M(Y)=(\Lambda V,d)\rightarrow (\Lambda V\otimes \Lambda W,D)\simeq M(X)$$ M ( Y ) = ( Λ V , d ) → ( Λ V ⊗ Λ W , D ) ≃ M ( X ) by Sullivan model theory (Félix et al., Rational homotopy theory, graduate texts in mathematics, Springer, Berlin, 2007). Let $$\mathrm{Baut}_1X$$ Baut 1 X be the Dold–Lashof classifying space of orientable fibrations with fiber X (Dold and Lashof, Ill J Math 3:285–305, 1959]). Its DGL (differential graded Lie algebra)-model is given by the derivations $$\mathrm{Der}M(X)$$ Der M ( X ) of the Sullivan minimal model M(X) of X. Then we consider the condition that the restriction $$b_f:\mathrm{Der} (\Lambda V\otimes \Lambda W,D)\rightarrow \mathrm{Der}(\Lambda V,d) $$ b f : Der ( Λ V ⊗ Λ W , D ) → Der ( Λ V , d ) is a DGL-map and the related topics.

The minimal model $$ \mathfrak{osp}\left(1|2\right) $$ osp 1 2 vertex operator superalgebras are the simple quotients of affine vertex operator superalgebras constructed from the affine Lie super algebra $$ \hat{\mathfrak{osp}}\left(1\left|2\right.\right) $$ osp ̂ 1 2 at certain rational values of the level k. We classify all isomorphism classes of ℤ2-graded simple relaxed highest weight modules over the minimal model $$ \mathfrak{osp}\left(1|2\right) $$ osp 1 2 vertex operator superalgebras in both the Neveu–Schwarz and Ramond sectors. To this end, we combine free field realisations, screening operators and the theory of symmetric functions in the Jack basis to compute explicit presentations for the Zhu algebras in both the Neveu–Schwarz and Ramond sectors. Two different free field realisations are used depending on the level. For k < −1, the free field realisation resembles the Wakimoto free field realisation of affine $$ \mathfrak{sl}(2) $$ sl 2 and is originally due to Bershadsky and Ooguri. It involves 1 free boson (or rank 1 Heisenberg vertex algebra), one βγ bosonic ghost system and one bc fermionic ghost system. For k > −1, the argument presented here requires the bosonisation of the βγ system by embedding it into an indefinite rank 2 lattice vertex algebra.

We show that the number of marked minimal models of an n-dimensional smooth complex projective variety of general type can be bounded in terms of its volume, and, if n=3, also in terms of its Betti numbers. For an n-dimensional projective klt pair (X,D) with $K_X+D$ big, we show more generally that the number of its weak log canonical models can be bounded in terms of the coefficients of D and the volume of $K_X+D$. We further show that all n-dimensional projective klt pairs (X,D), such that $K_X+D$ is big and nef of fixed volume and such that the coefficients of D are contained in a given DCC set, form a bounded family. It follows that in any dimension, minimal models of general type and bounded volume form a bounded family., 27 pages; final version; to appear in Annales scientifiques de l'ENS

In this dissertation I prove a number of results about the symplectic geometry of finite dimensional integrable Hamiltonian systems, especially those of semitoric type. Integrable systems are, roughly, dynamical systems with the maximal amount of conserved quantities. Though the study of integrable systems goes back hundreds of years, the earliest general result in this field is the action-angle theorem of Arnold in 1963, which was later extended to a global version by Duistermaat. The results of Atiyah, Guillemin-Sternberg, and Delzant in the 1980s classified toric integrable systems, which are those produced by effective Hamiltonian torus actions. Recently, Pelayo-Vu Ngoc classified semitoric integrable systems, which generalize toric systems in dimension four, in terms of five symplectic invariants. Using this classification, I construct a metric on the space of semitoric integrable systems. To study continuous paths in this space produced via symplectic semitoric blowups, I introduce an algebraic technique to study such systems by lifting matrix equations from the special linear group SL(2,Z) to its preimage in the universal cover of SL(2,R). With this method I determine the connected components of the space of semitoric integrable systems. Motivated by this algebraic technique, I introduce the notion of a semitoric helix; the natural combinatorial invariant of semitoric systems. By applying a refined version of the algebraic method to semitoric helixes I classify all possible minimal semitoric integrable systems, which are those that do not admit a symplectic semitoric blowdown. I also produce invariants of integrable systems designed to respect the natural symmetries of such systems, especially toric and semitoric ones. For any Lie group G, I construct a G-equivariant analogue of the Ekeland-Hofer symplectic capacities. I give examples when the capacity is an invariant of integrable systems, and I study the continuity of these capacities using the metric I defined on semitoric systems. Finally, as a first step towards constructing a meaningful metric on general integrable systems, I provide a framework to study convergence properties of families of maps between manifolds which have distinct domains by defining a metric on such a collection.

In this paper, we explicitly classify the minimal discriminants of all elliptic curves $E/\mathbb{Q}$ with a non-trivial torsion subgroup. This is done by considering various parameterized families of elliptic curves with the property that they parameterize all elliptic curves $E/\mathbb{Q}$ with a non-trivial torsion point. We follow this by giving admissible change of variables, which give a global minimal model for $E$. We also provide necessary and sufficient conditions on the parameters of these families to determine the primes at which $E$ has additive reduction. In addition, we use these parameterized families to give new proofs of results due to Frey and Flexor-Oesterl\'{e} pertaining to the primes at which an elliptic curve over a number field $K$ with a non-trivial $K$-torsion point can have additive reduction., Comment: 34 pages; incorporates suggestions by referees; results in section 3 have been strengthened; final version to appear in Research in Number Theory

International audience; We introduce an improved Gröbner basis completion algorithm for operads. To this end, we define operadic rewriting systems as a machinery to rewrite in operads, whose rewriting rules do not necessarily depend on an ambient monomial order. A Gröbner basis of an operadic ideal can be seen as a confluent and terminating operadic rewriting system; thus, the completion of a Gröbner basis is equivalent to the completion of a rewriting system. We improve the completion algorithm by filtering out redundant S-polynomials and testing only essential ones. Finally, we show how the notion of essential S-polynomials can be used to compute Gröbner bases for syzygy bimodules. This work is motivated by the computation of minimal models of associative algebras and symmetric operads. In this direction, we show how our completion algorithm extends to the case of shuffle operads. CCS Concepts • Computing methodologies → Combinatorial algorithms; Algebraic algorithms.

We give an elementary proof of the fact that any elliptic curve $E$ over an algebraically closed non-archimedean field $K$ with residue characteristic $\neq{2,3}$ and with $v(j(E)), Comment: 24 pages, 12 figures. This is a pre-print of an article published in "Arnold Mathematical Journal". The final authenticated version is available online at: https://doi.org/10.1007/s40598-019-00121-y

In this short note, we prove the existence of constant scalar curvature Kähler metrics on compact Kähler manifolds with semi-ample canonical bundles.

In this note, we provide an algorithm that, starting with a Sullivan algebra gives us its minimal model. More concretely, taking as input a (nonminimal) Sullivan algebra A with an ordered finite set of generators preserving the filtration defined on A, we obtain as output a minimal Sullivan algebra with the same rational cohomology as A. This algorithm is a kind of modified AT-model algorithm used, in the past, to compute a chain contraction providing other kinds of topological information such as (co)homology, cup products on cohomology and persistent homology. Ministerio de Ciencia, Innovación y Universidades PID2019-107339GB-100

This paper is a synthesis and extension of three earlier papers on PD4-complexes X such that π = π1(X) has one end and c.d.π = 2. The basic notion is that of strongly minimal PD4-complex, one for which the equivariant intersection pairing λX on π2(X) is null. The first main result is that two PD4-complexes with the same strongly minimal model are homotopy equivalent if and only if their intersection pairings are isometric. If c.d.π ≤ 2 every such complex has a strongly minimal model, and the second half of the paper focuses largely on determining the minimal models. In particular, if π is a surface group or is a semidirect product \( F(r) \rtimes {\mathbb{Z}} \) then the homotopy type of X is determined by π, the Stiefel-Whitney classes and λX. Although we expect that the strategy in the surface group case should extend to all π such that c.d.π = 2 and π has one end, we do not yet have a unified proof that covers the known cases. We conclude with an application to 2-knots and a short list of questions for further research.

Over any algebraically closed field of positive characteristic, we construct examples of fibrations violating subadditivity of Kodaira dimension., 11 pages

This thesis presents an algebraic study of two areas of conformal field theory. The first part extends the theory of Galilean conformal algebras. These algebras are extended conformal symmetry algebras for two-dimensional quantum field theories, formed through a process known as Galilean contraction. Analogous to an Inonu-Wigner contraction, the Galilean contraction procedure takes two conformal symmetry algebras, equivalent up to central charge, as input and produces a new conformal symmetry algebra. We extend the theory of Galilean contractions in several directions. First, we develop the theory to allow input of any number of symmetry algebras. These generalised algebras have a truncated graded structure. We develop a theory for multi-graded Galilean algebras, whereby we can extend structures which are graded by sequences. Finally, we present a comprehensive analysis of the possible algebras which arise when the input algebras are no longer required to be equivalent. We refer to this as the asymmetric Galilean contraction. For each stage of generalisation, we present several pertinent examples, discuss the Sugawara construction of a Galilean Virasoro algebra given an affine Lie algebra, and apply our results to the W-algebra W3.The second part of this thesis presents an exploratory study of reducible but indecomposable modules of the N=2 superconformal algebras, known as staggered modules. These modules are characterised by a non-diagonalisable action of L0, the Virasoro zero-mode. Using recent results on the coset construction of N=2 minimal models, we are able to construct the first examples of staggered modules for the N=2 algebras. We determine the structure of a family of such modules for all admissible values of the central charge. Furthermore, we investigate the action of symmetries such as spectral flow on these modules. We present the results along with a range of examples, and discuss possible paths towards a classification of such modules for the Neveu-Schwarz and Ramond N=2 superconformal algebras.

We show that there is a dual description of conformal blocks of $d=2$ rational CFT in terms of Hecke eigenfields and eigensheaves. In particular, partition functions, conformal characters and lattice theta functions may be reconstructed from the action of Hecke operators. This method can be applied to: 1) rings of integers of Galois number fields equipped with the trace (or anti-trace) form; 2) root lattices of affine Kac-Moody algebras and WZW-models; 3) minimal models of Belavin-Polyakov-Zamolodchikov and related $d=2$ spin-chain/lattice models; 4) vertex algebras of Leech and Niemeier lattices and others. We also use the original Witten's idea to construct the 3-dimensional quantum gravity as the AdS/CFT-dual of $c=24$ Monster vertex algebra of Frenkel-Lepowsky-Meurman. Concerning the geometric Langlands duality, we use results of Beilinson-Drinfeld, Frenkel-Ben-Zvi, Gukov-Kapustin-Witten and many others (cf. references)., Comment: 12 pages

We present an inductive strategy to show the existence of rational curves on compact Kaehler manifolds which are not minimal models but have a pseudoeffective canonical bundle. The tool for this inductive strategy is a weak subadjunction formula for lc centers associated to certain big cohomology classes. This subadjunction formula is based, as in the projective case, on positivity arguments for relative adjoint classes., Comment: 32 pages

Given a finitely presented Graded Commutative Differential Algebra (GCDA), we present a method to compute its minimal model, together with a map that is a quasi-isomorphism up to a given degree. The method works by adding generators one by one. We also provide a specific implementation of the method. We also provide two criteria for i-formality, one necessary and one sufficient., Comment: 12 pages, presented in MEGA 2019

We study the situation of an isolated fixed point with local index zero at a local separating point in a $2$-complex. This fixed point sometimes can be removed and sometimes not, either locally or globally. Criteria are given for local removability in dimension at most two. Results are applied to finding fixed point minimal models on $S^2\vee S^2$ and $S^1 \vee S^2$. A non-existence result is given in the case of a wedge in which both factors are surfaces with non-positive Euler characteristic.

We describe normal forms and minimal models of Picard curves, discussing various arithmetic aspects of these. We determine all so-called special Picard curves over $\mathbb{Q}$ with good reduction outside 2 and 3, and use this to determine the smallest possible conductor a special Picard curve may have. We also collect a database of Picard curves over $\mathbb{Q}$ of small conductor., 34 pages

We describe explicitly the chamber structure of the movable cone for a general smooth complete intersection Calabi-Yau threefold $X$ of Picard number two in certain Pr-ruled Fano manifold and hence verify the Morrison-Kawamata cone conjecture for such $X$. Moreover, all birational minimal models of such Calabi-Yau threefolds are found, whose number is finite up to isomorphism., 42 pages. Correct some typos in this new version. Comments are welcome

In this article, we prove a finiteness result on the number of log minimal models for 3-folds in $\operatorname{char}p>5$. We then use this result to prove a version of Batyrev’s conjecture on the structure of nef cone of curves on 3-folds in characteristic $p>5$. We also give a proof of the same conjecture in full generality in characteristic 0. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor $K_{X}$ of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on $X$.

Batterman and Rice ([2014]) argue that minimal models possess explanatory power that cannot be captured by what they call ‘common features’ approaches to explanation. Minimal models are exp...

Del Pezzo fibrations appear as minimal models of rationally connected varieties. The rationality of smooth del Pezzo fibrations is a well studied question but smooth fibrations are not dense in moduli. Little is known about the rationality of the singular models. We prove birational rigidity, hence non-rationality, of del Pezzo fibrations with simple non-Gorenstein singularities satisfying the famous $K^2$-condition. We then apply this result to study embeddings of $\operatorname{PSL}_2(7)$ into the Cremona group.

Studying the behavior of the Kahler–Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge–Ampere equations. In this article, the third of a series on this subject, we study the long term behavior of the normalized Kahler–Ricci flow on mildly singular varieties of positive Kodaira dimension, generalizing results of Song and Tian who dealt with smooth minimal models.

We prove that a quotient singularity ℂn/G by a finite subgroup G ⊂ SLn(ℂ) has a crepant resolution only if G is generated by junior elements. This is a generalization of the result of Verbitsky (Asian J. Math.4(3) (2000), 553–563). We also give a procedure to compute the Cox ring of a minimal model of a given ℂn/G explicitly from information of G. As an application, we investigate the smoothness of minimal models of some quotient singularities. Together with work of Bellamy and Schedler, this completes the classification of symplectically imprimitive quotient singularities that admit projective symplectic resolutions.

The asymptotics of characters $\chi_{k\lambda}(\exp(h/k))$ of irreducible representations of a compact Lie group $G$ for large values of the scaling factor $k$ are given by Duistermaat-Heckman (DH) integrals over coadjoint orbits of $G$. This phenomenon generalises to coadjoint orbits of central extensions of loop groups $\widehat{LG}$ and of diffeomorphisms of the circle $\widehat{\rm Diff}(S^1)$. We show that the asymptotics of characters of integrable modules of affine Kac-Moody algebras and of the Virasoro algebra factorize into a divergent contribution of the standard form and a convergent contribution which can be interpreted as a formal DH orbital integral. For some Virasoro modules, our results match the formal DH integrals recently computed by Stanford and Witten. In this case, the $k$-scaling has the same origin as the one which gives rise to classical conformal blocks. Furthermore, we consider reduced spaces of Virasoro coadjoint orbits and we suggest a new invariant which replaces symplectic volume in the infinite dimensional situation. We also consider other modules of the Virasoro algebra (in particular, the modules corresponding to minimal models) and we obtain DH-type expressions which do not correspond to any Virasoro coadjoint orbits. We study volume functions $V(x)$ corresponding to formal DH integrals over coadjoint orbits of the Virasoro algebra. We show that they are related by the Hankel transform to spectral densities $\rho(E)$ recently studied by Saad, Shenker and Stanford., Comment: 27 pages, new material is added, references added

Boolean Satisfiability (SAT) is arguably the archetypical NP-complete decision problem. Progress in SAT solving algorithms has motivated an ever increasing number of practical applications in recent years. However, many practical uses of SAT involve solving function as opposed to decision problems. Concrete examples include computing minimal unsatisfiable subsets, minimal correction subsets, prime implicates and implicants, minimal models, backbone literals, and autarkies, among several others. In most cases, solving a function problem requires a number of adaptive or non-adaptive calls to a SAT solver. Given the computational complexity of SAT, it is therefore important to develop algorithms that either require the smallest possible number of calls to the SAT solver, or that involve simpler instances. This paper addresses a number of representative function problems defined on Boolean formulae, and shows that all these function problems can be reduced to a generic problem of computing a minimal set subject to a monotone predicate. This problem is referred to as the Minimal Set over Monotone Predicate (MSMP) problem. This work provides new ways for solving well-known function problems, including prime implicates, minimal correction subsets, backbone literals, independent variables and autarkies, among several others. Moreover, this work also motivates the development of more efficient algorithms for the MSMP problem. Finally the paper outlines a number of areas of future research related with extensions of the MSMP problem.

Chaotic Set/Reset (RS) flip-flop circuits are investigated once again in the context of discrete planar dynamical system models of the threshold voltages, but this time starting with simple bilinear (minimal) component models derived from first principles. The dynamics of the minimal model is described in detail, and shown to exhibit some of the expected properties, but not the chaotic regimes typically found in simulations of physical realizations of chaotic flip-flop circuits. Any electronic physical realization of a chaotic logical circuit must necessarily involve small perturbations from the ideal - usually with large or even nonexistent derivatives in small diameter subsets of the phase space. Therefore, perturbed forms of the minimal model are also analyzed in considerable detail. It is proved that very slightly perturbed minimal models can exhibit chaotic regimes, sometimes associated with chaotic strange attractors, as well as some of the bifurcations present in most of the differential equations models for similar physical circuit realizations. In essence, this work is a mathematical exploration of simple models that reproduce the qualitative behavior of threshold control units of a chaotic RS flip-flop design. It is also shown that this method can be extended to other similar circuits. Validation of the approach developed is provided by some comparisons with (mainly simulated) dynamical results obtained from more traditional investigations.

Suppose that $$f:X\rightarrow \mathrm{Spec}\,R$$ is a minimal model of a complete local Gorenstein 3-fold, where the fibres of f are at most one dimensional, so by Van den Bergh (Duke Math J 122(3):423–455, 2004) there is a noncommutative ring $$\Lambda $$ derived equivalent to X. For any collection of curves above the origin, we show that this collection contracts to a point without contracting a divisor if and only if a certain factor of $$\Lambda $$ is finite dimensional, improving a result of Donovan and Wemyss (Contractions and deformations, arXiv:1511.00406 ). We further show that the mutation functor of Iyama and Wemyss (Invent Math 197(3):521–586, 2014, §6) is functorially isomorphic to the inverse of the Bridgeland–Chen flop functor in the case when the factor of $$\Lambda $$ is finite dimensional. These results then allow us to jump between all the minimal models of $$\mathrm{Spec}\,R$$ in an algorithmic way, without having to compute the geometry at each stage. We call this process the Homological MMP. This has several applications in GIT approaches to derived categories, and also to birational geometry. First, using mutation we are able to compute the full GIT chamber structure by passing to surfaces. We say precisely which chambers give the distinct minimal models, and also say which walls give flops and which do not, enabling us to prove the Craw–Ishii conjecture in this setting. Second, we are able to precisely count the number of minimal models, and also give bounds for both the maximum and the minimum numbers of minimal models based only on the dual graph enriched with scheme theoretic multiplicity. Third, we prove a bijective correspondence between maximal modifying R-module generators and minimal models, and for each such pair in this correspondence give a further correspondence linking the endomorphism ring and the geometry. This lifts the Auslander–McKay correspondence to dimension three.

Starting from a $d\times d$ rational Lax pair system of the form $\hbar \partial_x \Psi= L\Psi$ and $\hbar \partial_t \Psi=R\Psi$ we prove that, under certain assumptions (genus $0$ spectral curve and additional conditions on $R$ and $L$), the system satisfies the "topological type property". A consequence is that the formal $\hbar$-WKB expansion of its determinantal correlators, satisfy the topological recursion. This applies in particular to all $(p,q)$ minimal models reductions of the KP hierarchy, or to the six Painlev\'e systems., Comment: Published version in Annales Henri Poincar\'e

We prove the following results for projective klt pairs of dimension $3$ over an algebraically closed field of char $p>5$: the cone theorem, the base point free theorem, the contraction theorem, finiteness of minimal models, termination with scaling, existence of Mori fibre spaces, etc., 36 pages

In this paper, we introduce a new problem, called Top-k SAT, that consists in enumerating the Top-k models of a propositional formula. A Top-k model is defined as a model with less than k models preferred to it with respect to a preference relation. We show that Top-k SAT generalizes two well-known problems: the Partial MAX-SAT problem and the problem of computing minimal models. Moreover, we propose a general algorithm for Top-k SAT. Then, we give an application of our declarative framework in data mining, namely, the problem of mining Top-k motifs in the transaction databases and in the sequences. In the case of mining sequence data, we introduce a new mining task by considering the sequences of itemsets. Thanks to the flexibility and to the declarative aspects of our SAT-based approach, an encoding of this task is obtained by a very slight modification of mining motifs in the sequences of items.

We prove that one can run the log minimal model program for log canonical 3-fold pairs in characteristic $p>5$. In particular, we prove the cone theorem, contraction theorem, the existence of flips and the existence of log minimal models for pairs with log divisor numerically equivalent to an effective divisor. These follow from our main results, which are that certain log minimal models are good.

We show that minimal models of nondegenerated hypersufaces defined by Laurent polynomials with a $d$-dimensional Newton polytope $\Delta$ are Calabi-Yau varieties $X$ if and only if the Fine interior of $\Delta$ consists of a single lattice point. We give a combinatorial formula for computing the stringy Euler number of $X$. This formula allows to test mirror symmetry in cases when $\Delta$ is not a reflexive polytope. In particular we apply this formula to pairs of lattice polytopes $(\Delta, \Delta^{\vee})$ that appear in the Mavlyutov's generalization of the polar duality for reflexive polytopes. Some examples of Mavlyutov's dual pairs $(\Delta, \Delta^{\vee})$ show that the stringy Euler numbers of the corresponding Calabi-Yau varieties $X$ and $X^{\vee}$ may not satisfy the expected topological mirror symmetry test: $e_{\rm st}(X) = (-1)^{d-1} e_{\rm st}(X^{\vee})$. This shows the necessity of an additional condition on Mavlyutov's pairs $(\Delta, \Delta^\vee)$.

We show that the Malcev Lie algebra of the fundamental group of a compact 2n+ 1-dimensional Sasakian manifold with n ≥ 2 admits a quadratic presentation by using Morgan’s bigradings of minimal models of mixed-Hodge diagrams. By using bigradings of minimal models, we also simplify the proof of the result of Cappelletti-Montano, De Nicola, Marrero and Yudin on Sasakian nilmanifolds.

We study relations between two log minimal models of a fixed lc pair. For any two log minimal models of an lc pair constructed with log MMP, we prove that there are small birational models of the log minimal models which can be connected by a sequence of flops, and the two log minimal models share some properties. We also give examples of two log minimal models of an lc pair which have different properties., 20 pages, only minor revision. To appear in Internat. J. Math

We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for elliptic curves of Shimura type, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.

We study the limit of D-series minimal models when the central charge tends to a generic irrational value $c\in (-\infty, 1)$. We find that the limit theory's diagonal three-point structure constant differs from that of Liouville theory by a distribution factor, which is given by a divergent Verlinde formula. Nevertheless, correlation functions that involve both non-diagonal and diagonal fields are smooth functions of the diagonal fields' conformal dimensions. The limit theory is a non-trivial example of a non-diagonal, non-rational, solved two-dimensional conformal field theory., Comment: 27 pages, v4: Corrected global sign in Eq. (2.25)

We perform Monte-Carlo computations of four-point cluster connectivities in the critical 2d Potts model, for numbers of states $Q\in (0,4)$ that are not necessarily integer. We compare these connectivities to four-point functions in a CFT that interpolates between D-series minimal models. We find that 3 combinations of the 4 independent connectivities agree with CFT four-point functions, down to the $2$ to $4$ significant digits of our Monte-Carlo computations. However, we argue that the agreement is exact only in the special cases $Q=0, 3, 4$. We conjecture that the Potts model can be analytically continued to a double cover of the half-plane $\{\Re c, 53 pages, v2: small improvements and clarifications

We prove the Nonvanishing conjecture for uniruled projective log canonical pairs of dimension $n$, assuming the Nonvanishing conjecture for smooth projective varieties in dimension $n-1$. We also show that the existence of good minimal models for non-uniruled projective klt pairs in dimension $n$ implies the existence of good minimal models for projective log canonical pairs in dimension $n$., v2: this paper supersedes arXiv:1906.11451; minor changes; to appear in Electron. Res. Arch

Let F be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the DG-Lie algebra RHom(F,F) of derived endomorphisms of F is formal. The proof is based on the study of equivariant $L_{\infty}$ minimal models of DG-Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for K3 surfaces) on previous results on the same subject., Post-print version; accepted for publication in Compositio Mathematica

Given a countable group G G , we say that a metrizable flow Y Y is model-universal if by considering the various invariant measures on Y Y , we can recover every free measure-preserving G G -system up to isomorphism. Weiss in[Dynamical systems and group actions, American Mathematical Society, Providence, RI, 2012, pp. 249–264] constructs a minimal model-universal flow. In this note, we provide a new, streamlined construction, allowing us to show that a minimal model-universal flow is far from unique.

We study the birational geometry of moduli spaces of semistable sheaves on the projective plane via Bridgeland stability conditions. We show that the entire MMP of their moduli spaces can be run via wall-crossing. Via a description of the walls, we give a numerical description of their movable cones, along with its chamber decomposition corresponding to minimal models. As an application, we show that for primitive vectors, all birational models corresponding to open chambers in the movable cone are smooth and irreducible., 62 pages, 13 figures, comments are welcome

The objective is to show the construction of an Ulrich vector bundle on the blowing-up $\widetilde X$ of a nonsingular projective variety $X$ at a closed point, where the original variety is embedded by a very ample divisor $H$ and carries an Ulrich vector bundle. In order to achieve this result, we aim to find a suitable very ample divisor on $\widetilde X$, which is dependent on $H$. At the end, we take into consideration some applications to surfaces with regards to minimal models and their Kodaira dimension., Comment: v2: modified description of Thm.2, beginning of Section 3, Cor.3 and Cor.8. I thank G. Casnati and Y. Kim for their helpful comments

We prove a result that enables us to calculate the rational homotopy of a wide class of spaces by the theory of minimal models., Comment: 8 pages. arXiv admin note: text overlap with arXiv:0903.2926

We provide two criteria for establishing the non-formality of a differential graded Lie algebra in terms of higher Whitehead brackets, which are the Lie analogues of the Massey products of a differential graded associative algebra. We also show that formality of a differential graded Lie algebra is not equivalent to the collapse of its associated Quillen spectral sequence. Finally, we use L ∞ algebras and Quillen's formulation of rational homotopy theory to recover and improve a classical theorem for detecting higher Whitehead products in Sullivan minimal models, and give some applications.

We compare the minimal model of a log canonical pair with the minimal model of its reduced boundary. These results are then used to study the existence of the minimal model of a semi-log-canonical pair using its normalization.