Log Conifold Transitions

Abstract: We define log conifold transitions for Fano threefold pairs of index two and study their deformation theory. Relying on the recent solution to the relative Clemens conjectures in this setting, we construct rational curves with normal bundle $\OO(-1)\oplus \OO(-1)$ by blowing up anchored points on the boundary divisor. Contracting these curves yields a singular space with ordinary double points. We prove that local smoothings of the nodes can be lifted to global first-order deformations, and that the global deformation theory of both the log resolution space and the singular log pair is unconditionally unobstructed. Crucially, the geometry of the boundary del Pezzo surface guarantees this unobstructedness. Furthermore, unlike the classical Calabi-Yau case, the underlying Fano geometry forces the vanishing of global topological balancing conditions, allowing local first-order smoothings of the nodes to be lifted independently. As applications, we construct new non-Kähler threefolds via smoothings, we analyze the effective geometry of the smoothed threefolds by determining their Picard groups and proving the persistence of free curves. Finally, we study the Hodge theory of these non-Kähler threefolds.

• The paper develops a logarithmic Clemens–Friedman transition for half-anticanonical pairs of index-two Fano threefolds, using blowups, contractions of disjoint rational curves, and smoothings of the resulting nodes.
• It proves unconditional logarithmic unobstructedness through the vanishings H²(𝑋̃, T𝑋̃(−log Ỹ)) = 0 and 𝕋²ₓ₀(−log 𝑌₀) = 0, eliminating the homological balancing conditions required in the classical Calabi–Yau case.
• The construction produces non-Kähler threefolds with b₂ ≥ 1 and b₃ > 0, including examples with no effective surfaces but abundant free rational curves, while preserving a polarized pure weight-three Hodge structure on H³.

Overview

"Log Conifold Transitions" (2606.31072) develops a logarithmic analogue of the classical Clemens–Friedman conifold transition for Fano threefold pairs of index two. The setting is a half-anticanonical pair $(X,Y)$ with $K_X + 2Y \cong \mathscr{O}_X$, where $X$ is a smooth prime Fano threefold of index two and degree $d \in \{2,3,4,5,8\}$. The construction rests on the recent resolution of the relative Clemens conjectures for this class of pairs [A26b], which guarantees a finite positive number $N_e$ of smooth rational curves of degree $e$ through a generic point configuration $\xi \subset Y$, each with normal bundle $\mathscr{O}(e-1) \oplus \mathscr{O}(e-1)$.

The transition proceeds in three steps: blow up $X$ at the anchored intersection points $\xi$; contract the strict transforms of the curves, which acquire normal bundle $K_X + 2Y \cong \mathscr{O}_X$0 and are assumed disjoint; and smooth the resulting nodal pair $K_X + 2Y \cong \mathscr{O}_X$1. A key structural fact is that the log Calabi–Yau condition is preserved: since $K_X + 2Y \cong \mathscr{O}_X$2 and $K_X + 2Y \cong \mathscr{O}_X$3, one has $K_X + 2Y \cong \mathscr{O}_X$4, and this triviality descends through the contraction and persists on the general smoothing.

Deformation theory: lifting and unobstructedness

The central technical contribution is an unconditional unobstructedness theorem for both the log resolution and the singular log pair. The author defines a sheaf of logarithmic differentials $K_X + 2Y \cong \mathscr{O}_X$5 on the singular space by gluing Deligne's sheaf on the smooth locus with the ordinary Kähler differentials away from $K_X + 2Y \cong \mathscr{O}_X$6; because $K_X + 2Y \cong \mathscr{O}_X$7 avoids the node locus $K_X + 2Y \cong \mathscr{O}_X$8, no cotangent complex is required, and the local deformation sheaves $K_X + 2Y \cong \mathscr{O}_X$9 agree with Friedman's $X$0 near $X$1.

The main vanishing results are:

• $X$2,
• $X$3,

and consequently the map from global first-order deformations to local deformations of the nodes, $X$4, is surjective.

The proof strategy follows the philosophy articulated by Friedman [F26] that smoothability questions reduce to cohomology of the exceptional locus. For the absolute case, the chain of identifications $X$5 uses Nakano vanishing on the Fano threefold — a mechanism unavailable in the Calabi–Yau setting. For the logarithmic case, the key input is that $X$6 restricts to the anti-canonical bundle of a del Pezzo surface blown up at $X$7 points, so Kodaira vanishing gives $X$8. The surjectivity of $X$9 then follows from a commutative diagram comparing coboundary maps for the point ideals on $d \in \{2,3,4,5,8\}$0 and $d \in \{2,3,4,5,8\}$1.

A crucial contrast with the classical Calabi–Yau conifold transition deserves emphasis. In Friedman's framework, local first-order smoothings lift globally only if the fundamental classes of the contracted curves satisfy balancing relations in homology. Here, the vanishing of $d \in \{2,3,4,5,8\}$2 forces these topological balancing conditions to vanish identically: via the identification $d \in \{2,3,4,5,8\}$3 (valid because $d \in \{2,3,4,5,8\}$4 is disjoint from $d \in \{2,3,4,5,8\}$5), the local deformation space surjects onto the space spanned by the curve classes, and every combination of independent node smoothings lifts to an unobstructed global deformation. This independence of smoothings is the engine behind all subsequent applications.

Two caveats are stated plainly. First, disjointness of the strict transforms is assumed rather than proved in general (it is easy for low degrees). Second, $d \in \{2,3,4,5,8\}$6 is not expected to be a stable submanifold: once $d \in \{2,3,4,5,8\}$7, Riemann–Roch gives $d \in \{2,3,4,5,8\}$8, reflecting the fact that for large $d \in \{2,3,4,5,8\}$9 no divisor in $N_e$0 passes through $N_e$1. Stability holds precisely when $N_e$2, matching the dimension count $N_e$3 against the $N_e$4 imposed conditions.

Non-Kähler threefolds and the effective cone

The independent smoothings yield new families of non-Kähler threefolds with a distinctive profile. If a proper non-empty subset $N_e$5 of the nodes is smoothed while the rest are small-resolved, the surviving exceptional curves are forced into the trivial homology class (smoothing a node kills the class of its contracted curve), while remaining effective holomorphic curves. Any Kähler form would give them strictly positive volume, contradicting Stokes' theorem; hence the resulting $N_e$6 is non-Kähler whenever $N_e$7 and $N_e$8 is proper and non-empty.

More striking is the systematic construction using curves of degrees $N_e$9 and $e$0. Contracting a degree-$e$1 curve through all $e$2 points together with, for each $e$3, a degree-$e$4 curve through $e$5 imposes linear conditions on Cartier divisors $e$6 that force $e$7 for all $e$8, so $e$9 has rank one, generated by $\xi \subset Y$0. When $\xi \subset Y$1, Riemann–Roch gives $\xi \subset Y$2: the unique generator deforms as a line bundle $\xi \subset Y$3 to the smoothing, but its section is obstructed, so $\xi \subset Y$4 by semicontinuity. Since $\xi \subset Y$5 generates the Picard group, the smoothed threefold $\xi \subset Y$6 contains no effective surfaces at all. This coexists with an abundance of free rational curves: general very free curves on $\xi \subset Y$7 avoid the finite set $\xi \subset Y$8 and the codimension-two image of the contracted curves by a dominance argument on the evaluation morphism, retain their normal bundle under the contraction, and deform unobstructedly to $\xi \subset Y$9 because $\mathscr{O}(e-1) \oplus \mathscr{O}(e-1)$0 implies surjectivity onto $\mathscr{O}(e-1) \oplus \mathscr{O}(e-1)$1.

These examples occupy a previously unoccupied position among non-Kähler constructions: unlike Clemens–Friedman transitions ($\mathscr{O}(e-1) \oplus \mathscr{O}(e-1)$2) or Poon's small resolutions ($\mathscr{O}(e-1) \oplus \mathscr{O}(e-1)$3), they have $\mathscr{O}(e-1) \oplus \mathscr{O}(e-1)$4 together with $\mathscr{O}(e-1) \oplus \mathscr{O}(e-1)$5. Concretely, smoothing $\mathscr{O}(e-1) \oplus \mathscr{O}(e-1)$6 nodes spanning a one-dimensional homology subspace gives $\mathscr{O}(e-1) \oplus \mathscr{O}(e-1)$7, with Picard rank $\mathscr{O}(e-1) \oplus \mathscr{O}(e-1)$8 identified with the orthogonal complement $\mathscr{O}(e-1) \oplus \mathscr{O}(e-1)$9.

Hodge theory

Using Chen's theorem that a smoothing of an SNC variety with Kähler components satisfies the $X$0-lemma (Chen, 2024), the paper shows the general fiber $X$1 carries a polarized pure Hodge structure of weight three — consistent with related results of Friedman [F19] and Li [Li24]. As a sharp application, the paper rules out a twistor interpretation when $X$2 is a cubic threefold. Although blowing up a curve of genus $X$3 in a simply connected twistor space reproduces the third Betti number $X$4 exactly, the limiting mixed Hodge structures distinguish them: the compact abelian part of the limiting intermediate Jacobian of $X$5 is the intermediate Jacobian $X$6 of the cubic, whereas the corresponding object on the twistor side is a product of Jacobians of smooth curves. By the Clemens–Griffiths indecomposability theorem [CG72], these cannot coincide, so $X$7 is not even birational to a twistor space over a simply connected base.

Limitations and open questions

Several assumptions bound the scope of the results. The disjointness of the contracted strict transforms is an hypothesis, verified only in low-degree cases such as the six lines through a generic point of a cubic threefold. The existence of the degree-$X$8 and degree-$X$9 curves required for the Picard-rank-one construction is established by residuation only for the cubic threefold with $\xi$0; its validity across the full range of degrees $\xi$1 and arbitrary $\xi$2 remains open. The unobstructedness argument depends essentially on index two and the del Pezzo boundary; extension to other indices or to boundaries that are not del Pezzo is not addressed. Finally, the paper leaves open whether the absence of effective divisors combined with abundant free curves relates to twistor geometry in any positive sense — the cubic-threefold argument only excludes birational models over simply connected bases — and defers connections to the log minimal model program and to metric smoothings of the boundary to future work.

Conclusion

This paper transfers the conifold transition machinery to half-anticanonical Fano pairs and shows that the deformation theory is, in a precise sense, better behaved than in the Calabi–Yau case: Fano geometry eliminates the balancing conditions entirely, giving independent, unconditionally unobstructed smoothings. The geometric payoff is a systematic production of non-Kähler threefolds with no effective surfaces yet plentiful free rational curves, together with a polarized Hodge structure on $\xi$3 and a negative answer to the twistor question in the cubic case.