# The geometry of Landau-Ginzburg models

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1 Motivation Toric degeneration Hodge theory CY3s The Geometry of Landau-Ginzburg Models January 19, 2016

2 Motivation Toric degeneration Hodge theory CY3s Plan of talk 1. Landau-Ginzburg models and mirror symmetry: physics and mathematics 2. Degenerations and Laurent polynomials 3. Hodge theory of LG models 4. Tyurin degenerations and fibrations on compact Calabi-Yau varieties

3 Motivation Toric degeneration Hodge theory CY3s 1. Motivation Cartoon version of mirror symmetry in physics - Superstring theory predicts that we live in 10-dimensional space R 1,3 V where V is some manifold of (real) dimension 6. - In order for the resulting physical theory to be plausible V must be a (compact) Riemannian manifold with Ricci flat metric. - These manifolds are known (by the Calabi conjecture, proved by Yau) to be precisely Calabi-Yau manifolds. These are (complex) 3-dimensional Kähler manifolds with vanishing first Chern class.

4 Motivation Toric degeneration Hodge theory CY3s 1. Motivation - Precisely, associated to any Calabi-Yau threefold with complex structure I and B-field b, there is an N = 2 SCFT. - There are two topological twists of this SCFT, called the A- and B-twists respectively. - Mirror symmetry predicts there is a manifold (W, J, b ) so that A-twisted theory of (V, I, b) B-twisted theory of (W, J, b ) and vice versa. This is is an example of a string duality.

5 Motivation Toric degeneration Hodge theory CY3s 1. Motivation For the rest of this talk, we define a Calabi-Yau manifold to be a smooth projective d-dimensional variety V with trivial canonical bundle and h i,0 (V ) = 0 for i 0, d.

6 Motivation Toric degeneration Hodge theory CY3s 1. Motivation Mathematical mirror symmetry This can be restated in several ways as a correspondence in mathematics between Calabi-Yau threefolds V and W. 1. Hodge number mirror symmetry. h p,q (V ) = h 3 q,p (W ). 2. Enumerative mirror symmetry. The A-model connection constructed from GW invariants on V is equal to B-model connection on complex deformations of W. 3. Homological mirror symmetry. The derived Fukaya category of V is equivalent to the bounded derived category of coherent sheaves on W.

7 Motivation Toric degeneration Hodge theory CY3s 1. Motivation Fano and quasi-fano A Fano manifold is a smooth projective variety X with K X an ample divisor on X. A quasi-fano manifold is a smooth projective variety X so that there is a smooth Calabi-Yau divisor S in X so that S K X and h i,0 (X) = 0 for i 0. Eguchi-Hori-Xiong ( 96) (Physics!) noticed that there is an A-twist of the σ-model associated to a quasi-fano manifold. They showed that this theory is the same as the theory coming from a Landau-Ginzburg model, i.e. a pair (Y, w) where w : Y C.

8 Motivation Toric degeneration Hodge theory CY3s 1. Motivation Quasi-Fano mirror symmetry for mathematicians - Batyrev ( 93) proved that the quantum cohomology of a toric Fano variety can be reconstructed from the Jacobian ring of a specific Laurent polynomial. - Kontsevich ( 90s) formulated homological mirror symmetry for Fano manifolds. This is a relationship between the derived category of coherent sheaves on X and the directed Fukaya category of a LG model. - Katzarkov-Kontsevich-Pantev ( 08, 14) deduce conjectures from HMS about how mirror symmetry should be reflected by Hodge theory (related to the irregular Hodge filtration studied by Esnault, (Morihiko) Saito, Sabbah and Yu).

9 Motivation Toric degeneration Hodge theory CY3s 1. Motivation What are Landau-Ginzburg mirrors? Definition: (KKP 14) A Landau-Ginzburg (LG) mirror to a d-dimensional quasi-fano variety is a pair (Y, w) where Y is a d-dimensional Kähler manifold and w is a holomorphic function on Y so that: - The fibers of w are Calabi-Yau manifolds, and w is proper - The first Chern class of Y is zero If (Y, w) is the LG model of a Fano manifold, then in addition, we require that: - There is a compactification of Y to a projective variety Z so that D = Z \ Y is a normal crossings divisor and w extends to a function f : Z P 1 - There is a non-vanishing holomorphic dim Y -form with simple poles along D

10 Motivation Toric degeneration Hodge theory CY3s 1. Motivation Why study the geometry of LG models? The geometry of the LG/Fano correspondence is at least as rich as geometry of CY/CY mirror symmetry. - The moduli theory of LG models should be mirror to the birational geometry of quasi-fano varieties (and vice versa). - There is only a finite number of Fano varieties in each dimension. Therefore, there should be only a finite number of appropriate LG models. Their classification should be mirror to one another. - LG models should be glued together to produce compact Calabi-Yau varieties. - The Hodge theory of LG models is complex and interesting in its own right.

11 Motivation Toric degeneration Hodge theory CY3s 2. Degenerations and Laurent polynomials 2. Degenerations and Laurent polynomials Question: How does one construct an LG mirror to a quasi-fano variety? How is the geometry of the LG model related to the geometry of the mirror quasi-fano variety? Eguchi-Hori-Xiong claimed that the LG model of a d-dimensional Fano variety is a torus (C ) d equipped with a Laurent polynomial w. This is enough for enumerative mirror symmetry, however it does not have enough information for any other type of mirror symmetry. In general, LG models of many Fano varieties seem to be given by a bunch of tori glued together along specific birational maps (a sort of generalized or overdetermined cluster variety).

12 Motivation Toric degeneration Hodge theory CY3s 2. Degenerations and Laurent polynomials This sort of structure also appears in work of Auroux. What do these charts mean? To each chart there is a Laurent polynomial, and to each Laurent polynomial there is a polytope. Expectation: These polytopes are the moment polytopes of toric varieties to which X degenerates. This works well when we start with a smooth toric Fano variety.

13 Motivation Toric degeneration Hodge theory CY3s 2. Degenerations and Laurent polynomials How does this correspond with toric constructions? Let s describe this in the case of hypersurfaces in P n. Givental s prescription for hypersurfaces in P n : if X k is a degree k d hypersurface in P d then the LG model of X is Y k = {x 1 + +x k = 1} (C ) d 1, w = x k+1 + +x d +. x 1... x d X k degenerates to the toric varieties for a k a d+1 = k z 1... z k z a k+1 k+1... za d+1 d+1

14 Motivation Toric degeneration Hodge theory CY3s 2. Degenerations and Laurent polynomials Theorem: For each choice of a k a d+1 = k there is a choice of birational map φ : (C ) d 1 Y k so that φ w is a Laurent polynomial. Furthermore the Newton polytope of φ w is equal to the moment polytope of the toric variety determined by the equation above. Theorem: (Theorem ) An analogue of this is true for arbitrary complete intersections in toric varieties with ample enough anticanonical bundle. These two types of objects are mediated by a combinatorial structure called an amenable collection.

15 Motivation Toric degeneration Hodge theory CY3s 2. Degenerations and Laurent polynomials An existence result If a complete intersection in a toric Fano variety is Fano enough then it admits a toric degeneration. If X is a complete intersection in a toric Fano Gorenstein variety, L X = K X and L is ample on P then X admits a toric degeneration. A finer version of this statement holds for hypersurfaces in toric varieties. This partially addresses a conjecture of Przyjalkowski.

16 Motivation Toric degeneration Hodge theory CY3s 3. Hodge theory of LG models 3. Hodge theory of LG models Assume we have an LG model for a quasi-fano variety. What would we like to prove about this LG model? We will introduce Hodge-theoretic invariants of LG models and Hodge number mirror symmetry for Fano varieties. Definition: A holomorphic k-form α on Y with log poles along D is called f-adapted if df α has only log poles along D. Define the sheaf Ω k Z (log D, f) to be the sub-sheaf of Ω k Z (log D ) of f-adapted k-forms. Let h p,q (Y, w) be rank H q (Ω p Z (log D, f), Z). KKP show that the sheaves Ω k Z (log D, f) are the limit of relative cohomology sheaves Ω k Z (log D, rel V ) where V is a smooth fiber of w.

17 Motivation Toric degeneration Hodge theory CY3s 3. Hodge theory of LG models This leads to: Theorem: (Theorem 2.2.2) h p,q (Y, w) = gr F p H p+q (Y, V ) Here V is a smooth fiber of w and F is the natural Hodge filtration on the cohomology of the pair (Y, V ). If X and (Y, w) are mirror partners then KKP predict that h p,q (X) = h d q,p (Y, w). Thus h p,q (Y, w) should have the standard symmetries of the Hodge-diamond of an algebraic variety.

18 Motivation Toric degeneration Hodge theory CY3s 3. Hodge theory of LG models Hodge diamond dualities for LG models The following theorem assumes almost nothing about the fibers of the LG model (Y, w). Theorem: (Theorem 2.2.6, Theorem 2.2.9) If (Y, w) is an LG model, then its Hodge numbers are symmetric horizontally. I.e. h p,q (Y, w) = h q,p (Y, w). The Betti numbers of an LG model obey Poincaré duality. If we make assumptions on Y and V, then we can show that the full vertical symmetry holds. Theorem: If dim Y = 3, 4, V is Calabi-Yau and h i,0 (Z) = 0 for i 0 then the Hodge diamond of (Y, w) has vertical symmetry. The same result holds if we assume that h i,j (Z) = 0 for i j.

19 Motivation Toric degeneration Hodge theory CY3s 3. Hodge theory of LG models LG models of hypersurfaces in toric varieties If X is complete intersection in a toric variety X then Givental gives a combinatorial description of the LG model of X. This LG model is inappropriate for mirror symmetry, since it is not relatively compact. Theorem: (Theorem 3.2.6) There is a partial compactification of Givental s LG model which is as good as one could reasonably hope for. When X is 2- or 3-dimensional, this is precisely what the prescription of KKP calls for. When X had dimension greater than 3, we must allow mild singularities in the definition. Note: this compactification is not related to the possible compactifications mentioned before.

20 Motivation Toric degeneration Hodge theory CY3s 3. Hodge theory of LG models The Hodge number h 1,1 (Y, w) can be computed by counting components in each fiber of w (Theorem 3.3.1). If ρ t = # of components of w 1 (t), then t C (ρ t 1) = h 1,1 (Y, w). Theorem: (Theorem 3.4.9) If X is an ample enough hypersurface in X, then h 1,d 2 (X) = ρ 0 1 h 1,1 (Y, w). This gives a partial generalization of results of Przyjalkowski and Shramov ( 15).

21 Motivation Toric degeneration Hodge theory CY3s 4. Tyurin degenerations and gluing LG models 4. Tyurin degenerations and gluing LG models Assume we have a Calabi-Yau threefold V with a K3 surface fibration. In basic examples, one observes that the singular fibers of these fibrations are the same as the singular fibers of LG models. Example: The mirror quintic has a K3 surface fibration containing the singular which appears in the LG model of a quartic threefold. This is actually the fiber over 0 when we perform the construction in the previous section. Question: What is the relationship between this K3 surface fibration on V and the LG model of the quartic threefold? Is this just a coincidence?

22 Motivation Toric degeneration Hodge theory CY3s 4. Tyurin degenerations and gluing LG models Let V be a Calabi-Yau variety of dimension d. A Tyurin degeneration of V is a smooth (d + 1)-dimensional manifold V with a morphism π : V U with U a small disc in C containing 0, so that: - The fiber over t U is V for some t - π 1 (0) is a union of two smooth projective d-dimensional varieties X 1, X 2 - h i,0 (X 1 ) = h i,0 (X 2 ) = 0 if i 0 - X 1 X 2 = S meet transversally in a smooth Calabi-Yau variety S with so that O Xi (S) = K Xi for i = 1, 2. The existence of a Tyurin degeneration implies that N S/X1 N S/X2 = O S.

23 Motivation Toric degeneration Hodge theory CY3s 4. Tyurin degenerations and gluing LG models A question of Tyurin and a vague conjecture In the his final lecture, Tyurin raised the following question: Question: (Tyurin 02) Assume that V admits a Tyurin degeneration to X 1 S X 2. What is the relationship between the LG models of X 1, X 2 and the Calabi-Yau mirror of V? This has been addressed in special situations by Auroux ( 08). Conjecture: If V admits a Tyurin degeneration to X 1 S X 2, then the mirror W of V is constructed from (Y 1, w 1 ) and (Y 2, w 2 ) by a gluing construction.

24 Motivation Toric degeneration Hodge theory CY3s 4. Tyurin degenerations and gluing LG models Arts and crafts with LG models Remark: Mirror symmetry predicts that if S is the anticanonical hypersurface in a quasi-fano variety X and Q is a fiber of the LG model (Y, w) of X then Q and S are mirror Calabi-Yau varieties. According to homological mirror symmetry, the action of the tensor product with N S/Xi on D b (coh S) corresponds with the action of the monodromy symplectomorphism associated to a small counterclockwise loop around infinity on the derived Fukaya category of the fibers of the LG model of X i. Thus, heuristically, the condition that N S/X1 N S/X2 = O S means the monodromy symplectomorphisms ϕ 1 and ϕ 2 on the fibers of w 1 and w 2 respectively, satisfy the identity ϕ 1 = ϕ 1 2.

25 Motivation Toric degeneration Hodge theory CY3s 4. Tyurin degenerations and gluing LG models This means that we should have large real numbers r 1 and r 2 so that the w1 1 ({z C : z > r 1}) is diffeomorphic to w2 1 ({z C : z > r 2}). Thus we should be able to glue Y 1 to Y 2 along these open sets. We can do this in such a way that w 1 and w 2 can be extended to a fibration on W := Y 1 Y 2, f : W S 2 = CP 1.

26 Motivation Toric degeneration Hodge theory CY3s 4. Tyurin degenerations and gluing LG models Y 1 w 1 Y 1 B1 W w 1 Y1 B1 π diffeo Y 2 w 2 Y 2 B2 w 2 Y2 B2 Identify B 1 and B 2

27 Motivation Toric degeneration Hodge theory CY3s 4. Tyurin degenerations and gluing LG models Euler numbers We conjecture that this gluing construction is a construction of the mirror of V. We also conjecture that the topological fibration on W can be extended to a complex fibration. Let us check whether this conjecture is plausible. Since we only have topology (no Hodge theory) for the LG models of general quasi-fano varieties, we rephrase the Hodge number correspondence of KKP as a relationship between Euler numbers e(x) = ( 1) d e(y, w 1 (t)). Here d = dim X, t is a regular value of w and e(y, w 1 (t)) = 2d i=1 ( 1) i h i (Y, w 1 (t)).

28 Motivation Toric degeneration Hodge theory CY3s 4. Tyurin degenerations and gluing LG models Theorem: (Theorem 6.2.1) Assume that e(x i ) = ( 1) d e(y i, wi 1 (t)), V admits a Tyurin degeneration to X 1 S X 2. Then if W is constructed from Y 1 and Y 2 as in the previous slide, then e(v ) = ( 1) d e(w ).

29 Motivation Toric degeneration Hodge theory CY3s 4. Tyurin degenerations and gluing LG models Examples Example (Chapter 7): The quintic admits a Tyurin degeneration where X 1 is the blow up of P 3 in the intersection of a generic quartic and a generic quintic and where X 2 is a quartic threefold. There is a K3 surface fibration on the mirror quintic which is topologically equivalent to the LG model of X 1 glued to the LG model of X 2 glued as described. Example (Chapter 6): If V is an anticanonical hypersurface in a toric variety, then there is combinatorial data which (when it exists) equips V with a Tyurin degeneration, and equips a Calabi-Yau threefold birational to the Batyrev dual W of V with a K3 surface fibration. This K3 surface fibration has singular fibers which are closely related to the singular fibers of the LG models of the two components of the mirror Tyurin degeneration.

30 Motivation Toric degeneration Hodge theory CY3s 4. Tyurin degenerations and gluing LG models Thank you for your attention!

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