We associate a ring R R to a log Calabi-Yau pair ( X , D ) (X,D) or a degeneration of Calabi-Yau manifolds X → B X\to B . The vector space underlying R R is determined by the tropicalization of ( X , D ) (X,D) or X → B X\to B , while the product rule is defined using punctured Gromov-Witten invariants, defined in joint work with Abramovich and Chen. In the log Calabi-Yau case, if D D is maximally degenerate, then we propose that Spec ⁡ R \operatorname {Spec} R is the mirror to X ∖ D X\setminus D , while in the Calabi-Yau degeneration case, if the degeneration is maximally unipotent, the mirror is expected to be Proj ⁡ R \operatorname {Proj} R . The main result in this paper is that R R as defined is an associative, commutative ring with unit, with associativity the most difficult part.

Let E be a smooth cubic in the projective plane P2. Nobuyoshi Takahashi formulated a conjecture that expresses counts of rational curves of varying degree in P2\ E as the Taylor coefficients of a particular period integral of a pencil of affine plane cubics after reparametrizing the pencil using the exponential of a second period integral. The intrinsic mirror construction introduced by Mark Gross and the third author associates to a degeneration of (P2,E) a canonical wall structure from which one constructs a family of projective plane cubics that is birational to Takahashi's pencil in its reparametrized form. By computing the period integral of the positive real locus explicitly, we find that it equals the logarithm of the product of all asymptotic wall functions. The coefficients of these asymptotic wall functions are logarithmic Gromov-Witten counts of the central fiber of the degeneration that agree with the algebraic curve counts in (P2,E) in question. We conclude that Takahashi's conjecture is a natural consequence of intrinsic mirror symmetry. Our method generalizes to give similar results for log Calabi-Yau varieties of arbitrary dimension.

This paper, largely written in 2009/2010, fits Landau–Ginzburg models into the mirror symmetry program pursued by the last author jointly with Mark Gross since 2001. This point of view transparently brings in tropical disks of Maslov index 2 via the notion of broken lines, previously introduced in two dimensions by Mark Gross in his study of mirror symmetry for ℙ 2 . A major insight is the equivalence of properness of the Landau–Ginzburg potential with smoothness of the anticanonical divisor on the mirror side. We obtain proper superpotentials which agree on an open part with those classically known for toric varieties. Examples include mirror LG models for non-singular and singular del Pezzo surfaces, Hirzebruch surfaces and some Fano threefolds.

As announced "Intrinsic mirror symmetry and punctured invariants" in 2016, we construct and prove consistency of the canonical wall structure. This construction starts with a log Calabi-Yau pair (X,D) and produces a wall structure, as defined by Gross-Hacking-Siebert. Roughly put, the canonical wall structure is a data structure which encodes an algebro-geometric analogue of counts of Maslov index zero disks. These enumerative invariants are defined in terms of the punctured invariants of Abramovich-Chen-Gross-Siebert. There are then two main theorems of the paper. First, we prove consistency of the canonical wall structure, so that the canonical wall structure gives rise to a mirror family. Second, we prove that this mirror family coincides with the intrinsic mirror constructed in our paper "Intrinsic mirror symmetry". While the setup of this paper is narrower than that of the latter paper, it gives a more detailed description of the mirror.

This paper expands on a remark in the paper Mirror Symmetry for Log Calabi-Yau Surfaces I of the first three authors of this paper, explaining fully how various constructions of the authors apply to give the mirror to the cubic surface. We give a full description of the scattering diagram associated to the cubic surface: this is a particularly nice diagram in which rays of every rational slope occur, but they may all be described. The equation of the mirror cubic family is then derived in two ways, first by using broken lines and then by using more recent constructions involving a direct calculation of Gromov-Witten invariants.

We show that a large class of maximally degenerating families of n n -dimensional polarized varieties comes with a canonical basis of sections of powers of the ample line bundle. The families considered are obtained by smoothing a reducible union of toric varieties governed by a wall structure on a real n n -(pseudo-)manifold. Wall structures have previously been constructed inductively for cases with locally rigid singularities [Gross and Siebert, From real affine geometry to complex geometry (2011)] and by Gromov-Witten theory for mirrors of log Calabi-Yau surfaces and K 3 K3 surfaces [Gross, Pandharipande and Siebert, The tropical vertex; Gross, Hacking and Keel, Mirror symmetry for log Calabi-Yau surfaces (2015); Gross, Hacking, Keel, and Siebert, Theta functions and K 3 K3 surfaces (In preparation)]. For trivial wall structures on the n n -torus we retrieve the classical theta functions. We anticipate that wall structures can be constructed quite generally from maximal degenerations. The construction given here then provides the homogeneous coordinate ring of the mirror degeneration along with a canonical basis. The appearance of a canonical basis of sections for certain degenerations points towards a good compactification of moduli of certain polarized varieties via stable pairs, generalizing the picture for K3 surfaces [Gross, Hacking, Keel, and Siebert, Theta functions and K 3 K3 surfaces (In preparation)]. Another possible application apart from mirror symmetry may be to geometric quantization of varieties with effective anti-canonical class.

We give a simple expression for the integral of the canonical holomorphic volume form in degenerating families of varieties constructed from wall structures and with central fiber a union of toric varieties. The cycles to integrate over are constructed from tropical 1-cycles in the intersection complex of the central fiber. One application is a proof that the mirror map for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author is trivial. We also show that these families are the completion of an analytic family, without reparametrization, and that they are formally versal as deformations of logarithmic schemes. Other applications include canonical one-parameter type III degenerations of K3 surfaces with prescribed Picard groups. As a technical result of independent interest we develop a theory of period integrals with logarithmic poles on finite order deformations of normal crossing analytic spaces.

We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $X \longrightarrow B$ with singular fibre over $b_0\in B$ yields a family $\mathscr {M}(X/B,\beta ) \longrightarrow B$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $b_0$ in terms of rigid tropical maps to the tropicalization of $X/B$ . This generalizes one aspect of known results in the case that the fibre $X_{b_0}$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.

Mirror symmetry suggests to study families of varieties with a certain maximal degeneration behaviour [CdGP91], [Mo93], [De93], [HKTY95]. In the important CalabiYau case this means that the monodromy transformation along a general loop around the critical locus is unipotent of maximally possible exponent [Mo93, §2]. The limiting mixed Hodge structure on the cohomology of a nearby smooth fibre is then of Hodge-Tate type [De93]. An important insight in this situation is the existence of a distinguished class of holomorphic coordinates on the base space of the maximal degeneration [Mo93], [De93]. Explicitly, these canonical coordinates are computed as exp of those period integrals of the holomorphic n-form Ω over n-cycles that have a logarithmic pole at the degenerate fibre. For an algebraic family they are often determined as certain solutions of the Picard-Fuchs equation solving the parallel transport with respect to the Gauß-Manin connection. For complete intersections in toric varieties these solutions can be written as hyper-geometric series. In particular, canonical coordinates are typically transcendental functions of the algebraic parameters. The coordinate change from the algebraic

This contribution to the 2015 AMS Summer Institute in Algebraic Geometry (Salt Lake City) announces a general mirror construction. This construction applies to log Calabi-Yau pairs (X,D) with maximal boundary D or to maximally unipotent degenerations of Calabi-Yau manifolds. The new ingredient is a notion of "punctured Gromov-Witten invariant", currently in progress with Abramovich and Chen. The mirror to a pair (X,D) is constructed as the spectrum of a ring defined using the punctured invariants of (X,D). An analogous construction leads to mirrors of Calabi-Yau manifolds. This can be viewed as a generalization of constructions developed jointly with Hacking and Keel in the case of log CY surfaces and K3 surfaces.

Using Brauer type obstructions and the degeneration method due to Voisin, ColliotThelene, Pirutka et al., we discuss rationality properties of certain conic bundles over P^2 and P^3. This is joint work (in progress) with Hans-Christian von Bothmer, Asher Auel and Alena Pirutka. Tom Coates (Imperial College London) Title: Mirror symmetry and Fano manifolds Abstract: I will outline a program -joint work with Corti, Galkin, Golyshev, Kasprzyk, and others -to find and classify Fano manifolds using mirror symmetry. I will describe recent progress in this direction, including in dimension 4. The latter is work by and joint work with Prince, Kasprzyk, and Kalashnikov. Kento Fujita (Kyoto University) I will outline a program -joint work with Corti, Galkin, Golyshev, Kasprzyk, and others -to find and classify Fano manifolds using mirror symmetry. I will describe recent progress in this direction, including in dimension 4. The latter is work by and joint work with Prince, Kasprzyk, and Kalashnikov. Kento Fujita (Kyoto University) Title: A valuative criterion for uniform K-stability of log Fano pairs Abstract: It's a difficult problem whether a given Fano manifold is K-(semi)stable or not. In this talk, I will introduce "valuative criteria", of those stability conditions, which seem relatively easy. Mark Gross (University of Cambridge) and Bernd Siebert (Universität Hamburg) It's a difficult problem whether a given Fano manifold is K-(semi)stable or not. In this talk, I will introduce "valuative criteria", of those stability conditions, which seem relatively easy. Mark Gross (University of Cambridge) and Bernd Siebert (Universität Hamburg) Title: An intrinsic mirror symmetry construction. Abstract: We will survey recent progress in our joint program for understanding mirror symmetry, leading to a general mirror symmetry construction for maximal log Calabi-Yau pairs and maximally unipotent degenerations of Calabi-Yau varieties. We do this by introducing a variant of log GromovWitten invariants called "punctured invariants" developed in joint work with Abramovich, Chen. We will then explain how to use these invariants to give a general construction of mirrors by building the coordinate ring of the mirror explicitly in terms of these invariants. Dominic Joyce (University of Oxford) We will survey recent progress in our joint program for understanding mirror symmetry, leading to a general mirror symmetry construction for maximal log Calabi-Yau pairs and maximally unipotent degenerations of Calabi-Yau varieties. We do this by introducing a variant of log GromovWitten invariants called "punctured invariants" developed in joint work with Abramovich, Chen. We will then explain how to use these invariants to give a general construction of mirrors by building the coordinate ring of the mirror explicitly in terms of these invariants. Dominic Joyce (University of Oxford) Title: New Donaldson-Thomas style counting invariants for Calabi-Yau 4-folds Abstract: Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209 introduced the notion of "k-shifted symplectic derived schemes and stacks" in Derived Algebraic Geometry. They showed that moduli stacks of coherent sheaves and complexes on a Calabi-Yau m-fold Y are (2-m)-shifted symplectic. So in particular Calabi-Yau 3-fold moduli stacks are -1-shifted symplectic, and Calabi-Yau 4-fold moduli stacks are -2-shifted symplectic. In previous work with Ben-Bassat, Brav, Bussi, Dupont, Meinhardt, and Szendroi we studied -1-shifted (3-Calabi-Yau) geometry and generalizations of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209 introduced the notion of "k-shifted symplectic derived schemes and stacks" in Derived Algebraic Geometry. They showed that moduli stacks of coherent sheaves and complexes on a Calabi-Yau m-fold Y are (2-m)-shifted symplectic. So in particular Calabi-Yau 3-fold moduli stacks are -1-shifted symplectic, and Calabi-Yau 4-fold moduli stacks are -2-shifted symplectic. In previous work with Ben-Bassat, Brav, Bussi, Dupont, Meinhardt, and Szendroi we studied -1-shifted (3-Calabi-Yau) geometry and generalizations of Donaldson-Thomas theory. Today we move on to the -2-shifted case. Using a "shifted symplectic Darboux Theorem" by Brav, Bussi and Joyce, we prove that a -2-shifted symplectic derived scheme X over C can be given the structure of a "derived smooth manifold" (dmanifold, or M-Kuranishi space) X*, uniquely up to bordisms of X* fixing the underlying topological space. If X is proper and has an "orientation" (similar to Kontsevich-Soibelman orientation data in the 3Calabi-Yau case), then X* is a compact, oriented derived manifold, and so has a virtual class (e.g. in bordism), which is an integer if vdim X = 0. This should give virtual classes for proper Calabi-Yau 4-fold moduli schemes, and lead to new Donaldson-Thomas style invariants "counting" (semi)stable coherent sheaves on a Calabi-Yau 4fold Y, which will be unchanged under continuous deformations of Y. This is joint work with Dennis Borisov. It is related to work of Cao and Leung in arXiv:1407.7659. Anne-Sophie Kaloghiros (Brunel University) Title: The Sarkisov program for Mori fibred Calabi-Yau pairs Abstract: A Calabi-Yau (CY) pair (X,D) is a pair of a normal variety X and a reduced divisor D such that K+D is a Cartier divisor linearly equivalent to 0; two such pairs will be considered equivalent when there is a volume preserving birational map between them. If X itself has a structure of Mori fibre space-that is if X is a ‘good’ family of Fano varieties— then (X,D) is a Mori fibred CY pair. Such a pair is the end product of two distinct Minimal Model Programs: on the one hand, it is a K+D-minimal model, and on the other it is the end product of a classical MMP. In this talk, I will present results on volume preserving maps between CY pairs, and in particular, a Sarkisov-type factorisation theorem for maps between Mori fibered CY pairs. Conan Leung (Chinese University of Hong Kong) A Calabi-Yau (CY) pair (X,D) is a pair of a normal variety X and a reduced divisor D such that K+D is a Cartier divisor linearly equivalent to 0; two such pairs will be considered equivalent when there is a volume preserving birational map between them. If X itself has a structure of Mori fibre space-that is if X is a ‘good’ family of Fano varieties— then (X,D) is a Mori fibred CY pair. Such a pair is the end product of two distinct Minimal Model Programs: on the one hand, it is a K+D-minimal model, and on the other it is the end product of a classical MMP. In this talk, I will present results on volume preserving maps between CY pairs, and in particular, a Sarkisov-type factorisation theorem for maps between Mori fibered CY pairs. Conan Leung (Chinese University of Hong Kong) Title: Categorical Plucker formula and homological projective dual Abstract: A generalised Plucker formula describes changes of intersection numbers of complex Lagrangian under Mukai flop. In a recent joint work with Jiang and Xie, we generalise this to the level of derived category of coherent sheaves. Yuchen Liu (Princeton University) A generalised Plucker formula describes changes of intersection numbers of complex Lagrangian under Mukai flop. In a recent joint work with Jiang and Xie, we generalise this to the level of derived category of coherent sheaves. Yuchen Liu (Princeton University) Title: The volume of Kähler-Einstein Q-Fano varieties Abstract: A complex projective variety is Q-Fano if it has klt singularities and the anti-canonical divisor is Q-Cartier and ample. Starting from dimension 2, the anti-canonical volume of a Q-Fano variety can be arbitrarily large, e.g. weighted projective spaces. Recently, Fujita showed that an ndimensional Kähler-Einstein Q-Fano variety has volume at most (n+1)^n. In this talk, I will discuss a refinement of Fujita's volume upper bounds involving invariants of the local singularities. If time permits, I will also talk about an equivalent relation between K-semistability and de Fernex-EinMustaţă type inequalities. Part of this work is joint with Chi Li. Eduard Looijenga (Utrecht University, Tsinghua University) A complex projective variety is Q-Fano if it has klt singularities and the anti-canonical divisor is Q-Cartier and ample. Starting from dimension 2, the anti-canonical volume of a Q-Fano variety can be arbitrarily large, e.g. weighted projective spaces. Recently, Fujita showed that an ndimensional Kähler-Einstein Q-Fano variety has volume at most (n+1)^n. In this talk, I will discuss a refinement of Fujita's volume upper bounds involving invariants of the local singularities. If time permits, I will also talk about an equivalent relation between K-semistability and de Fernex-EinMustaţă type inequalities. Part of this work is joint with Chi Li. Eduard Looijenga (Utrecht University, Tsinghua University) Title: Basic Tate extensions in the cohomology of Baily-Borel compactifications Abstract: An automorphic vector bundle on a Shimura variety will in general not extend to as a bundle to its Baily-Borel compactification. Yet Goresky and Pardon showed that the Chern classes of such a bundle do so in a natural manner and asked whether these classes lie in rational cohomology. We show that these extensions have a natural description in the setting of mixed Hodge theory and may not lie in rational cohomology. We apply this to the primitive part of the stable cohomology of the Baily-Borel compactification of the ppav's to show that they yield all the basic Tate extensions. Emanuele Macri (Northeastern University) Title: Tilting of bounded t-structures and applications Abstract: We will review the construction of tilting for bounded t-structures in the derived category of coherent sheaves on a smooth projective variety. We will discuss a few applications of this: Bayer's proof of the Brill-Noether Theorem, the description of nef cones of Hilbert schemes of points on surfaces, and the proof for a conjecture of Huybrechts on derived categ

I am interested in algebraic geometry with applications to mirror symmetry. I use methods of logarithmic geometry, in particular log Gromov-Witten theory, tropical geometry and symplectic geometry. The three projects I worked on during my PhD under the supervision of Prof. Bernd Siebert are summarised below. The previous research I conducted before changing my direction towards algebraic geometry was on differential and symplectic topology and can also be found on the arxiv ([AK],[AKO]).

We study the real loci of toric degenerations of complex varieties with reducible central fibre, as introduced in the joint work of the second author with Mark Gross on mirror symmetry. The topology of such degenerations can be explicitly described via the Kato-Nakayama space of the central fibre as a log space. The paper provides generalities of real structures in log geometry and their lift to Kato-Nakayama spaces, the description of the Kato-Nakayama space of a toric degeneration and its real locus, as bundles determined by tropical data. Examples include real toric degenerations of K3-surfaces.

This is a survey covering aspects of varied work of the authors with Mohammed Abouzaid, Paul Hacking, and Sean Keel. While theta functions are traditionally canonical sections of ample line bundles on abelian varieties, we motivate, using mirror symmetry, the idea that theta functions exist in much greater generality. This suggestion originates with the work of the late Andrei Tyurin. We outline how to construct theta functions on the degenerations of varieties constructed in previous work of the authors, and then explain applications of this construction to homological mirror symmetry and constructions of broad classes of mirror varieties.

We study the real loci of toric degenerations of complex varieties with reducible central fibre, as introduced in the joint work of the second author with Mark Gross on mirror symmetry. The topology of such degenerations can be explicitly described via the Kato-Nakayama space of the central fibre as a log space. The paper provides generalities of real structures in log geometry and their lift to Kato-Nakayama spaces, the description of the Kato-Nakayama space of a toric degeneration and its real locus, as bundles determined by tropical data. Examples include real toric degenerations of K3-surfaces.

The workshop covered a broad variety of areas in algebraic geometry and was the occasion to report on recent advances and works in progress. Special emphasis was put on the role of derived categories and various stability concepts for sheaves, varieties, complexes, etc. The mix of people working in areas like classification theory, mirror symmetry, derived categories, moduli spaces, $p$-adic geometry, characteristic $p$ methods, singularity theory led to stimulating discussions.

We prove that the mirror map is trivial for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author. In other words, the natural coordinate in a canonical Calabi-Yau family is a canonical coordinate in the sense of Hodge theory. This implies that the higher weight periods directly carry enumerative information with no further gauging necessary as opposed to the classical case. A side result is that the canonical formal families lift to analytic families. We compute the relevant period integrals explicitly. The cycles to integrate over are constructed from tropical 1-cycles in the intersection complex of the degenerate Calabi-Yau.

We discuss how the reconstruction theorem of [20] applies to local mirror symmetry [11]. This theorem associates to certain combinatorial data a degeneration of (log) Calabi-Yau varieties. While in this case most of the subtleties of the construction are absent, an important normalization condition already introduces rich geometry. This condition guarantees the parameters of the construction are canonical coordinates in the sense of mirror symmetry. The normalization condition is also related to a count of holomorphic disks and cylinders, as conjectured in [20] and partially proved in [7-9]. We sketch a possible alternative proof of these counts via logarithmic Gromov-Witten theory.There is also a surprisingly simple interpretation via rooted trees marked by monomials, which points to an underlying rich algebraic structure both in the relevant period integrals and the counting of holomorphic disks.

The Conference focused on several classical and novel theories in the realm of complex algebraic geometry, such as Algebraic surfaces, Moduli theory, Minimal Model Program, Abelian Varieties, Holomorphic Symplectic Varieties, Homological algebra, Kähler manifolds theory, Holomorphic dynamics, Quantum cohomology.

The goal of this paper is to give a general theory of logarithmic Gromov-Witten invariants. This gives a vast generalization of the theory of relative Gromov-Witten invariants introduced by Li-Ruan, Ionel-Parker, and Jun Li, and completes a program first proposed by the second named author in 2002. One considers target spaces X carrying a log structure. Domains of stable log curves are log smooth curves. Algebraicity of the stack of such stable log maps is proven, subject only to the hypothesis that the log structure on X is fine, saturated, and Zariski. A notion of basic stable log map is introduced; all stable log maps are pull-backs of basic stable log maps via base-change. With certain additional hypotheses, the stack of basic stable log maps is proven to be proper. Under these hypotheses and the additional hypothesis that X is log smooth, one obtains a theory of log Gromov-Witten invariants.